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Allen B. Downey
Think Python

[LSI]Think Python by Allen B. Downey
Copyright 2016 Allen Downey. All rights reserved.
Printed in the United States of America.
Published by OReilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA 95472.
OReilly books may be purchased for educational, business, or sales promotional use. Online editions are also available for most titles ( For more information, contact our corporate/
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Production Editor: Kristen Brown
Copyeditor: Nan Reinhardt
Proofreader: Amanda KerseyIndexer: Allen Downey
Interior Designer: David Futato
Cover Designer: Karen Montgomery
Illustrator: Rebecca DemarestAugust 2012:
First Edition
December 2015:
Second Edition
Revision History for the Second Edition
2015-11-20: First Release
See for release details.
The OReilly logo is a registered trademark of OReilly Media, Inc.
ink Python , the cover image of a
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ink Python is available under the Creative Commons Attribution-NonCommercial 3.0 Unported
License. The author maintains an online version at

Table of Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . xi
The Way of the Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
What Is a Program? 1
Running Python 2
The First Program 3
Arithmetic Operators 3 Values and Types 4Formal and Natural Languages 5Debugging
Glossary 8
Variables, Expressions and Statements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Assignment Statements 11
Variable Names 12
Expressions and Statements 12
Script Mode 13
Order of Operations 14
String Operations 15
Comments 15
Glossary 17
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 21
Function Calls 21
Math Functions 22

Composition 23Adding New Functions 23
Definitions and Uses 25
Flow of Execution 25
Parameters and Arguments 26
Variables and Parameters Are Local 27
Stack Diagrams 28
Fruitful Functions and Void Functions 29
Why Functions? 30
Glossary 31
32 4.
Case Study: Interface Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
The turtle Module 35
Simple Repetition 37
Encapsulation 38
Generalization 39
Interface Design 40
A Development Plan 42
Glossary 44
Conditionals and Recursion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Floor Division and Modulus 47
Boolean Expressions 48
Logical Operators 49
Conditional Execution 49
Alternative Execution 49
Chained Conditionals 50
Nested Conditionals 50
Stack Diagrams for Recursive Functions 53
Infinite Recursion 53
Keyboard Input 54
Glossary 56
iv | Table of Contents

6. Fruitful Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Return Values 61Incremental Development 62
Composition 64
Boolean Functions 65
More Recursion 66
Leap of Faith 68
One More Example 68
Checking Types 69
Glossary 71
72 7.
Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 75
Reassignment 75
Updating Variables 76
The while Statement 77
Square Roots
79 Algorithms
Glossary 82
Strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 85
A String Is a Sequence 85
Traversal with a for Loop 86
String Slices
87 Strings Are Immutable 88
Looping and Counting 89
String Methods 90
The in Operator 91
String Comparison 92
Glossary 94
Case Study: Word Play. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Reading Word Lists 99Exercises
Table of Contents | v

Looping with Indices 103
Debugging 104
Glossary 105
105 10.
Lists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 107
A List Is a Sequence 107
Lists Are Mutable 108
Traversing a List 109
List Operations 110
List Slices
List Methods 111
Map, Filter and Reduce 111
Deleting Elements 113
Lists and Strings 113 Objects and Values 114
List Arguments 116
Debugging 118
Glossary 119
Dictionaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 125
A Dictionary Is a Mapping 125
Dictionary as a Collection of Counters 127 Looping and Dictionaries 128
Reverse Lookup 129
Dictionaries and Lists 130
Global Variables 133
Debugging 134
Glossary 135
Tuples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 139
Tuples Are Immutable 139
Tuple Assignment 141
Tuples as Return Values 141
Variable-Length Argument Tuples 142
Lists and Tuples 143
Dictionaries and Tuples 144
vi | Table of Contents

Sequences of Sequences 146
Debugging 147
Glossary 148
148 13.
Case Study: Data Structure Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Word Frequency Analysis 151
Random Numbers 152
Word Histogram 153
Most Common Words 155
Optional Parameters 155
Dictionary Subtraction 156
Random Words 157
Markov Analysis 158
Data Structures 159
Debugging 161
Glossary 162
Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 165
165 Reading and Writing 166
Format Operator 166
Filenames and Paths 167
Catching Exceptions 169 Databases 169
171 Writing Modules 172
Debugging 173
Glossary 174
Classes and Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Programmer-Defined Types 177 Attributes
Instances as Return Values 181 Objects Are Mutable 181Copying
Debugging 183
Glossary 184
Table of Contents | vii

185 16.
Classes and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Pure Functions 188 Modifiers
189Prototyping versus Planning 190
Debugging 192
Glossary 192
Classes and Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Object-Oriented Features 195
Printing Objects 196
Another Example 198
A More Complicated Example 198
The init Method 199
The __str__ Method 200
Operator Overloading 200
Type-Based Dispatch 201
Polymorphism 202
Interface and Implementation 203
Debugging 204
Glossary 204
Inheritance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Card Objects 207
Class Attributes 208
Comparing Cards 210
Printing the Deck 211
Add, Remove, Shuffle and Sort 212
Class Diagrams 214
Data Encapsulation 215
Debugging 217
Glossary 218
The Goodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 223
Conditional Expressions 223
viii | Table of Contents

List Comprehensions 224
Generator Expressions 225
any and all 226
Counters 228
defaultdict 229
Named Tuples 230
Gathering Keyword Args 232
Glossary 233
233 20.
Debugging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 235
Syntax Errors 235 I keep making changes and it makes no difference. 237
Runtime Errors 237 My program does absolutely nothing. 237
My program hangs. 238
When I run the program I get an exception. 239
I added so many print statements I get inundated with output. 240
Semantic Errors 241 My program doesnt work. 241
Ive got a big hairy expression and it doesnt do what I expect. 242
Ive got a function that doesnt return what I expect. 243 Im really, really stuck and I need help. 243
No, I really need help. 243
Analysis of Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Order of Growth 246
Analysis of Basic Python Operations 248
Analysis of Search Algorithms 250
Hashtables 251 Glossary 255
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 257
Table of Contents | ix

The Strange History of This Book In January 1999 I was preparing to teach an introductory programming class in Java.
I had taught it three times and I was getting frustrated. The failure rate in the class
was too high and, even for students who succeeded, the overall level of achievement was too low.
One of the problems I saw was the books. They were too big, with too much unneces
sary detail about Java, and not enough high-level guidance about how to program.
And they all suffered from the trapdoor effect: they would start out easy, proceed
gradually, and then somewhere around Chapter 5 the bottom would fall out. The stu dents would get too much new material, too fast, and I would spend the rest of the
semester picking up the pieces.
Two weeks before the first day of classes, I decided to write my own book. My goals

Keep it short. It is better for students to read 10 pages than not read 50 pages.

Be careful with vocabulary. I tried to minimize jargon and define each term at first use.

Build gradually. To avoid trapdoors, I took the most difficult topics and splitthem into a series of small steps.

Focus on programming, not the programming language. I included the mini mum useful subset of Java and left out the rest.
I needed a title, so on a whim I chose How to
ink Like a Computer Scientist.
My first version was rough, but it worked. Students did the reading, and they under
stood enough that I could spend class time on the hard topics, the interesting topics
and (most important) letting the students practice.

I released the book under the GNU Free Documentation License, which allows users
to copy, modify, and distribute the book.
What happened next is the cool part. Jeff Elkner, a high school teacher in Virginia, adopted my book and translated it into Python. He sent me a copy of his translation,
and I had the unusual experience of learning Python by reading my own book. As
Green Tea Press, I published the first Python version in 2001.
In 2003 I started teaching at Olin College and I got to teach Python for the first time.
The contrast with Java was striking. Students struggled less, learned more, worked on more interesting projects, and generally had a lot more fun.
Since then Ive continued to develop the book, correcting errors, improving some of
the examples and adding material, especially exercises.
The result is this book, now with the less grandiose title
ink Python . Some of the
changes are:

I added a section about debugging at the end of each chapter. These sections present general techniques for finding and avoiding bugs, and warnings about
Python pitfalls.

I added more exercises, ranging from short tests of understanding to a few sub stantial projects. Most exercises include a link to my solution.

I added a series of case studieslonger examples with exercises, solutions, anddiscussion.

I expanded the discussion of program development plans and basic designpatterns.

I added appendices about debugging and analysis of algorithms.
The second edition of
ink Python has these new features:

The book and all supporting code have been updated to Python 3.

I added a few sections, and more details on the Web, to help beginners get started
running Python in a browser, so you dont have to deal with installing Python
until you want to.

For “The turtle Module” on page 35 I switched from my own turtle graphics
package, called Swampy, to a more standard Python module, turtle, which is
easier to install and more powerful.

I added a new chapter called “The Goodies”, which introduces some additional
Python features that are not strictly necessary, but sometimes handy.
xii | Preface

I hope you enjoy working with this book, and that it helps you learn to program and
think like a computer scientist, at least a little bit.
Allen B. DowneyOlin College
Conventions Used in This Book
The following typographical conventions are used in this book:
Italic Indicates new terms, URLs, email addresses, filenames, and file extensions.
Bold Indicates terms defined in the Glossary.
Constant width Used for program listings, as well as within paragraphs to refer to program elements such as variable or function names, databases, data types, environment
variables, statements, and keywords.
Constant width bold Shows commands or other text that should be typed literally by the user.
Constant width italic Shows text that should be replaced with user-supplied values or by values deter
mined by context.
Using Code Examples Supplemental material (code examples, exercises, etc.) is available for download at .
This book is here to help you get your job done. In general, if example code is offered
with this book, you may use it in your programs and documentation. You do not
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ink Python , 2nd Edition, by Allen
B. Downey (OReilly). Copyright 2016 Allen Downey, 978-1-4919-3936-9.”
Preface | xiii

If you feel your use of code examples falls outside fair use or the permission given
above, feel free to contact us at
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site at .xiv | Preface

Find us on Facebook:
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Watch us on YouTube:
Many thanks to Jeff Elkner, who translated my Java book into Python, which got this
project started and introduced me to what has turned out to be my favorite language.
Thanks also to Chris Meyers, who contributed several sections to How to
ink Like a
Computer Scientist .
Thanks to the Free Software Foundation for developing the GNU Free Documenta tion License, which helped make my collaboration with Jeff and Chris possible, and
Creative Commons for the license I am using now.
Thanks to the editors at Lulu who worked on How to
ink Like a Computer Scientist.
Thanks to the editors at OReilly Media who worked on
ink Python .
Thanks to all the students who worked with earlier versions of this book and all the
contributors (listed below) who sent in corrections and suggestions.
Contributor List More than 100 sharp-eyed and thoughtful readers have sent in suggestions and cor
rections over the past few years. Their contributions, and enthusiasm for this project,
have been a huge help.
If you have a suggestion or correction, please send email to feedback@thinkpy . If I make a change based on your feedback, I will add you to the contribu
tor list (unless you ask to be omitted).
If you include at least part of the sentence the error appears in, that makes it easy for
me to search. Page and section numbers are fine, too, but not quite as easy to work
with. Thanks!

Lloyd Hugh Allen sent in a correction to Section 8.4.

Yvon Boulianne sent in a correction of a semantic error in Chapter 5.

Fred Bremmer submitted a correction in Section 2.1.

Jonah Cohen wrote the Perl scripts to convert the LaTeX source for this book into beauti
ful HTML.

Michael Conlon sent in a grammar correction in Chapter 2 and an improvement in style in Chapter 1, and he initiated discussion on the technical aspects of interpreters.

Benoit Girard sent in a correction to a humorous mistake in Section 5.6.
Preface | xv

Courtney Gleason and Katherine Smith wrote, which was used as a case
study in an earlier version of the book. Their program can now be found on the website.
Lee Harr submitted more corrections than we have room to list here, and indeed he should be listed as one of the principal editors of the text.

James Kaylin is a student using the text. He has submitted numerous corrections.

David Kershaw fixed the broken catTwice function in Section 3.10.

Eddie Lam has sent in numerous corrections to Chapters 1, 2, and 3. He also fixed the
Makefile so that it creates an index the first time it is run and helped us set up a version ing scheme.

Man-Yong Lee sent in a correction to the example code in Section 2.4.

David Mayo pointed out that the word “unconsciously” in Chapter 1 needed to be
changed to “subconsciously”.

Chris McAloon sent in several corrections to Sections 3.9 and 3.10.

Matthew J. Moelter has been a long-time contributor who sent in numerous corrections
and suggestions to the book.

Simon Dicon Montford reported a missing function definition and several typos in Chap
ter 3. He also found errors in the increment function in Chapter 13.

John Ouzts corrected the definition of “return value” in Chapter 3.

Kevin Parks sent in valuable comments and suggestions as to how to improve the distri
bution of the book.

David Pool sent in a typo in the glossary of Chapter 1, as well as kind words of encourage

Michael Schmitt sent in a correction to the chapter on files and exceptions.

Robin Shaw pointed out an error in Section 13.1, where the printTime function was used
in an example without being defined.

Paul Sleigh found an error in Chapter 7 and a bug in Jonah Cohens Perl script that gener ates HTML from LaTeX.

Craig T. Snydal is testing the text in a course at Drew University. He has contributed several valuable suggestions and corrections.

Ian Thomas and his students are using the text in a programming course. They are the first ones to test the chapters in the latter half of the book, and they have made numerous
corrections and suggestions.

Keith Verheyden sent in a correction in Chapter 3.

Peter Winstanley let us know about a longstanding error in our Latin in Chapter 3.

Chris Wrobel made corrections to the code in the chapter on file I/O and exceptions.

Moshe Zadka has made invaluable contributions to this project. In addition to writing the
first draft of the chapter on Dictionaries, he provided continual guidance in the early
stages of the book.

Christoph Zwerschke sent several corrections and pedagogic suggestions, and explained
the difference between gleich and selbe.
xvi | Preface

James Mayer sent us a whole slew of spelling and typographical errors, including two inthe contributor list.
Hayden McAfee caught a potentially confusing inconsistency between two examples.

Angel Arnal is part of an international team of translators working on the Spanish version
of the text. He has also found several errors in the English version.

Tauhidul Hoque and Lex Berezhny created the illustrations in Chapter 1 and improved
many of the other illustrations.

Dr. Michele Alzetta caught an error in Chapter 8 and sent some interesting pedagogic
comments and suggestions about Fibonacci and Old Maid.

Andy Mitchell caught a typo in Chapter 1 and a broken example in Chapter 2.

Kalin Harvey suggested a clarification in Chapter 7 and caught some typos.

Christopher P. Smith caught several typos and helped us update the book for Python 2.2.

David Hutchins caught a typo in the Foreword.

Gregor Lingl is teaching Python at a high school in Vienna, Austria. He is working on a German translation of the book, and he caught a couple of bad errors in Chapter 5.

Julie Peters caught a typo in the Preface.

Florin Oprina sent in an improvement in makeTime, a correction in printTime, and a nice

D. J. Webre suggested a clarification in Chapter 3.

Ken found a fistful of errors in Chapters 8, 9 and 11.

Ivo Wever caught a typo in Chapter 5 and suggested a clarification in Chapter 3.

Curtis Yanko suggested a clarification in Chapter 2.

Ben Logan sent in a number of typos and problems with translating the book into HTML.

Jason Armstrong saw the missing word in Chapter 2.

Louis Cordier noticed a spot in Chapter 16 where the code didnt match the text.

Brian Cain suggested several clarifications in Chapters 2 and 3.

Rob Black sent in a passel of corrections, including some changes for Python 2.2.

Jean-Philippe Rey at Ecole Centrale Paris sent a number of patches, including some updates for Python 2.2 and other thoughtful improvements.

Jason Mader at George Washington University made a number of useful suggestions andcorrections.

Jan Gundtofte-Bruun reminded us that “a error” is an error.

Abel David and Alexis Dinno reminded us that the plural of “matrix” is “matrices”, not
“matrixes”. This error was in the book for years, but two readers with the same initials reported it on the same day. Weird.

Charles Thayer encouraged us to get rid of the semicolons we had put at the ends of some
statements and to clean up our use of “argument” and “parameter”.

Roger Sperberg pointed out a twisted piece of logic in Chapter 3.

Sam Bull pointed out a confusing paragraph in Chapter 2.
Preface | xvii

Andrew Cheung pointed out two instances of “use before def ”.
C. Corey Capel spotted a missing word and a typo in Chapter 4.

Alessandra helped clear up some Turtle confusion.

Wim Champagne found a braino in a dictionary example.

Douglas Wright pointed out a problem with floor division in arc.

Jared Spindor found some jetsam at the end of a sentence.

Lin Peiheng sent a number of very helpful suggestions.

Ray Hagtvedt sent in two errors and a not-quite-error.

Torsten Hbsch pointed out an inconsistency in Swampy.

Inga Petuhhov corrected an example in Chapter 14.

Arne Babenhauserheide sent several helpful corrections.

Mark E. Casida is is good at spotting repeated words.

Scott Tyler filled in a that was missing. And then sent in a heap of corrections.

Gordon Shephard sent in several corrections, all in separate emails.

Andrew Turner spotted an error in Chapter 8.

Adam Hobart fixed a problem with floor division in arc.

Daryl Hammond and Sarah Zimmerman pointed out that I served up math.pi too early.
And Zim spotted a typo.

George Sass found a bug in a Debugging section.

Brian Bingham suggested Exercise 11-5.

Leah Engelbert-Fenton pointed out that I used tuple as a variable name, contrary to my
own advice. And then found a bunch of typos and a “use before def ”.

Joe Funke spotted a typo.

Chao-chao Chen found an inconsistency in the Fibonacci example.

Jeff Paine knows the difference between space and spam.

Lubos Pintes sent in a typo.

Gregg Lind and Abigail Heithoff suggested Exercise 14-3.

Max Hailperin has sent in a number of corrections and suggestions. Max is one of the
authors of the extraordinary Concrete Abstractions (Course Technology, 1998), which you
might want to read when you are done with this book.

Chotipat Pornavalai found an error in an error message.

Stanislaw Antol sent a list of very helpful suggestions.

Eric Pashman sent a number of corrections for Chapters 411.

Miguel Azevedo found some typos.

Jianhua Liu sent in a long list of corrections.

Nick King found a missing word.

Martin Zuther sent a long list of suggestions.
xviii | Preface

Adam Zimmerman found an inconsistency in my instance of an “instance” and severalother errors.
Ratnakar Tiwari suggested a footnote explaining degenerate triangles.

Anurag Goel suggested another solution for is_abecedarian and sent some additional
corrections. And he knows how to spell Jane Austen.

Kelli Kratzer spotted one of the typos.

Mark Griffiths pointed out a confusing example in Chapter 3.

Roydan Ongie found an error in my Newtons method.

Patryk Wolowiec helped me with a problem in the HTML version.

Mark Chonofsky told me about a new keyword in Python 3.

Russell Coleman helped me with my geometry.

Wei Huang spotted several typographical errors.

Karen Barber spotted the the oldest typo in the book.

Nam Nguyen found a typo and pointed out that I used the Decorator pattern but didnt mention it by name.

Stphane Morin sent in several corrections and suggestions.

Paul Stoop corrected a typo in uses_only.

Eric Bronner pointed out a confusion in the discussion of the order of operations.

Alexandros Gezerlis set a new standard for the number and quality of suggestions he sub
mitted. We are deeply grateful!

Gray Thomas knows his right from his left.

Giovanni Escobar Sosa sent a long list of corrections and suggestions.

Alix Etienne fixed one of the URLs.

Kuang He found a typo.

Daniel Neilson corrected an error about the order of operations.

Will McGinnis pointed out that polyline was defined differently in two places.

Swarup Sahoo spotted a missing semicolon.

Frank Hecker pointed out an exercise that was under-specified, and some broken links.

Animesh B helped me clean up a confusing example.

Martin Caspersen found two round-off errors.

Gregor Ulm sent several corrections and suggestions.

Dimitrios Tsirigkas suggested I clarify an exercise.

Carlos Tafur sent a page of corrections and suggestions.

Martin Nordsletten found a bug in an exercise solution.

Lars O.D. Christensen found a broken reference.

Victor Simeone found a typo.

Sven Hoexter pointed out that a variable named input shadows a build-in function.

Viet Le found a typo.
Preface | xix

Not surprisingly, integers belong to the type int, strings belong to str, and floating-
point numbers belong to float.
What about values like 2 and 42.0 ? They look like numbers, but they are in quo
tation marks like strings:
>>> type(2)

>>> type(42.0)

Theyre strings.
When you type a large integer, you might be tempted to use commas between groups
of digits, as in 1,000,000. This is not a legal integer in Python, but it is legal:
>>> 1,000,000 (1, 0, 0)
Thats not what we expected at all! Python interprets 1,000,000 as a comma-
separated sequence of integers. Well learn more about this kind of sequence later.
Formal and Natural Languages Natural languages are the languages people speak, such as English, Spanish, and
French. They were not designed by people (although people try to impose some order on them); they evolved naturally.
Formal languages are languages that are designed by people for specific applications.
For example, the notation that mathematicians use is a formal language that is partic ularly good at denoting relationships among numbers and symbols. Chemists use a
formal language to represent the chemical structure of molecules. And most impor
Programming languages are formal languages that have been designed to express
Formal languages tend to have strict syntax rules that govern the structure of state
ments. For example, in mathematics the statement 3 + 3 = 6
has correct syntax, but
3 + = 3$6
does not. In chemistry H
is a syntactically correct formula, but
Syntax rules come in two flavors, pertaining to tokens and structure. Tokens are the
basic elements of the language, such as words, numbers, and chemical elements. One of the problems with
3 + = 3$6
is that
is not a legal token in mathematics (at least
as far as I know). Similarly, 2Zz
is not legal because there is no element with the
abbreviation Zz.
Formal and Natural Languages | 5

The second type of syntax rule pertains to the way tokens are combined. The equa
3 + = 3
is illegal because even though + and = are legal tokens, you cant have
one right after the other. Similarly, in a chemical formula the subscript comes after
the element name, not before.
This is @ well-structured Engli$h sentence with invalid t*kens in it. This sentence all
valid tokens has, but invalid structure with.
When you read a sentence in English or a statement in a formal language, you have to
figure out the structure (although in a natural language you do this subconsciously). This process is called parsing.
Although formal and natural languages have many features in commontokens,
structure, and syntaxthere are some differences:
ambiguity: Natural languages are full of ambiguity, which people deal with by using contextual clues and other information. Formal languages are designed to be nearly or
completely unambiguous, which means that any statement has exactly one mean ing, regardless of context.
redundancy: In order to make up for ambiguity and reduce misunderstandings, natural languages employ lots of redundancy. As a result, they are often verbose. Formal
languages are less redundant and more concise.
literalness: Natural languages are full of idiom and metaphor. If I say, “The penny dropped”,there is probably no penny and nothing dropping (this idiom means that some
one understood something after a period of confusion). Formal languages mean exactly what they say.
Because we all grow up speaking natural languages, it is sometimes hard to adjust to
formal languages. The difference between formal and natural language is like the dif
ference between poetry and prose, but more so:
Poetry: Words are used for their sounds as well as for their meaning, and the wholepoem together creates an effect or emotional response. Ambiguity is not only common but often deliberate.
Prose: The literal meaning of words is more important, and the structure contributes
more meaning. Prose is more amenable to analysis than poetry but still often ambiguous.
6 | Chapter 1: The Way of the Program

Programs:The meaning of a computer program is unambiguous and literal, and can beunderstood entirely by analysis of the tokens and structure.
Formal languages are more dense than natural languages, so it takes longer to read
them. Also, the structure is important, so it is not always best to read from top to bot
tom, left to right. Instead, learn to parse the program in your head, identifying the
tokens and interpreting the structure. Finally, the details matter. Small errors in spell ing and punctuation, which you can get away with in natural languages, can make a
big difference in a formal language.
Debugging Programmers make mistakes. For whimsical reasons, programming errors are called
bugs and the process of tracking them down is called debugging.
Programming, and especially debugging, sometimes brings out strong emotions. If
you are struggling with a difficult bug, you might feel angry, despondent, or embar
There is evidence that people naturally respond to computers as if they were people. When they work well, we think of them as teammates, and when they are obstinate or
rude, we respond to them the same way we respond to rude, obstinate people (Reeves
and Nass,
e Media Equation: How People Treat Computers, Television, and New
Media Like Real People and Places ).
Preparing for these reactions might help you deal with them. One approach is to
think of the computer as an employee with certain strengths, like speed and preci
sion, and particular weaknesses, like lack of empathy and inability to grasp the big
Your job is to be a good manager: find ways to take advantage of the strengths and mitigate the weaknesses. And find ways to use your emotions to engage with the
problem, without letting your reactions interfere with your ability to work effectively.
Learning to debug can be frustrating, but it is a valuable skill that is useful for many
activities beyond programming. At the end of each chapter there is a section, like this one, with my suggestions for debugging. I hope they help!
Debugging | 7

Glossaryproblem solving: The process of formulating a problem, finding a solution, and expressing it.
high-level language: A programming language like Python that is designed to be easy for humans to
read and write.
low-level language: A programming language that is designed to be easy for a computer to run; also
called “machine language” or “assembly language”.
portability: A property of a program that can run on more than one kind of computer.
interpreter: A program that reads another program and executes it.
prompt: Characters displayed by the interpreter to indicate that it is ready to take inputfrom the user.
program: A set of instructions that specifies a computation.
print statement: An instruction that causes the Python interpreter to display a value on the screen.
operator: A special symbol that represents a simple computation like addition, multiplica
tion, or string concatenation.
value: One of the basic units of data, like a number or string, that a programmanipulates.
type: A category of values. The types we have seen so far are integers (type int),
floating-point numbers (type float), and strings (type str).
integer: A type that represents whole numbers.
A type that represents numbers with fractional parts.
8 | Chapter 1: The Way of the Program

string:A type that represents sequences of characters.
natural language: Any one of the languages that people speak that evolved naturally.
formal language: Any one of the languages that people have designed for specific purposes, such asrepresenting mathematical ideas or computer programs; all programming lan
guages are formal languages.
token: One of the basic elements of the syntactic structure of a program, analogous to aword in a natural language.
syntax: The rules that govern the structure of a program.
parse: To examine a program and analyze the syntactic structure.
bug: An error in a program.
debugging: The process of finding and correcting bugs.
Exercises Exercise 1-1.
It is a good idea to read this book in front of a computer so you can try out the exam ples as you go.
Whenever you are experimenting with a new feature, you should try to make mis
takes. For example, in the “Hello, world!” program, what happens if you leave out one
of the quotation marks? What if you leave out both? What if you spell print wrong?
This kind of experiment helps you remember what you read; it also helps when you
are programming, because you get to know what the error messages mean. It is better to make mistakes now and on purpose than later and accidentally.1.
In a print statement, what happens if you leave out one of the parentheses, orboth?
If you are trying to print a string, what happens if you leave out one of the quotation marks, or both?
Exercises | 9

3. You can use a minus sign to make a negative number like -2. What happens if
you put a plus sign before a number? What about 2++2? 4.
In math notation, leading zeros are okay, as in 02. What happens if you try this in
What happens if you have two values with no operator between them?
Exercise 1-2.
Start the Python interpreter and use it as a calculator.
How many seconds are there in 42 minutes 42 seconds?
How many miles are there in 10 kilometers? Hint: there are 1.61 kilometers in a mile.
If you run a 10 kilometer race in 42 minutes 42 seconds, what is your average pace (time per mile in minutes and seconds)? What is your average speed in
miles per hour?
10 | Chapter 1: The Way of the Program

Variables, Expressions and Statements
One of the most powerful features of a programming language is the ability to manipulate variables. A variable is a name that refers to a value.
Assignment Statements
An assignment statement creates a new variable and gives it a value:
>>> message = And now for something completely different
>>> n = 17
>>> pi = 3.141592653589793
This example makes three assignments. The first assigns a string to a new variable
named message ; the second gives the integer 17 to n; the third assigns the (approxi
mate) value of to pi.
A common way to represent variables on paper is to write the name with an arrow pointing to its value. This kind of figure is called a state diagram because it shows
what state each of the variables is in (think of it as the variables state of mind).
Figure 2-1 shows the result of the previous example.
Figure 2-1. State diagram.

Variable Names
Programmers generally choose names for their variables that are meaningfulthey document what the variable is used for.
Variable names can be as long as you like. They can contain both letters and numbers, but they cant begin with a number. It is legal to use uppercase letters, but it is conventional to use only lowercase for variables names.
The underscore character, _, can appear in a name. It is often used in names with
multiple words, such as your_name or airspeed_of_unladen_swallow .
If you give a variable an illegal name, you get a syntax error: >>> 76trombones = big parade
SyntaxError: invalid syntax
>>> more@ = 1000000
SyntaxError: invalid syntax
>>> class = Advanced Theoretical Zymurgy
SyntaxError: invalid syntax
76trombones is illegal because it begins with a number. more@ is illegal because it con
tains an illegal character, @. But whats wrong with class?
It turns out that class is one of Pythons keywords. The interpreter uses keywords to
recognize the structure of the program, and they cannot be used as variable names.
Python 3 has these keywords: False class finally is return
None continue for lambda try True def from nonlocal whileand del global not with
as elif if or yield
assert else import pass
break except in raise
You dont have to memorize this list. In most development environments, keywords are displayed in a different color; if you try to use one as a variable name, youll know.
Expressions and Statements
An expression is a combination of values, variables, and operators. A value all by
itself is considered an expression, and so is a variable, so the following are all legal
expressions:12 | Chapter 2: Variables, Expressions and Statements

>>> 42
>>> n
>>> n + 25
When you type an expression at the prompt, the interpreter evaluates it, which
means that it finds the value of the expression. In this example, n has the value 17 and
n + 25 has the value 42.
A statement is a unit of code that has an effect, like creating a variable or displaying a
>>> n = 17
>>> print(n)
The first line is an assignment statement that gives a value to n. The second line is a
print statement that displays the value of n.
When you type a statement, the interpreter executes it, which means that it does
whatever the statement says. In general, statements dont have values.
Script Mode
So far we have run Python in interactive mode, which means that you interact
directly with the interpreter. Interactive mode is a good way to get started, but if you
are working with more than a few lines of code, it can be clumsy.
The alternative is to save code in a file called a script and then run the interpreter in
script mode to execute the script. By convention, Python scripts have names that end
with .py.
If you know how to create and run a script on your computer, you are ready to go.
Otherwise I recommend using PythonAnywhere again. I have posted instructions for running in script mode at .
Because Python provides both modes, you can test bits of code in interactive mode
before you put them in a script. But there are differences between interactive mode
and script mode that can be confusing.
For example, if you are using Python as a calculator, you might type: >>> miles = 26.2
>>> miles * 1.61
The first line assigns a value to miles, but it has no visible effect. The second line is
an expression, so the interpreter evaluates it and displays the result. It turns out that a
marathon is about 42 kilometers. Script Mode | 13

But if you type the same code into a script and run it, you get no output at all. In
script mode an expression, all by itself, has no visible effect. Python actually evaluates the expression, but it doesnt display the value unless you tell it to:
miles = 26.2
print(miles * 1.61)
This behavior can be confusing at first.
A script usually contains a sequence of statements. If there is more than one state
ment, the results appear one at a time as the statements execute.
For example, the script print(1)
x = 2
produces the output
The assignment statement produces no output.
To check your understanding, type the following statements in the Python interpreter
and see what they do:
x = 5
x + 1
Now put the same statements in a script and run it. What is the output? Modify the
script by transforming each expression into a print statement and then run it again.
Order of Operations
When an expression contains more than one operator, the order of evaluation
depends on the order of operations . For mathematical operators, Python follows
mathematical convention. The acronym PEMDAS is a useful way to remember the
P arentheses have the highest precedence and can be used to force an expression
to evaluate in the order you want. Since expressions in parentheses are evaluated
first, 2 * (3-1) is 4, and (1+1)**(5-2) is 8. You can also use parentheses to
make an expression easier to read, as in (minute * 100) / 60, even if it doesnt
change the result.

E xponentiation has the next highest precedence, so 1 + 2**3 is 9, not 27, and 2
* 3**2 is 18, not 36.
14 | Chapter 2: Variables, Expressions and Statements

Multiplication and Division have higher precedence than Addition and Subtrac
tion. So 2*3-1 is 5, not 4, and 6+4/2 is 8, not 5.
Operators with the same precedence are evaluated from left to right (except
exponentiation). So in the expression degrees / 2 * pi, the division happens
first and the result is multiplied by pi. To divide by
, you can use parentheses
or write degrees / 2 / pi .
I dont work very hard to remember the precedence of operators. If I cant tell by
looking at the expression, I use parentheses to make it obvious.
String Operations
In general, you cant perform mathematical operations on strings, even if the strings look like numbers, so the following are illegal:
2-1 eggs/easy third*a charm
But there are two exceptions, + and *.
The + operator performs string concatenation , which means it joins the strings by
linking them end-to-end. For example:
>>> first = throat
>>> second = warbler
>>> first + second
The * operator also works on strings; it performs repetition. For example, Spam*3 is
SpamSpamSpam . If one of the values is a string, the other has to be an integer.
This use of + and * makes sense by analogy with addition and multiplication. Just as
4*3 is equivalent to 4+4+4, we expect Spam*3 to be the same as
Spam+Spam+Spam , and it is. On the other hand, there is a significant way in
which string concatenation and repetition are different from integer addition and
multiplication. Can you think of a property that addition has that string concatena
tion does not?
As programs get bigger and more complicated, they get more difficult to read. Formal languages are dense, and it is often difficult to look at a piece of code and figure out
what it is doing, or why.
For this reason, it is a good idea to add notes to your programs to explain in natural
language what the program is doing. These notes are called comments, and they start
with the # symbol:
String Operations | 15

# compute the percentage of the hour that has elapsedpercentage = (minute * 100) / 60
In this case, the comment appears on a line by itself. You can also put comments at
the end of a line:
percentage = (minute * 100) / 60 # percentage of an hour
Everything from the # to the end of the line is ignoredit has no effect on the execu
tion of the program.
Comments are most useful when they document non-obvious features of the code. It
is reasonable to assume that the reader can figure out what the code does; it is more
useful to explain why.
This comment is redundant with the code and useless: v = 5 # assign 5 to v
This comment contains useful information that is not in the code:
v = 5 # velocity in meters/second.
Good variable names can reduce the need for comments, but long names can make
complex expressions hard to read, so there is a trade-off.
Debugging Three kinds of errors can occur in a program: syntax errors, runtime errors, and
semantic errors. It is useful to distinguish between them in order to track them down
more quickly.
Syntax error: “Syntax” refers to the structure of a program and the rules about that structure.
For example, parentheses have to come in matching pairs, so (1 + 2) is legal, but
8) is a syntax error .
If there is a syntax error anywhere in your program, Python displays an error
message and quits, and you will not be able to run the program. During the first few weeks of your programming career, you might spend a lot of time tracking
down syntax errors. As you gain experience, you will make fewer errors and find them faster.
Runtime error: The second type of error is a runtime error, so called because the error does notappear until after the program has started running. These errors are also called
exceptions because they usually indicate that something exceptional (and bad)
has happened. 16 | Chapter 2: Variables, Expressions and Statements

Runtime errors are rare in the simple programs you will see in the first few chapters, so it might be a while before you encounter one.
Semantic error: The third type of error is “semantic”, which means related to meaning. If there is
a semantic error in your program, it will run without generating error messages, but it will not do the right thing. It will do something else. Specifically, it will dowhat you told it to do.
Identifying semantic errors can be tricky because it requires you to work back
ward by looking at the output of the program and trying to figure out what it is doing.
variable: A name that refers to a value.
assignment: A statement that assigns a value to a variable.
state diagram: A graphical representation of a set of variables and the values they refer to.
keyword: A reserved word that is used to parse a program; you cannot use keywords like
if , def , and while as variable names.
operand: One of the values on which an operator operates.
expression: A combination of variables, operators, and values that represents a single result.
evaluate: To simplify an expression by performing the operations in order to yield a single
statement: A section of code that represents a command or action. So far, the statements wehave seen are assignments and print statements.
execute: To run a statement and do what it says.
interactive mode: A way of using the Python interpreter by typing code at the prompt. Glossary | 17

script mode:A way of using the Python interpreter to read code from a script and run it.
script: A program stored in a file.
order of operations: Rules governing the order in which expressions involving multiple operators andoperands are evaluated.
concatenate: To join two operands end-to-end.
comment: Information in a program that is meant for other programmers (or anyone read
ing the source code) and has no effect on the execution of the program.
syntax error: An error in a program that makes it impossible to parse (and therefore impossible to interpret).
exception: An error that is detected while the program is running.
semantics: The meaning of a program.
semantic error: An error in a program that makes it do something other than what the programmer intended.
Exercises Exercise 2-1.
Repeating my advice from the previous chapter, whenever you learn a new feature, you should try it out in interactive mode and make errors on purpose to see what
goes wrong.
Weve seen that n = 42 is legal. What about 42 = n?

How about x = y = 1?

In some languages every statement ends with a semicolon, ;. What happens if
you put a semicolon at the end of a Python statement?
18 | Chapter 2: Variables, Expressions and Statements

What if you put a period at the end of a statement?
In math notation you can multiply x and y like this:
x y
. What happens if you try
that in Python?
Exercise 2-2.
Practice using the Python interpreter as a calculator:
The volume of a sphere with radius r is
3 r 3
. What is the volume of a sphere with
radius 5?
Suppose the cover price of a book is $24.95, but bookstores get a 40% discount. Shipping costs $3 for the first copy and 75 cents for each additional copy. What is
the total wholesale cost for 60 copies?
If I leave my house at 6:52 am and run 1 mile at an easy pace (8:15 per mile), then 3 miles at tempo (7:12 per mile) and 1 mile at an easy pace again, what time do I
get home for breakfast?
Exercises | 19

In the context of programming, a function is a named sequence of statements that
performs a computation. When you define a function, you specify the name and the
sequence of statements. Later, you can “call” the function by name.
Function Calls
We have already seen one example of a function call:
>>> type(42)

The name of the function is type. The expression in parentheses is called the argu
ment of the function. The result, for this function, is the type of the argument.
It is common to say that a function “takes” an argument and “returns” a result. The result is also called the return value.
Python provides functions that convert values from one type to another. The int
function takes any value and converts it to an integer, if it can, or complains other
>>> int(32)
>>> int(Hello)
ValueError: invalid literal for int(): Hello
int can convert floating-point values to integers, but it doesnt round off; it chops off
the fraction part:
>>> int(3.99999)
>>> int(-2.3)

float converts integers and strings to floating-point numbers:
>>> float(32)
>>> float(3.14159)
Finally, str converts its argument to a string:
>>> str(32)
>>> str(3.14159)
Math Functions
Python has a math module that provides most of the familiar mathematical functions. A module is a file that contains a collection of related functions.
Before we can use the functions in a module, we have to import it with an import
statement :
>>> import math
This statement creates a module object named math. If you display the module
object, you get some information about it:
>>> math

The module object contains the functions and variables defined in the module. To
access one of the functions, you have to specify the name of the module and the name
of the function, separated by a dot (also known as a period). This format is called dot
notation .
>>> ratio = signal_power / noise_power >>> decibels = 10 * math.log10(ratio)
>>> radians = 0.7
>>> height = math.sin(radians)
The first example uses math.log10 to compute a signal-to-noise ratio in decibels
(assuming that signal_power and noise_power are defined). The math module also
provides log, which computes logarithms base e.
The second example finds the sine of radians. The name of the variable is a hint that
sin and the other trigonometric functions ( cos, tan , etc.) take arguments in radians.
To convert from degrees to radians, divide by 180 and multiply by :22 | Chapter 3: Functions

>>> degrees = 45
>>> radians = degrees / 180.0 * math.pi
>>> math.sin(radians)
The expression math.pi gets the variable pi from the math module. Its value is a
floating-point approximation of , accurate to about 15 digits.
If you know trigonometry, you can check the previous result by comparing it to the
square root of 2 divided by 2:
>>> math.sqrt(2) / 2.0
So far, we have looked at the elements of a programvariables, expressions, and statementsin isolation, without talking about how to combine them.
One of the most useful features of programming languages is their ability to take
small building blocks and compose them. For example, the argument of a function
can be any kind of expression, including arithmetic operators:
x = math.sin(degrees / 360.0 * 2 * math.pi)
And even function calls:
x = math.exp(math.log(x+1))
Almost anywhere you can put a value, you can put an arbitrary expression, with one
exception: the left side of an assignment statement has to be a variable name. Any
other expression on the left side is a syntax error (we will see exceptions to this rule
>>> minutes = hours * 60 # right>>> hours * 60 = minutes # wrong!
SyntaxError: cant assign to operator
Adding New Functions So far, we have only been using the functions that come with Python, but it is also
possible to add new functions. A function
definition specifies the name of a new
function and the sequence of statements that run when the function is called.
Here is an example: def print_lyrics():
print("Im a lumberjack, and Im okay.")
print("I sleep all night and I work all day.")
Composition | 23

def is a keyword that indicates that this is a function definition. The name of the
function is print_lyrics . The rules for function names are the same as for variable
names: letters, numbers and underscore are legal, but the first character cant be a
number. You cant use a keyword as the name of a function, and you should avoid
having a variable and a function with the same name.
The empty parentheses after the name indicate that this function doesnt take any
The first line of the function definition is called the header; the rest is called the
body . The header has to end with a colon and the body has to be indented. By con
vention, indentation is always four spaces. The body can contain any number of state
The strings in the print statements are enclosed in double quotes. Single quotes and double quotes do the same thing; most people use single quotes except in cases like
this where a single quote (which is also an apostrophe) appears in the string.
All quotation marks (single and double) must be “straight quotes”, usually located
next to Enter on the keyboard. “Curly quotes”, like the ones in this sentence, are not
legal in Python.
If you type a function definition in interactive mode, the interpreter prints dots ( ...)
to let you know that the definition isnt complete:
>>> def print_lyrics():
... print("Im a lumberjack, and Im okay.")
... print("I sleep all night and I work all day.")
To end the function, you have to enter an empty line.
Defining a function creates a function object, which has type function:
>>> print(print_lyrics)
>>> type(print_lyrics)

The syntax for calling the new function is the same as for built-in functions:
>>> print_lyrics()
Im a lumberjack, and Im okay. I sleep all night and I work all day.
Once you have defined a function, you can use it inside another function. For exam
ple, to repeat the previous refrain, we could write a function called repeat_lyrics:
def repeat_lyrics(): print_lyrics()
print_lyrics()24 | Chapter 3: Functions

And then call repeat_lyrics :
>>> repeat_lyrics()
Im a lumberjack, and Im okay.
I sleep all night and I work all day.
Im a lumberjack, and Im okay.
I sleep all night and I work all day.
But thats not really how the song goes.
Definitions and Uses
Pulling together the code fragments from the previous section, the whole program
looks like this:
def print_lyrics():
print("Im a lumberjack, and Im okay.")
print("I sleep all night and I work all day.")
def repeat_lyrics():
This program contains two function definitions: print_lyrics and repeat_lyrics .
Function definitions get executed just like other statements, but the effect is to create
function objects. The statements inside the function do not run until the function is
called, and the function definition generates no output.
As you might expect, you have to create a function before you can run it. In other words, the function definition has to run before the function gets called.
As an exercise, move the last line of this program to the top, so the function call
appears before the definitions. Run the program and see what error message you get.
Now move the function call back to the bottom and move the definition of print_lyrics after the definition of repeat_lyrics. What happens when you run
this program?
Flow of Execution
To ensure that a function is defined before its first use, you have to know the order statements run in, which is called the
ow of execution .
Execution always begins at the first statement of the program. Statements are run one
at a time, in order from top to bottom.
Function definitions do not alter the flow of execution of the program, but remember
that statements inside the function dont run until the function is called.
Definitions and Uses | 25

A function call is like a detour in the flow of execution. Instead of going to the next
statement, the flow jumps to the body of the function, runs the statements there, and then comes back to pick up where it left off.
That sounds simple enough, until you remember that one function can call another.
While in the middle of one function, the program might have to run the statements
in another function. Then, while running that new function, the program might have
to run yet another function!
Fortunately, Python is good at keeping track of where it is, so each time a function completes, the program picks up where it left off in the function that called it. When
it gets to the end of the program, it terminates.
In summary, when you read a program, you dont always want to read from top to
bottom. Sometimes it makes more sense if you follow the flow of execution.
Parameters and Arguments
Some of the functions we have seen require arguments. For example, when you call math.sin you pass a number as an argument. Some functions take more than one
argument: math.pow takes two, the base and the exponent.
Inside the function, the arguments are assigned to variables called parameters. Here
is a definition for a function that takes an argument:
def print_twice(bruce):
This function assigns the argument to a parameter named bruce. When the function
is called, it prints the value of the parameter (whatever it is) twice.
This function works with any value that can be printed: >>> print_twice(Spam)
>>> print_twice(42)
>>> print_twice(math.pi)
The same rules of composition that apply to built-in functions also apply to
programmer-defined functions, so we can use any kind of expression as an argument
for print_twice :26 | Chapter 3: Functions

>>> print_twice(Spam *4)
Spam Spam Spam Spam
Spam Spam Spam Spam
>>> print_twice(math.cos(math.pi))
The argument is evaluated before the function is called, so in the examples the
expressions Spam *4 and math.cos(math.pi) are only evaluated once.
You can also use a variable as an argument: >>> michael = Eric, the half a bee.
>>> print_twice(michael)
Eric, the half a bee.
Eric, the half a bee.
The name of the variable we pass as an argument ( michael) has nothing to do with
the name of the parameter ( bruce). It doesnt matter what the value was called back
home (in the caller); here in print_twice, we call everybody bruce.
Variables and Parameters Are Local
When you create a variable inside a function, it is local, which means that it only
exists inside the function. For example:
def cat_twice(part1, part2):
cat = part1 + part2
This function takes two arguments, concatenates them, and prints the result twice. Here is an example that uses it:
>>> line1 = Bing tiddle
>>> line2 = tiddle bang.
>>> cat_twice(line1, line2)
Bing tiddle tiddle bang.
Bing tiddle tiddle bang.
When cat_twice terminates, the variable cat is destroyed. If we try to print it, we get
an exception:
>>> print(cat)
NameError: name cat is not defined
Parameters are also local. For example, outside print_twice, there is no such thing as
bruce . Variables and Parameters Are Local | 27

Stack DiagramsTo keep track of which variables can be used where, it is sometimes useful to draw a
stack diagram . Like state diagrams, stack diagrams show the value of each variable,
but they also show the function each variable belongs to.
Each function is represented by a frame. A frame is a box with the name of a function
beside it and the parameters and variables of the function inside it. The stack diagram
for the previous example is shown in Figure 3-1.
Figure 3-1. Stack diagram.
The frames are arranged in a stack that indicates which function called which, and so
on. In this example, print_twice was called by cat_twice, and cat_twice was called
by __main__ , which is a special name for the topmost frame. When you create a vari
able outside of any function, it belongs to __main__.
Each parameter refers to the same value as its corresponding argument. So, part1 has
the same value as line1, part2 has the same value as line2, and bruce has the same
value as cat.
If an error occurs during a function call, Python prints the name of the function, the
name of the function that called it, and the name of the function that called that, all
the way back to __main__.
For example, if you try to access cat from within print_twice, you get a NameError :
Traceback (innermost last): File "", line 13, in __main__
cat_twice(line1, line2)
File "", line 5, in cat_twice
File "", line 9, in print_twice
NameError: name cat is not defined
28 | Chapter 3: Functions

This list of functions is called a traceback. It tells you what program file the error
occurred in, and what line, and what functions were executing at the time. It also shows the line of code that caused the error.
The order of the functions in the traceback is the same as the order of the frames in
the stack diagram. The function that is currently running is at the bottom.
Fruitful Functions and Void Functions Some of the functions we have used, such as the math functions, return results; for lack of a better name, I call them fruitful functions. Other functions, like
print_twice , perform an action but dont return a value. They are called void func
tions .
When you call a fruitful function, you almost always want to do something with the
result; for example, you might assign it to a variable or use it as part of an expression:
x = math.cos(radians)
golden = (math.sqrt(5) + 1) / 2
When you call a function in interactive mode, Python displays the result:
>>> math.sqrt(5)
But in a script, if you call a fruitful function all by itself, the return value is lost for
This script computes the square root of 5, but since it doesnt store or display the
result, it is not very useful.
Void functions might display something on the screen or have some other effect, but they dont have a return value. If you assign the result to a variable, you get a special
value called None:
>>> result = print_twice(Bing)
>>> print(result)
The value None is not the same as the string None. It is a special value that has its
own type:
>>> print(type(None))

The functions we have written so far are all void. We will start writing fruitful func
tions in a few chapters. Fruitful Functions and Void Functions | 29

Why Functions?
It may not be clear why it is worth the trouble to divide a program into functions. There are several reasons:
Creating a new function gives you an opportunity to name a group of statements,which makes your program easier to read and debug.

Functions can make a program smaller by eliminating repetitive code. Later, ifyou make a change, you only have to make it in one place.

Dividing a long program into functions allows you to debug the parts one at atime and then assemble them into a working whole.

Well-designed functions are often useful for many programs. Once you write and
debug one, you can reuse it.
One of the most important skills you will acquire is debugging. Although it can be frustrating, debugging is one of the most intellectually rich, challenging, and interest
ing parts of programming.
In some ways debugging is like detective work. You are confronted with clues and
you have to infer the processes and events that led to the results you see.
Debugging is also like an experimental science. Once you have an idea about what is going wrong, you modify your program and try again. If your hypothesis was correct,
you can predict the result of the modification, and you take a step closer to a working
program. If your hypothesis was wrong, you have to come up with a new one. As
Sherlock Holmes pointed out, “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.” (A. Conan Doyle,
e Sign of Four ).
For some people, programming and debugging are the same thing. That is, program
ming is the process of gradually debugging a program until it does what you want.
The idea is that you should start with a working program and make small modifica tions, debugging them as you go.
For example, Linux is an operating system that contains millions of lines of code, but
it started out as a simple program Linus Torvalds used to explore the Intel 80386 chip. According to Larry Greenfield, “One of Linuss earlier projects was a program that
would switch between printing AAAA and BBBB. This later evolved to Linux.” (
Linux Users Guide Beta Version 1).
30 | Chapter 3: Functions

Glossaryfunction: A named sequence of statements that performs some useful operation. Functions
may or may not take arguments and may or may not produce a result.
A statement that creates a new function, specifying its name, parameters, and the statements it contains.
function object: A value created by a function definition. The name of the function is a variable
that refers to a function object.
header: The first line of a function definition.
body: The sequence of statements inside a function definition.
parameter: A name used inside a function to refer to the value passed as an argument.
function call: A statement that runs a function. It consists of the function name followed by an
argument list in parentheses.
argument: A value provided to a function when the function is called. This value is assignedto the corresponding parameter in the function.
local variable: A variable defined inside a function. A local variable can only be used inside its
return value: The result of a function. If a function call is used as an expression, the returnvalue is the value of the expression.
fruitful function: A function that returns a value.
void function: A function that always returns None.
None :
A special value returned by void functions.
Glossary | 31

module:A file that contains a collection of related functions and other definitions.
import statement: A statement that reads a module file and creates a module object.
module object: A value created by an import statement that provides access to the values defined
in a module.
dot notation: The syntax for calling a function in another module by specifying the module
name followed by a dot (period) and the function name.
composition: Using an expression as part of a larger expression, or a statement as part of alarger statement.
ow of execution: The order statements run in.
stack diagram: A graphical representation of a stack of functions, their variables, and the valuesthey refer to.
frame: A box in a stack diagram that represents a function call. It contains the local vari
ables and parameters of the function.
traceback: A list of the functions that are executing, printed when an exception occurs.
Exercises Exercise 3-1.
Write a function named right_justify that takes a string named s as a parameter
and prints the string with enough leading spaces so that the last letter of the string is
in column 70 of the display:
>>> right_justify(monty)
Hint: Use string concatenation and repetition. Also, Python provides a built-in func tion called len that returns the length of a string, so the value of len(monty) is 5.
32 | Chapter 3: Functions

Exercise 3-2.
A function object is a value you can assign to a variable or pass as an argument. For
example, do_twice is a function that takes a function object as an argument and calls
it twice:
def do_twice(f):
Heres an example that uses do_twice to call a function named print_spam twice:
def print_spam():
Type this example into a script and test it.
Modify do_twice so that it takes two arguments, a function object and a value,
and calls the function twice, passing the value as an argument.
Copy the definition of print_twice from earlier in this chapter to your script.
Use the modified version of do_twice to call print_twice twice, passing spam
as an argument.
Define a new function called do_four that takes a function object and a value and
calls the function four times, passing the value as a parameter. There should be
only two statements in the body of this function, not four.
Solution: .
Exercise 3-3.
Note: This exercise should be done using only the statements and other features we
have learned so far.
Write a function that draws a grid like the following:
+ - - - - + - - - - +
| | |
| | |
| | |
| | |
+ - - - - + - - - - +
| | |
| | |
| | |
| | |
+ - - - - + - - - - +
Exercises | 33

Hint: to print more than one value on a line, you can print a comma-separatedsequence of values:
print(+, -)
By default, print advances to the next line, but you can override that behavior
and put a space at the end, like this:
print(+, end= )
The output of these statements is + -.
A print statement with no argument ends the current line and goes to the next
line. 2.
Write a function that draws a similar grid with four rows and four columns.
Solution: . Credit: This exercise is based on an
exercise in Oualline, Practical C Programming,
ird Edition, OReilly Media, 1997.
34 | Chapter 3: Functions

Case Study: Interface Design
This chapter presents a case study that demonstrates a process for designing func tions that work together.
It introduces the turtle module, which allows you to create images using turtle
graphics. The turtle module is included in most Python installations, but if you are
running Python using PythonAnywhere, you wont be able to run the turtle examples (at least you couldnt when I wrote this).
If you have already installed Python on your computer, you should be able to run the
examples. Otherwise, now is a good time to install. I have posted instructions at .
Code examples from this chapter are available from .
The turtle Module
To check whether you have the turtle module, open the Python interpreter and type:
>>> import turtle
>>> bob = turtle.Turtle()
When you run this code, it should create a new window with a small arrow that rep
resents the turtle. Close the window.
Create a file named and type in the following code:
import turtle
bob = turtle.Turtle()

The turtle module (with a lowercase t) provides a function called Turtle (with an
uppercase T) that creates a Turtle object, which we assign to a variable named bob.
Printing bob displays something like:

This means that bob refers to an object with type Turtle as defined in module
turtle .
mainloop tells the window to wait for the user to do something, although in this case
theres not much for the user to do except close the window.
Once you create a Turtle, you can call a method to move it around the window. A
method is similar to a function, but it uses slightly different syntax. For example, to
move the turtle forward:
The method, fd, is associated with the turtle object were calling bob. Calling a
method is like making a request: you are asking bob to move forward.
The argument of fd is a distance in pixels, so the actual size depends on your display.
Other methods you can call on a Turtle are bk to move backward, lt for left turn, and
rt right turn. The argument for lt and rt is an angle in degrees.
Also, each Turtle is holding a pen, which is either down or up; if the pen is down, the Turtle leaves a trail when it moves. The methods pu and pd stand for “pen up” and
“pen down”.
To draw a right angle, add these lines to the program (after creating bob and before
calling mainloop ):
When you run this program, you should see bob move east and then north, leaving
two line segments behind.
Now modify the program to draw a square. Dont go on until youve got it working!36 | Chapter 4: Case Study: Interface Design

Simple Repetition
Chances are you wrote something like this: bob.fd(100)
We can do the same thing more concisely with a for statement. Add this example to and run it again:
for i in range(4):
You should see something like this:
This is the simplest use of the for statement; we will see more later. But that should
be enough to let you rewrite your square-drawing program. Dont go on until you do.
Here is a for statement that draws a square:
for i in range(4):
The syntax of a for statement is similar to a function definition. It has a header that
ends with a colon and an indented body. The body can contain any number of state
A for statement is also called a loop because the flow of execution runs through the
body and then loops back to the top. In this case, it runs the body four times.
This version is actually a little different from the previous square-drawing code
because it makes another turn after drawing the last side of the square. The extra turn takes more time, but it simplifies the code if we do the same thing every time through
the loop. This version also has the effect of leaving the turtle back in the starting posi
tion, facing in the starting direction.
Simple Repetition | 37

ExercisesThe following is a series of exercises using TurtleWorld. They are meant to be fun, but
they have a point, too. While you are working on them, think about what the point is.
The following sections have solutions to the exercises, so dont look until you have
finished (or at least tried).1.
Write a function called square that takes a parameter named t, which is a turtle.
It should use the turtle to draw a square.
Write a function call that passes bob as an argument to square, and then run the
program again.
Add another parameter, named length, to square . Modify the body so length of
the sides is length, and then modify the function call to provide a second argu
ment. Run the program again. Test your program with a range of values for
length .
Make a copy of square and change the name to polygon. Add another parameter
named n and modify the body so it draws an n-sided regular polygon.
Hint: The exterior angles of an n-sided regular polygon are 360/n degrees.
Write a function called circle that takes a turtle, t, and radius, r, as parameters
and that draws an approximate circle by calling polygon with an appropriate
length and number of sides. Test your function with a range of values of r.
Hint: figure out the circumference of the circle and make sure that length * n =
circumference .
Make a more general version of circle called arc that takes an additional
parameter angle, which determines what fraction of a circle to draw. angle is in
units of degrees, so when angle=360, arc should draw a complete circle.
The first exercise asks you to put your square-drawing code into a function definition
and then call the function, passing the turtle as a parameter. Here is a solution:
def square(t):
for i in range(4):
38 | Chapter 4: Case Study: Interface Design

The innermost statements, fd and lt, are indented twice to show that they are inside
the for loop, which is inside the function definition. The next line, square(bob), is
flush with the left margin, which indicates the end of both the for loop and the func
tion definition.
Inside the function, t refers to the same turtle bob, so has the same effect as . In that case, why not call the parameter bob? The idea is that t can be
any turtle, not just bob, so you could create a second turtle and pass it as an argument
to square :
alice = Turtle()
Wrapping a piece of code up in a function is called encapsulation. One of the bene
fits of encapsulation is that it attaches a name to the code, which serves as a kind of
documentation. Another advantage is that if you reuse the code, it is more concise to
call a function twice than to copy and paste the body!
The next step is to add a length parameter to square. Here is a solution:
def square(t, length):
for i in range(4):
square(bob, 100)
Adding a parameter to a function is called generalization because it makes the func
tion more general: in the previous version, the square is always the same size; in this
version it can be any size.
The next step is also a generalization. Instead of drawing squares, polygon draws reg
ular polygons with any number of sides. Here is a solution:
def polygon(t, n, length):
angle = 360 / n
for i in range(n):
polygon(bob, 7, 70)
This example draws a 7-sided polygon with side length 70.
If you are using Python 2, the value of angle might be off because of integer division.
A simple solution is to compute angle = 360.0 / n. Because the numerator is a
floating-point number, the result is floating point. Generalization | 39

When a function has more than a few numeric arguments, it is easy to forget what
they are, or what order they should be in. In that case it is often a good idea to include
the names of the parameters in the argument list:
polygon(bob, n=7, length=70)
These are called keyword arguments because they include the parameter names as
“keywords” (not to be confused with Python keywords like while and def).
This syntax makes the program more readable. It is also a reminder about how argu
ments and parameters work: when you call a function, the arguments are assigned to the parameters.
Interface Design
The next step is to write circle, which takes a radius, r, as a parameter. Here is a
simple solution that uses polygon to draw a 50-sided polygon:
import math
def circle(t, r):
circumference = 2 * math.pi * r
n = 50
length = circumference / n
polygon(t, n, length)
The first line computes the circumference of a circle with radius r using the formula
2 r
. Since we use math.pi, we have to import math. By convention, import state
ments are usually at the beginning of the script.
n is the number of line segments in our approximation of a circle, so length is the
length of each segment. Thus, polygon draws a 50-sided polygon that approximates a
circle with radius r.
One limitation of this solution is that n is a constant, which means that for very big
circles, the line segments are too long, and for small circles, we waste time drawing
very small segments. One solution would be to generalize the function by taking n as
a parameter. This would give the user (whoever calls circle) more control, but the
interface would be less clean.
The interface of a function is a summary of how it is used: what are the parameters?
What does the function do? And what is the return value? An interface is “clean” if it
allows the caller to do what they want without dealing with unnecessary details.
40 | Chapter 4: Case Study: Interface Design

In this example, r belongs in the interface because it specifies the circle to be drawn. n
is less appropriate because it pertains to the details of how the circle should be ren
Rather than clutter up the interface, it is better to choose an appropriate value of n
depending on circumference :
def circle(t, r): circumference = 2 * math.pi * r
n = int(circumference / 3) + 1
length = circumference / n
polygon(t, n, length)
Now the number of segments is an integer near circumference/3, so the length of
each segment is approximately 3, which is small enough that the circles look good,
but big enough to be efficient, and acceptable for any size circle.
When I wrote circle, I was able to reuse polygon because a many-sided polygon is a
good approximation of a circle. But arc is not as cooperative; we cant use polygon or
circle to draw an arc.
One alternative is to start with a copy of polygon and transform it into arc. The
result might look like this:
def arc(t, r, angle):
arc_length = 2 * math.pi * r * angle / 360
n = int(arc_length / 3) + 1
step_length = arc_length / n
step_angle = angle / n

for i in range(n):
The second half of this function looks like polygon, but we cant reuse polygon
without changing the interface. We could generalize polygon to take an angle as a
third argument, but then polygon would no longer be an appropriate name! Instead,
lets call the more general function polyline:
def polyline(t, n, length, angle):
for i in range(n):
Now we can rewrite polygon and arc to use polyline : Refactoring | 41

def polygon(t, n, length):
angle = 360.0 / n
polyline(t, n, length, angle)
def arc(t, r, angle):
arc_length = 2 * math.pi * r * angle / 360
n = int(arc_length / 3) + 1
step_length = arc_length / n
step_angle = float(angle) / n
polyline(t, n, step_length, step_angle)
Finally, we can rewrite circle to use arc:
def circle(t, r):
arc(t, r, 360)
This processrearranging a program to improve interfaces and facilitate code reuse
is called refactoring . In this case, we noticed that there was similar code in arc and
polygon , so we “factored it out” into polyline.
If we had planned ahead, we might have written polyline first and avoided refactor
ing, but often you dont know enough at the beginning of a project to design all the interfaces. Once you start coding, you understand the problem better. Sometimes
refactoring is a sign that you have learned something.
A Development Plan A development plan is a process for writing programs. The process we used in this
case study is “encapsulation and generalization”. The steps of this process are: 1.
Start by writing a small program with no function definitions.
Once you get the program working, identify a coherent piece of it, encapsulate the piece in a function and give it a name.
Generalize the function by adding appropriate parameters.
Repeat steps 13 until you have a set of working functions. Copy and paste working code to avoid retyping (and re-debugging).
Look for opportunities to improve the program by refactoring. For example, if you have similar code in several places, consider factoring it into an appropriately
general function.
This process has some drawbackswe will see alternatives laterbut it can be useful
if you dont know ahead of time how to divide the program into functions. This
approach lets you design as you go along.
42 | Chapter 4: Case Study: Interface Design

docstringA docstring is a string at the beginning of a function that explains the interface (“doc”
is short for “documentation”). Here is an example:
def polyline(t, n, length, angle):
"""Draws n line segments with the given length and
angle (in degrees) between them. t is a turtle.
for i in range(n):
By convention, all docstrings are triple-quoted strings, also known as multiline
strings because the triple quotes allow the string to span more than one line.
It is terse, but it contains the essential information someone would need to use this
function. It explains concisely what the function does (without getting into the details of how it does it). It explains what effect each parameter has on the behavior of the
function and what type each parameter should be (if it is not obvious).
Writing this kind of documentation is an important part of interface design. A well-
designed interface should be simple to explain; if you have a hard time explaining one of your functions, maybe the interface could be improved.
Debugging An interface is like a contract between a function and a caller. The caller agrees to
provide certain parameters and the function agrees to do certain work.
For example, polyline requires four arguments: t has to be a Turtle; n has to be an
integer; length should be a positive number; and angle has to be a number, which is
understood to be in degrees.
These requirements are called preconditions because they are supposed to be true
before the function starts executing. Conversely, conditions at the end of the function
are postconditions . Postconditions include the intended effect of the function (like
drawing line segments) and any side effects (like moving the Turtle or making other
Preconditions are the responsibility of the caller. If the caller violates a (properly doc
umented!) precondition and the function doesnt work correctly, the bug is in the caller, not the function.
If the preconditions are satisfied and the postconditions are not, the bug is in the
function. If your pre- and postconditions are clear, they can help with debugging. docstring | 43

Glossarymethod: A function that is associated with an object and called using dot notation.
loop: A part of a program that can run repeatedly.
encapsulation: The process of transforming a sequence of statements into a function definition.
generalization: The process of replacing something unnecessarily specific (like a number) with
something appropriately general (like a variable or parameter).
keyword argument: An argument that includes the name of the parameter as a “keyword”.
interface: A description of how to use a function, including the name and descriptions ofthe arguments and return value.
refactoring: The process of modifying a working program to improve function interfaces andother qualities of the code.
development plan: A process for writing programs.
docstring: A string that appears at the top of a function definition to document the func
tions interface.
precondition: A requirement that should be satisfied by the caller before a function starts.
postcondition: A requirement that should be satisfied by the function before it ends.
Exercises Exercise 4-1.
Download the code in this chapter from
44 | Chapter 4: Case Study: Interface Design

1. Draw a stack diagram that shows the state of the program while executing cir
cle(bob, radius) . You can do the arithmetic by hand or add print statements
to the code. 2.
The version of arc in “Refactoring” on page 41 is not very accurate because the
linear approximation of the circle is always outside the true circle. As a result, the
Turtle ends up a few pixels away from the correct destination. My solution shows a way to reduce the effect of this error. Read the code and see if it makes sense toyou. If you draw a diagram, you might see how it works.
Exercise 4-2.
Write an appropriately general set of functions that can draw flowers as in Figure 4-1.
Figure 4-1. Turtle owers.
Solution: , also requires http://thinkpy .
Exercise 4-3.
Write an appropriately general set of functions that can draw shapes as in Figure 4-2.
Figure 4-2. Turtle pies.
Solution: .
Exercises | 45

Exercise 4-4.
The letters of the alphabet can be constructed from a moderate number of basic ele
ments, like vertical and horizontal lines and a few curves. Design an alphabet that can
be drawn with a minimal number of basic elements and then write functions that
draw the letters.
You should write one function for each letter, with names draw_a, draw_b , etc., and
put your functions in a file named You can download a “turtle type
writer” from to help you test your code.
You can get a solution from ; it also requires .
Exercise 4-5.
Read about spirals at ; then write a program that
draws an Archimedian spiral (or one of the other kinds).
Solution: .46 | Chapter 4: Case Study: Interface Design

Conditionals and Recursion
The main topic of this chapter is the if statement, which executes different code
depending on the state of the program. But first I want to introduce two new opera
tors: floor division and modulus.
Floor Division and ModulusThe
oor division operator, //, divides two numbers and rounds down to an integer.
For example, suppose the run time of a movie is 105 minutes. You might want to
know how long that is in hours. Conventional division returns a floating-point
>>> minutes = 105
>>> minutes / 60
But we dont normally write hours with decimal points. Floor division returns the
integer number of hours, dropping the fraction part:
>>> minutes = 105
>>> hours = minutes // 60
>>> hours
To get the remainder, you could subtract off one hour in minutes:
>>> remainder = minutes - hours * 60
>>> remainder
An alternative is to use the modulus operator, %, which divides two numbers and
returns the remainder:

>>> remainder = minutes % 60
>>> remainder
The modulus operator is more useful than it seems. For example, you can check whether one number is divisible by anotherif x % y is zero, then x is divisible by y.
Also, you can extract the right-most digit or digits from a number. For example, x %
10 yields the right-most digit of x (in base 10). Similarly x % 100 yields the last two
If you are using Python 2, division works differently. The division operator, /, per
forms floor division if both operands are integers, and floating-point division if either
operand is a float.
Boolean Expressions
A boolean expression is an expression that is either true or false. The following
examples use the operator ==, which compares two operands and produces True if
they are equal and False otherwise:
>>> 5 == 5
>>> 5 == 6
True and False are special values that belong to the type bool; they are not strings:
>>> type(True)

>>> type(False)

The == operator is one of the relational operators; the others are:
x != y # x is not equal to y
x > y # x is greater than y
x < y # x is less than y
x >= y # x is greater than or equal to y x <= y # x is less than or equal to y
Although these operations are probably familiar to you, the Python symbols are dif
ferent from the mathematical symbols. A common error is to use a single equal sign
( = ) instead of a double equal sign ( ==). Remember that = is an assignment operator
and == is a relational operator. There is no such thing as =< or =>. 48 | Chapter 5: Conditionals and Recursion

Logical OperatorsThere are three logical operators : and , or , and not. The semantics (meaning) of these
operators is similar to their meaning in English. For example, x > 0 and x < 10 is
true only if x is greater than 0 and less than 10.
n%2 == 0 or n%3 == 0 is true if either or both of the conditions is true, that is, if the
number is divisible by 2 or 3.
Finally, the not operator negates a boolean expression, so not (x > y) is true if x >
y is false, that is, if x is less than or equal to y.
Strictly speaking, the operands of the logical operators should be boolean expres
sions, but Python is not very strict. Any nonzero number is interpreted as True:
>>> 42 and True True
This flexibility can be useful, but there are some subtleties to it that might be confus
ing. You might want to avoid it (unless you know what you are doing).
Conditional Execution In order to write useful programs, we almost always need the ability to check condi
tions and change the behavior of the program accordingly. Conditional statements
give us this ability. The simplest form is the if statement:
if x > 0:
print(x is positive)
The boolean expression after if is called the condition. If it is true, the indented
statement runs. If not, nothing happens.
if statements have the same structure as function definitions: a header followed by
an indented body. Statements like this are called compound statements.
There is no limit on the number of statements that can appear in the body, but there
has to be at least one. Occasionally, it is useful to have a body with no statements (usually as a place keeper for code you havent written yet). In that case, you can use
the pass statement, which does nothing.
if x < 0:
pass # TODO: need to handle negative values!
Alternative Execution
A second form of the if statement is “alternative execution”, in which there are two
possibilities and the condition determines which one runs. The syntax looks like this: Logical Operators | 49

if x % 2 == 0:
print(x is even)
print(x is odd)
If the remainder when x is divided by 2 is 0, then we know that x is even, and the
program displays an appropriate message. If the condition is false, the second set of statements runs. Since the condition must be true or false, exactly one of the alterna
tives will run. The alternatives are called branches, because they are branches in the
flow of execution.
Chained Conditionals
Sometimes there are more than two possibilities and we need more than two
branches. One way to express a computation like that is a chained conditional:
if x < y:
print(x is less than y)
elif x > y:
print(x is greater than y)
print(x and y are equal)
elif is an abbreviation of “else if ”. Again, exactly one branch will run. There is no
limit on the number of elif statements. If there is an else clause, it has to be at the
end, but there doesnt have to be one.
if choice == a:
elif choice == b:
elif choice == c:
Each condition is checked in order. If the first is false, the next is checked, and so on. If one of them is true, the corresponding branch runs and the statement ends. Even if
more than one condition is true, only the first true branch runs.
Nested Conditionals
One conditional can also be nested within another. We could have written the exam ple in the previous section like this:
if x == y:
print(x and y are equal)
if x < y:
print(x is less than y) else:
print(x is greater than y) 50 | Chapter 5: Conditionals and Recursion

The outer conditional contains two branches. The first branch contains a simple
statement. The second branch contains another if statement, which has two
branches of its own. Those two branches are both simple statements, although they
could have been conditional statements as well.
Although the indentation of the statements makes the structure apparent, nested
conditionals become difficult to read very quickly. It is a good idea to avoid them
when you can.
Logical operators often provide a way to simplify nested conditional statements. For example, we can rewrite the following code using a single conditional:
if 0 < x:
if x < 10:
print(x is a positive single-digit number.)
The print statement runs only if we make it past both conditionals, so we can get the
same effect with the and operator:
if 0 < x and x < 10:
print(x is a positive single-digit number.)
For this kind of condition, Python provides a more concise option:
if 0 < x < 10:
print(x is a positive single-digit number.)
Recursion It is legal for one function to call another; it is also legal for a function to call itself. It
may not be obvious why that is a good thing, but it turns out to be one of the most
magical things a program can do. For example, look at the following function:
def countdown(n):
if n <= 0:
If n is 0 or negative, it outputs the word, “Blastoff !” Otherwise, it outputs n and then
calls a function named countdownitselfpassing n-1 as an argument.
What happens if we call this function like this? >>> countdown(3) Recursion | 51

The execution of countdown begins with n=3, and since n is greater than 0, it outputs
the value 3, and then calls itself...
The execution of countdown begins with n=2, and since n is greater than 0, it outputs
the value 2, and then calls itself...
The execution of countdown begins with n=1, and since n is greater than 0, it
outputs the value 1, and then calls itself...
The execution of countdown begins with n=0, and since n is not
greater than 0, it outputs the word, “Blastoff !” and then returns.
The countdown that got n=1 returns.
The countdown that got n=2 returns.
The countdown that got n=3 returns.
And then youre back in __main__. So, the total output looks like this:
A function that calls itself is recursive; the process of executing it is called recursion.
As another example, we can write a function that prints a string n times:
def print_n(s, n): if n <= 0:
print_n(s, n-1)
If n <= 0 the return statement exits the function. The flow of execution immediately
returns to the caller, and the remaining lines of the function dont run.
The rest of the function is similar to countdown: it displays s and then calls itself to
display s n-1 additional times. So the number of lines of output is 1 + (n - 1),
which adds up to n.
For simple examples like this, it is probably easier to use a for loop. But we will see
examples later that are hard to write with a for loop and easy to write with recursion,
so it is good to start early.52 | Chapter 5: Conditionals and Recursion

Stack Diagrams for Recursive Functions
In “Stack Diagrams” on page 28, we used a stack diagram to represent the state of a
program during a function call. The same kind of diagram can help interpret a recur
sive function.
Every time a function gets called, Python creates a frame to contain the functions
local variables and parameters. For a recursive function, there might be more than one frame on the stack at the same time.
Figure 5-1 shows a stack diagram for countdown called with n = 3.
Figure 5-1. Stack diagram.
As usual, the top of the stack is the frame for __main__. It is empty because we did
not create any variables in __main__ or pass any arguments to it.
The four countdown frames have different values for the parameter n. The bottom of
the stack, where n=0, is called the base case. It does not make a recursive call, so there
are no more frames.
As an exercise, draw a stack diagram for print_n called with s = Hello and n=2.
Then write a function called do_n that takes a function object and a number, n, as
arguments, and that calls the given function n times.
Infinite Recursion
If a recursion never reaches a base case, it goes on making recursive calls forever, and
the program never terminates. This is known as
infinite recursion , and it is generally
not a good idea. Here is a minimal program with an infinite recursion:
def recurse():
Stack Diagrams for Recursive Functions | 53

In most programming environments, a program with infinite recursion does not
really run forever. Python reports an error message when the maximum recursion depth is reached:
File "", line 2, in recurse File "", line 2, in recurse File "", line 2, in recurse
File "", line 2, in recurse
RuntimeError: Maximum recursion depth exceeded
This traceback is a little bigger than the one we saw in the previous chapter. When the
error occurs, there are 1,000 recurse frames on the stack!
If you write an infinite recursion by accident, review your function to confirm that
there is a base case that does not make a recursive call. And if there is a base case,
check whether you are guaranteed to reach it.
Keyboard Input
The programs we have written so far accept no input from the user. They just do the
same thing every time.
Python provides a built-in function called input that stops the program and waits for
the user to type something. When the user presses Return or Enter, the program
resumes and input returns what the user typed as a string. In Python 2, the same
function is called raw_input.
>>> text = input()
What are you waiting for?
>>> text
What are you waiting for?
Before getting input from the user, it is a good idea to print a prompt telling the user
what to type. input can take a prompt as an argument:
>>> name = input( your name?\n) your name?
Arthur, King of the Britons!
>>> name
Arthur, King of the Britons!
The sequence \n at the end of the prompt represents a newline, which is a special
character that causes a line break. Thats why the users input appears below the
If you expect the user to type an integer, you can try to convert the return value to int :54 | Chapter 5: Conditionals and Recursion

>>> prompt = the airspeed velocity of an unladen swallow?\n
>>> speed = input(prompt) the airspeed velocity of an unladen swallow? 42
>>> int(speed)
But if the user types something other than a string of digits, you get an error:
>>> speed = input(prompt) the airspeed velocity of an unladen swallow?
What do you mean, an African or a European swallow? >>> int(speed)
ValueError: invalid literal for int() with base 10
We will see how to handle this kind of error later.
When a syntax or runtime error occurs, the error message contains a lot of informa tion, but it can be overwhelming. The most useful parts are usually:
What kind of error it was

Where it occurred
Syntax errors are usually easy to find, but there are a few gotchas. Whitespace errors
can be tricky because spaces and tabs are invisible and we are used to ignoring them.
>>> x = 5
>>> y = 6
File "", line 1
y = 6
IndentationError: unexpected indent
In this example, the problem is that the second line is indented by one space. But the error message points to y, which is misleading. In general, error messages indicate
where the problem was discovered, but the actual error might be earlier in the code,
sometimes on a previous line.
The same is true of runtime errors. Suppose you are trying to compute a signal-to-
noise ratio in decibels. The formula is
SN R db= 10 log
10 Psig nal /
. In Python,
you might write something like this:
import math
signal_power = 9
noise_power = 10
ratio = signal_power // noise_power
decibels = 10 * math.log10(ratio)print(decibels)
Debugging | 55

When you run this program, you get an exception:Traceback (most recent call last):
File "", line 5, in ?
decibels = 10 * math.log10(ratio)
ValueError: math domain error
The error message indicates line 5, but there is nothing wrong with that line. To find
the real error, it might be useful to print the value of ratio, which turns out to be 0.
The problem is in line 4, which uses floor division instead of floating-point division.
You should take the time to read error messages carefully, but dont assume that
everything they say is correct.
oor division:
An operator, denoted //, that divides two numbers and rounds down (toward
zero) to an integer.
modulus operator: An operator, denoted with a percent sign ( %), that works on integers and returns
the remainder when one number is divided by another.
boolean expression: An expression whose value is either True or False .
relational operator: One of the operators that compares its operands: ==, != , > , < , >= , and <=.
logical operator: One of the operators that combines boolean expressions: and, or , and not.
conditional statement: A statement that controls the flow of execution depending on some condition.
condition: The boolean expression in a conditional statement that determines which branchruns.
compound statement: A statement that consists of a header and a body. The header ends with a colon
(:). The body is indented relative to the header.
branch: One of the alternative sequences of statements in a conditional statement.
56 | Chapter 5: Conditionals and Recursion

chained conditional:A conditional statement with a series of alternative branches.
nested conditional: A conditional statement that appears in one of the branches of another conditional statement.
return statement: A statement that causes a function to end immediately and return to the caller.
recursion: The process of calling the function that is currently executing.
base case: A conditional branch in a recursive function that does not make a recursive call.
infinite recursion:
A recursion that doesnt have a base case, or never reaches it. Eventually, an infin
ite recursion causes a runtime error.
Exercise 5-1.
The time module provides a function, also named time, that returns the current
Greenwich Mean Time in “the epoch”, which is an arbitrary time used as a reference
point. On UNIX systems, the epoch is 1 January 1970.
>>> import time
>>> time.time()
Write a script that reads the current time and converts it to a time of day in hours,
minutes, and seconds, plus the number of days since the epoch.
Exercise 5-2.
Fermats Last Theorem says that there are no positive integers a, b , and c such that
a n
+ bn
= cn
for any values of
n greater than 2.
Exercises | 57

1. Write a function named check_fermat that takes four parameters a, b, c and n
and checks to see if Fermats theorem holds. If n is greater than 2 and an
+ bn
= cn
the program should print, “Holy smokes, Fermat was wrong!” Otherwise the pro
gram should print, “No, that doesnt work.”
Write a function that prompts the user to input values for a, b, c and n, converts
them to integers, and uses check_fermat to check whether they violate Fermats
Exercise 5-3.
If you are given three sticks, you may or may not be able to arrange them in a trian
gle. For example, if one of the sticks is 12 inches long and the other two are one inch long, you will not be able to get the short sticks to meet in the middle. For any three
lengths, there is a simple test to see if it is possible to form a triangle:
If any of the three lengths is greater than the sum of the other two, then you cannot
form a triangle. Otherwise, you can. (If the sum of two lengths equals the third, they
form what is called a “degenerate” triangle.)
Write a function named is_triangle that takes three integers as arguments, and
that prints either “Yes” or “No”, depending on whether you can or cannot form a
triangle from sticks with the given lengths.
Write a function that prompts the user to input three stick lengths, converts them
to integers, and uses is_triangle to check whether sticks with the given lengths
can form a triangle.
Exercise 5-4.
What is the output of the following program? Draw a stack diagram that shows the
state of the program when it prints the result.
def recurse(n, s):
if n == 0:
recurse(n-1, n+s)
recurse(3, 0)
What would happen if you called this function like this: recurse(-1, 0)?
58 | Chapter 5: Conditionals and Recursion

2. Write a docstring that explains everything someone would need to know in orderto use this function (and nothing else).
The following exercises use the turtle module, described in Chapter 4:
Exercise 5-5.
Read the following function and see if you can figure out what it does (see the exam
ples in Chapter 4 ). Then run it and see if you got it right.
def draw(t, length, n):
if n == 0:
angle = 50
draw(t, length, n-1)
draw(t, length, n-1)
Figure 5-2. A Koch curve.
Exercise 5-6.
The Koch curve is a fractal that looks something like Figure 5-2. To draw a Koch
curve with length x, all you have to do is:
Draw a Koch curve with length x/3.
Turn left 60 degrees.
Draw a Koch curve with length x/3.
Turn right 120 degrees.
Draw a Koch curve with length x/3.
Exercises | 59

6. Turn left 60 degrees. 7.
Draw a Koch curve with length x/3.
The exception is if x is less than 3: in that case, you can just draw a straight line with
length x.
Write a function called koch that takes a turtle and a length as parameters, and
that uses the turtle to draw a Koch curve with the given length.
Write a function called snowflake that draws three Koch curves to make the out
line of a snowflake.
Solution: .
The Koch curve can be generalized in several ways. See
for examples and implement your favorite.
60 | Chapter 5: Conditionals and Recursion

Fruitful Functions
Many of the Python functions we have used, such as the math functions, produce return values. But the functions weve written are all void: they have an effect, like
printing a value or moving a turtle, but they dont have a return value. In this chapter
you will learn to write fruitful functions.
Return Values Calling the function generates a return value, which we usually assign to a variable or
use as part of an expression.
e = math.exp(1.0)
height = radius * math.sin(radians)
The functions we have written so far are void. Speaking casually, they have no return
value; more precisely, their return value is None.
In this chapter, we are (finally) going to write fruitful functions. The first example is area , which returns the area of a circle with the given radius:
def area(radius):
a = math.pi * radius**2
return a
We have seen the return statement before, but in a fruitful function the return state
ment includes an expression. This statement means: “Return immediately from this
function and use the following expression as a return value.” The expression can be
arbitrarily complicated, so we could have written this function more concisely:
def area(radius):
return math.pi * radius**2
On the other hand, temporary variables like a can make debugging easier.

Sometimes it is useful to have multiple return statements, one in each branch of a
def absolute_value(x):
if x < 0:
return -x
return x
Since these return statements are in an alternative conditional, only one runs.
As soon as a return statement runs, the function terminates without executing any
subsequent statements. Code that appears after a return statement, or any other place
the flow of execution can never reach, is called dead code.
In a fruitful function, it is a good idea to ensure that every possible path through the program hits a return statement. For example:
def absolute_value(x):
if x < 0:
return -x
if x > 0:
return x
This function is incorrect because if x happens to be 0, neither condition is true, and
the function ends without hitting a return statement. If the flow of execution gets to
the end of a function, the return value is None, which is not the absolute value of 0:
>>> absolute_value(0)
By the way, Python provides a built-in function called abs that computes absolute
As an exercise, write a compare function takes two values, x and y, and returns 1 if x
> y , 0 if x == y , and -1 if x < y .
Incremental Development
As you write larger functions, you might find yourself spending more time debug ging.
To deal with increasingly complex programs, you might want to try a process called
incremental development . The goal of incremental development is to avoid long
debugging sessions by adding and testing only a small amount of code at a time.
As an example, suppose you want to find the distance between two points, given by the coordinates
x 2,
. By the Pythagorean theorem, the distance is:
62 | Chapter 6: Fruitful Functions

distance = x2
1 2
+ y2
1 2
The first step is to consider what a distance function should look like in Python. In
other words, what are the inputs (parameters) and what is the output (return value)?
In this case, the inputs are two points, which you can represent using four numbers.
The return value is the distance represented by a floating-point value.
Immediately you can write an outline of the function: def distance(x1, y1, x2, y2):
return 0.0
Obviously, this version doesnt compute distances; it always returns zero. But it is syn tactically correct, and it runs, which means that you can test it before you make it
more complicated.
To test the new function, call it with sample arguments: >>> distance(1, 2, 4, 6)
I chose these values so that the horizontal distance is 3 and the vertical distance is 4;
that way, the result is 5, the hypotenuse of a 3-4-5 triangle. When testing a function, it is useful to know the right answer.
At this point we have confirmed that the function is syntactically correct, and we can
start adding code to the body. A reasonable next step is to find the differences
. The next version stores those values in temporary variables and prints
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
print(dx is, dx)
print(dy is, dy)
return 0.0
If the function is working, it should display dx is 3 and dy is 4 . If so, we know that
the function is getting the right arguments and performing the first computation cor
rectly. If not, there are only a few lines to check.
Next we compute the sum of squares of dx and dy:
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx**2 + dy**2
print(dsquared is: , dsquared)
return 0.0
Incremental Development | 63

Again, you would run the program at this stage and check the output (which should
be 25). Finally, you can use math.sqrt to compute and return the result:
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx**2 + dy**2
result = math.sqrt(dsquared)
return result
If that works correctly, you are done. Otherwise, you might want to print the value of
result before the return statement.
The final version of the function doesnt display anything when it runs; it only returns a value. The print statements we wrote are useful for debugging, but once you get the
function working, you should remove them. Code like that is called
because it is helpful for building the program but is not part of the final product.
When you start out, you should add only a line or two of code at a time. As you gain
more experience, you might find yourself writing and debugging bigger chunks. Either way, incremental development can save you a lot of debugging time.
The key aspects of the process are:
Start with a working program and make small incremental changes. At any point, if there is an error, you should have a good idea where it is.
Use variables to hold intermediate values so you can display and check them.
Once the program is working, you might want to remove some of the scaffoldingor consolidate multiple statements into compound expressions, but only if it does
not make the program difficult to read.
As an exercise, use incremental development to write a function called hypotenuse
that returns the length of the hypotenuse of a right triangle given the lengths of the
other two legs as arguments. Record each stage of the development process as you go.
As you should expect by now, you can call one function from within another. As an example, well write a function that takes two points, the center of the circle and a
point on the perimeter, and computes the area of the circle.
Assume that the center point is stored in the variables xc and yc, and the perimeter
point is in xp and yp. The first step is to find the radius of the circle, which is the
distance between the two points. We just wrote a function, distance, that does that:
radius = distance(xc, yc, xp, yp)
64 | Chapter 6: Fruitful Functions

The next step is to find the area of a circle with that radius; we just wrote that, too:result = area(radius)
Encapsulating these steps in a function, we get:
def circle_area(xc, yc, xp, yp):
radius = distance(xc, yc, xp, yp)
result = area(radius)
return result
The temporary variables radius and result are useful for development and debug
ging, but once the program is working, we can make it more concise by composing the function calls:
def circle_area(xc, yc, xp, yp):
return area(distance(xc, yc, xp, yp))
Boolean Functions
Functions can return booleans, which is often convenient for hiding complicated tests
inside functions. For example:
def is_divisible(x, y):
if x % y == 0:
return True
return False
It is common to give boolean functions names that sound like yes/no questions; is_divisible returns either True or False to indicate whether x is divisible by y.
Here is an example: >>> is_divisible(6, 4)
>>> is_divisible(6, 3)
The result of the == operator is a boolean, so we can write the function more con
cisely by returning it directly:
def is_divisible(x, y):
return x % y == 0
Boolean functions are often used in conditional statements:
if is_divisible(x, y):
print(x is divisible by y)
It might be tempting to write something like:
if is_divisible(x, y) == True: print(x is divisible by y) Boolean Functions | 65

But the extra comparison is unnecessary.As an exercise, write a function is_between(x, y, z) that returns True if x y z
or False otherwise.
More Recursion
We have only covered a small subset of Python, but you might be interested to know that this subset is a complete programming language, which means that anything that
can be computed can be expressed in this language. Any program ever written could
be rewritten using only the language features you have learned so far (actually, you
would need a few commands to control devices like the mouse, disks, etc., but thats all).
Proving that claim is a nontrivial exercise first accomplished by Alan Turing, one of
the first computer scientists (some would argue that he was a mathematician, but a
lot of early computer scientists started as mathematicians). Accordingly, it is known as the Turing Thesis. For a more complete (and accurate) discussion of the Turing
Thesis, I recommend Michael Sipsers book Introduction to the
eory of Computation
(Course Technology, 2012).
To give you an idea of what you can do with the tools you have learned so far, well evaluate a few recursively defined mathematical functions. A recursive definition is
similar to a circular definition, in the sense that the definition contains a reference to
the thing being defined. A truly circular definition is not very useful:
vorpal: An adjective used to describe something that is vorpal.
If you saw that definition in the dictionary, you might be annoyed. On the other hand, if you looked up the definition of the factorial function, denoted with the sym
bol !, you might get something like this:
0 ! = 1
n ! = n
n 1 !
This definition says that the factorial of 0 is 1, and the factorial of any other value, n,
is n multiplied by the factorial of n-1.
So 3! is 3 times 2!, which is 2 times 1!, which is 1 times 0!. Putting it all together, 3!
equals 3 times 2 times 1 times 1, which is 6.
If you can write a recursive definition of something, you can write a Python program
to evaluate it. The first step is to decide what the parameters should be. In this case it
should be clear that factorial takes an integer:
66 | Chapter 6: Fruitful Functions

def factorial(n):
If the argument happens to be 0, all we have to do is return 1:
def factorial(n):
if n == 0:
return 1
Otherwise, and this is the interesting part, we have to make a recursive call to find the
factorial of n-1 and then multiply it by n:
def factorial(n):
if n == 0:
return 1
recurse = factorial(n-1)
result = n * recurse
return result
The flow of execution for this program is similar to the flow of countdown in “Recur
sion” on page 51 . If we call factorial with the value 3:
Since 3 is not 0, we take the second branch and calculate the factorial of n-1...
Since 2 is not 0, we take the second branch and calculate the factorial of n-1...
Since 1 is not 0, we take the second branch and calculate the factorial of n-1...
Since 0 equals 0, we take the first branch and return 1 without mak
ing any more recursive calls.
The return value, 1, is multiplied by n, which is 1, and the result is returned.
The return value, 1, is multiplied by n, which is 2, and the result is returned.
The return value (2) is multiplied by n, which is 3, and the result, 6, becomes the
return value of the function call that started the whole process.
Figure 6-1 shows what the stack diagram looks like for this sequence of function calls.
Figure 6-1. Stack diagram.
More Recursion | 67

The return values are shown being passed back up the stack. In each frame, the
return value is the value of result, which is the product of n and recurse .
In the last frame, the local variables recurse and result do not exist, because the
branch that creates them does not run.
Leap of Faith Following the flow of execution is one way to read programs, but it can quickly
become overwhelming. An alternative is what I call the “leap of faith”. When you come to a function call, instead of following the flow of execution, you assume that
the function works correctly and returns the right result.
In fact, you are already practicing this leap of faith when you use built-in functions.
When you call math.cos or math.exp , you dont examine the bodies of those func
tions. You just assume that they work because the people who wrote the built-in func tions were good programmers.
The same is true when you call one of your own functions. For example, in “Boolean
Functions” on page 65, we wrote a function called is_divisible that determines
whether one number is divisible by another. Once we have convinced ourselves that this function is correctby examining the code and testingwe can use the function
without looking at the body again.
The same is true of recursive programs. When you get to the recursive call, instead of
following the flow of execution, you should assume that the recursive call works
(returns the correct result) and then ask yourself, “Assuming that I can find the facto rial of n-1, can I compute the factorial of n?” It is clear that you can, by multiplying
by n.
Of course, its a bit strange to assume that the function works correctly when you
havent finished writing it, but thats why its called a leap of faith!
One More Example After factorial , the most common example of a recursively defined mathematical
function is fibonacci, which has the following definition (see
wiki/Fibonacci_number ):
fibonacci 0 = 0
1 = 1
n = fibonacci n 1 + fibonacci n 2
Translated into Python, it looks like this:
68 | Chapter 6: Fruitful Functions

def fibonacci (n):
if n == 0:
return 0
elif n == 1:
return 1
return fibonacci(n-1) + fibonacci(n-2)
If you try to follow the flow of execution here, even for fairly small values of n, your
head explodes. But according to the leap of faith, if you assume that the two recursive
calls work correctly, then it is clear that you get the right result by adding them together.
Checking Types What happens if we call factorial and give it 1.5 as an argument?
>>> factorial(1.5)
RuntimeError: Maximum recursion depth exceeded
It looks like an infinite recursion. How can that be? The function has a base case when n == 0 . But if n is not an integer, we can miss the base case and recurse forever.
In the first recursive call, the value of n is 0.5. In the next, it is -0.5. From there, it gets
smaller (more negative), but it will never be 0.
We have two choices. We can try to generalize the factorial function to work with
floating-point numbers, or we can make factorial check the type of its argument.
The first option is called the gamma function and its a little beyond the scope of this
book. So well go for the second.
We can use the built-in function isinstance to verify the type of the argument.
While were at it, we can also make sure the argument is positive:
def factorial (n):
if not isinstance(n, int):
print(Factorial is only defined for integers.)
return None
elif n < 0:
print(Factorial is not defined for negative integers.) return None
elif n == 0:
return 1
return n * factorial(n-1)
The first base case handles nonintegers; the second handles negative integers. In both
cases, the program prints an error message and returns None to indicate that some
thing went wrong: Checking Types | 69

>>> factorial(fred)
Factorial is only defined for integers.
>>> factorial(-2)
Factorial is not defined for negative integers. None
If we get past both checks, we know that n is positive or zero, so we can prove that the
recursion terminates.
This program demonstrates a pattern sometimes called a guardian. The first two
conditionals act as guardians, protecting the code that follows from values that might
cause an error. The guardians make it possible to prove the correctness of the code.
In “Reverse Lookup” on page 129 we will see a more flexible alternative to printing an
error message: raising an exception.
Breaking a large program into smaller functions creates natural checkpoints for
debugging. If a function is not working, there are three possibilities to consider:
There is something wrong with the arguments the function is getting; a precon dition is violated.

There is something wrong with the function; a postcondition is violated.

There is something wrong with the return value or the way it is being used.
To rule out the first possibility, you can add a print statement at the beginning of the
function and display the values of the parameters (and maybe their types). Or you
can write code that checks the preconditions explicitly.
If the parameters look good, add a print statement before each return statement and
display the return value. If possible, check the result by hand. Consider calling the
function with values that make it easy to check the result (as in “Incremental Devel
opment” on page 62 ).
If the function seems to be working, look at the function call to make sure the return
value is being used correctly (or used at all!).
Adding print statements at the beginning and end of a function can help make the flow of execution more visible. For example, here is a version of factorial with print
def factorial(n):
space = * (4 * n)
print(space, factorial, n)
if n == 0:
print(space, returning 1)
70 | Chapter 6: Fruitful Functions

return 1
recurse = factorial(n-1)
result = n * recurse
print(space, returning, result)
return result
space is a string of space characters that controls the indentation of the output. Here
is the result of factorial(4) :
factorial 4
factorial 3
factorial 2
factorial 1
factorial 0
returning 1
returning 1
returning 2
returning 6
returning 24
If you are confused about the flow of execution, this kind of output can be helpful. It
takes some time to develop effective scaffolding, but a little bit of scaffolding can save a lot of debugging.
temporary variable: A variable used to store an intermediate value in a complex calculation.
dead code: Part of a program that can never run, often because it appears after a return
incremental development: A program development plan intended to avoid debugging by adding and testingonly a small amount of code at a time.
Code that is used during program development but is not part of the final
guardian: A programming pattern that uses a conditional statement to check for and handle circumstances that might cause an error.
Glossary | 71

ExercisesExercise 6-1.
Draw a stack diagram for the following program. What does the program print? def b(z):
prod = a(z, z)
print(z, prod)
return prod
def a(x, y):
x = x + 1
return x * y
def c(x, y, z):
total = x + y + z
square = b(total)**2
return square
x = 1
y = x + 1
print(c(x, y+3, x+y))
Exercise 6-2.
The Ackermann function,
A m ,n
, is defined:
A m,n =
n + 1 if m= 0
m 1, 1 if m > 0 and n= 0
m 1, A m,n 1 if m > 0 and n> 0 .
See . Write a function named ack
that evaluates the Ackermann function. Use your function to evaluate ack(3, 4),
which should be 125. What happens for larger values of m and n?
Solution: .
Exercise 6-3.
A palindrome is a word that is spelled the same backward and forward, like “noon” and “redivider”. Recursively, a word is a palindrome if the first and last letters are the
same and the middle is a palindrome.
The following are functions that take a string argument and return the first, last, and
middle letters:
72 | Chapter 6: Fruitful Functions

def first(word):
return word[0]
def last(word):
return word[-1]
def middle(word):
return word[1:-1]
Well see how they work in Chapter 8. 1.
Type these functions into a file named and test them out. What
happens if you call middle with a string with two letters? One letter? What about
the empty string, which is written and contains no letters?
Write a function called is_palindrome that takes a string argument and returns
True if it is a palindrome and False otherwise. Remember that you can use the
built-in function len to check the length of a string.
Solution: .
Exercise 6-4.
A number, a, is a power of b if it is divisible by b and a/b is a power of b. Write a
function called is_power that takes parameters a and b and returns True if a is a
power of b. Note: you will have to think about the base case.
Exercise 6-5.
The greatest common divisor (GCD) of a and b is the largest number that divides
both of them with no remainder.
One way to find the GCD of two numbers is based on the observation that if r is the
remainder when a is divided by b, then
g cd a ,b = g cd b,r
. As a base case, we can
g cd a , 0 = a
Write a function called gcd that takes parameters a and b and returns their greatest
common divisor.
Credit: This exercise is based on an example from Abelson and Sussmans Structure
and Interpretation of Computer Programs (MIT Press, 1996).
Exercises | 73

This chapter is about iteration, which is the ability to run a block of statements repeatedly. We saw a kind of iteration, using recursion, in “Recursion” on page 51. We
saw another kind, using a for loop, in “Simple Repetition” on page 37. In this chapter
well see yet another kind, using a while statement. But first I want to say a little more
about variable assignment.
Reassignment As you may have discovered, it is legal to make more than one assignment to the
same variable. A new assignment makes an existing variable refer to a new value (and
stop referring to the old value).
>>> x = 5
>>> x
>>> x = 7
>>> x
The first time we display x, its value is 5; the second time, its value is 7.
Figure 7-1 shows what reassignment looks like in a state diagram.
At this point I want to address a common source of confusion. Because Python uses
the equal sign ( =) for assignment, it is tempting to interpret a statement like a = b as
a mathematical proposition of equality; that is, the claim that a and b are equal. But
this interpretation is wrong.
First, equality is a symmetric relationship and assignment is not. For example, in
mathematics, if a=7 then 7=a. But in Python, the statement a = 7 is legal and 7 = a
is not.

Also, in mathematics, a proposition of equality is either true or false for all time. If
a=b now, then a will always equal b. In Python, an assignment statement can make
two variables equal, but they dont have to stay that way:
>>> a = 5
>>> b = a # a and b are now equal
>>> a = 3 # a and b are no longer equal
>>> b
The third line changes the value of a but does not change the value of b, so they are
no longer equal.
Reassigning variables is often useful, but you should use it with caution. If the values
of variables change frequently, it can make the code difficult to read and debug.
Figure 7-1. State diagram.
Updating Variables
A common kind of reassignment is an update, where the new value of the variable
depends on the old.
>>> x = x + 1
This means “get the current value of x, add one, and then update x with the new
If you try to update a variable that doesnt exist, you get an error, because Python evaluates the right side before it assigns a value to x:
>>> x = x + 1
NameError: name x is not defined
Before you can update a variable, you have to initialize it, usually with a simple
>>> x = 0
>>> x = x + 1
Updating a variable by adding 1 is called an increment; subtracting 1 is called a
decrement .
76 | Chapter 7: Iteration

The while Statement
Computers are often used to automate repetitive tasks. Repeating identical or similar
tasks without making errors is something that computers do well and people do
poorly. In a computer program, repetition is also called iteration.
We have already seen two functions, countdown and print_n , that iterate using recur
sion. Because iteration is so common, Python provides language features to make it
easier. One is the for statement we saw in “Simple Repetition” on page 37. Well get
back to that later.
Another is the while statement. Here is a version of countdown that uses a while
def countdown(n):
while n > 0:
n = n - 1
You can almost read the while statement as if it were English. It means, “While n is
greater than 0, display the value of n and then decrement n. When you get to 0, dis
play the word Blastoff!”
More formally, here is the flow of execution for a while statement:1.
Determine whether the condition is true or false.
If false, exit the while statement and continue execution at the next statement.
If the condition is true, run the body and then go back to step 1.
This type of flow is called a loop because the third step loops back around to the top. The body of the loop should change the value of one or more variables so that the
condition becomes false eventually and the loop terminates. Otherwise the loop will repeat forever, which is called an
infinite loop . An endless source of amusement for
computer scientists is the observation that the directions on shampoo, “Lather, rinse,
repeat”, are an infinite loop.
In the case of countdown, we can prove that the loop terminates: if n is zero or nega
tive, the loop never runs. Otherwise, n gets smaller each time through the loop, so
eventually we have to get to 0.
For some other loops, it is not so easy to tell. For example:
The while Statement | 77

def sequence(n):
while n != 1:
if n % 2 == 0: # n is even
n = n / 2
else: # n is odd
n = n*3 + 1
The condition for this loop is n != 1, so the loop will continue until n is 1, which
makes the condition false.
Each time through the loop, the program outputs the value of n and then checks
whether it is even or odd. If it is even, n is divided by 2. If it is odd, the value of n is
replaced with n*3 + 1. For example, if the argument passed to sequence is 3, the
resulting values of n are 3, 10, 5, 16, 8, 4, 2, 1.
Since n sometimes increases and sometimes decreases, there is no obvious proof that
n will ever reach 1, or that the program terminates. For some particular values of n,
we can prove termination. For example, if the starting value is a power of two, n will
be even every time through the loop until it reaches 1. The previous example ends
with such a sequence, starting with 16.
The hard question is whether we can prove that this program terminates for all posi
tive values of n. So far, no one has been able to prove it or disprove it! (See http:// .)
As an exercise, rewrite the function print_n from “Recursion” on page 51 using itera
tion instead of recursion.
Sometimes you dont know its time to end a loop until you get halfway through the body. In that case you can use the break statement to jump out of the loop.
For example, suppose you want to take input from the user until they type done. You
could write:
while True:
line = input(> )
if line == done:
print(Done!) 78 | Chapter 7: Iteration

The loop condition is True, which is always true, so the loop runs until it hits the
break statement.
Each time through, it prompts the user with an angle bracket. If the user types done,
the break statement exits the loop. Otherwise the program echoes whatever the user
types and goes back to the top of the loop. Heres a sample run:
> not done
not done
> done
This way of writing while loops is common because you can check the condition
anywhere in the loop (not just at the top) and you can express the stop condition
affirmatively (“stop when this happens”) rather than negatively (“keep going until
that happens”).
Square Roots
Loops are often used in programs that compute numerical results by starting with an
approximate answer and iteratively improving it.
For example, one way of computing square roots is Newtons method. Suppose that you want to know the square root of a. If you start with almost any estimate, x, you
can compute a better estimate with the following formula:y = x+ a/x
For example, if a is 4 and x is 3:
>>> a = 4
>>> x = 3
>>> y = (x + a/x) / 2
>>> y
The result is closer to the correct answer (
4 = 2
). If we repeat the process with the
new estimate, it gets even closer:
>>> x = y
>>> y = (x + a/x) / 2
>>> y
Square Roots | 79

After a few more updates, the estimate is almost exact:>>> x = y
>>> y = (x + a/x) / 2
>>> y
>>> x = y
>>> y = (x + a/x) / 2
>>> y
In general we dont know ahead of time how many steps it takes to get to the right
answer, but we know when we get there because the estimate stops changing:
>>> x = y
>>> y = (x + a/x) / 2
>>> y
>>> x = y
>>> y = (x + a/x) / 2
>>> y
When y == x , we can stop. Here is a loop that starts with an initial estimate, x, and
improves it until it stops changing:
while True:
y = (x + a/x) / 2
if y == x:
x = y
For most values of a this works fine, but in general it is dangerous to test float equal
ity. Floating-point values are only approximately right: most rational numbers, like
1/3 , and irrational numbers, like
, cant be represented exactly with a float.
Rather than checking whether x and y are exactly equal, it is safer to use the built-in
function abs to compute the absolute value, or magnitude, of the difference between
if abs(y-x) < epsilon:
Where epsilon has a value, like 0.0000001, that determines how close is close
80 | Chapter 7: Iteration

Newtons method is an example of an algorithm: it is a mechanical process for solv
ing a category of problems (in this case, computing square roots).
To understand what an algorithm is, it might help to start with something that is not
an algorithm. When you learned to multiply single-digit numbers, you probably
memorized the multiplication table. In effect, you memorized 100 specific solutions. That kind of knowledge is not algorithmic.
But if you were “lazy”, you might have learned a few tricks. For example, to find the
product of n and 9, you can write n-1 as the first digit and 10-n as the second digit.
This trick is a general solution for multiplying any single-digit number by 9. Thats an
Similarly, the techniques you learned for addition with carrying, subtraction with borrowing, and long division are all algorithms. One of the characteristics of algo
rithms is that they do not require any intelligence to carry out. They are mechanical
processes where each step follows from the last according to a simple set of rules.
Executing algorithms is boring, but designing them is interesting, intellectually chal lenging, and a central part of computer science.
Some of the things that people do naturally, without difficulty or conscious thought,
are the hardest to express algorithmically. Understanding natural language is a good
example. We all do it, but so far no one has been able to explain how we do it, at least
not in the form of an algorithm.
As you start writing bigger programs, you might find yourself spending more time
debugging. More code means more chances to make an error and more places for bugs to hide.
One way to cut your debugging time is “debugging by bisection”. For example, if there
are 100 lines in your program and you check them one at a time, it would take 100
Instead, try to break the problem in half. Look at the middle of the program, or near it, for an intermediate value you can check. Add a print statement (or something else
that has a verifiable effect) and run the program.
If the mid-point check is incorrect, there must be a problem in the first half of the
program. If it is correct, the problem is in the second half.
Algorithms | 81

Every time you perform a check like this, you halve the number of lines you have to
search. After six steps (which is fewer than 100), you would be down to one or two lines of code, at least in theory.
In practice it is not always clear what the “middle of the program” is and not always
possible to check it. It doesnt make sense to count lines and find the exact midpoint.
Instead, think about places in the program where there might be errors and places
where it is easy to put a check. Then choose a spot where you think the chances are about the same that the bug is before or after the check.
Glossary reassignment: Assigning a new value to a variable that already exists.
update: An assignment where the new value of the variable depends on the old.
initialization: An assignment that gives an initial value to a variable that will be updated.
increment: An update that increases the value of a variable (often by one).
decrement: An update that decreases the value of a variable.
iteration: Repeated execution of a set of statements using either a recursive function call or
a loop.
infinite loop:
A loop in which the terminating condition is never satisfied.
algorithm: A general process for solving a category of problems.
Exercise 7-1.
Copy the loop from “Square Roots” on page 79 and encapsulate it in a function called
mysqrt that takes a as a parameter, chooses a reasonable value of x, and returns an
estimate of the square root of a.
To test it, write a function named test_square_root that prints a table like this:
82 | Chapter 7: Iteration

a mysqrt(a) math.sqrt(a) diff
- --------- ------------ ----
1.0 1.0 1.0 0.0
2.0 1.41421356237 1.41421356237 2.22044604925e-16
3.0 1.73205080757 1.73205080757 0.0
4.0 2.0 2.0 0.0
5.0 2.2360679775 2.2360679775 0.0
6.0 2.44948974278 2.44948974278 0.0
7.0 2.64575131106 2.64575131106 0.0
8.0 2.82842712475 2.82842712475 4.4408920985e-16
9.0 3.0 3.0 0.0
The first column is a n
umber, a; the second column is the square root of a computed
with mysqrt; the third column is the square root computed by math.sqrt; the fourth
column is the absolute value of the difference between the two estimates.
Exercise 7-2.
The built-in function eval takes a string and evaluates it using the Python interpreter.
For example:
>>> eval(1 + 2 * 3)
>>> import math
>>> eval(math.sqrt(5))
>>> eval(type(math.pi))

Write a function called eval_loop that iteratively prompts the user, takes the result
ing input and evaluates it using eval, and prints the result.
It should continue until the user enters done, and then return the value of the last
expression it evaluated.
Exercise 7-3.
The mathematician Srinivasa Ramanujan found an infinite series that can be used to
generate a numerical approximation of
1 / :1
= 2 2
k =

4k ! 1103 + 26390k
! 4
396 4k
Write a function called estimate_pi tha
t uses this formula to compute and return an
estimate of . It should use a while loop to compute terms of the summation until the
Exercises | 83

last term is smaller than 1e-15 (which is Python notation for
10 15
). You can check
the result by comparing it to math.pi.
Solution: . 84 | Chapter 7: Iteration

CHAPTER 8Strings
Strings are not like integers, floats, and booleans. A string is a sequence, which means
it is an ordered collection of other values. In this chapter youll see how to access the
characters that make up a string, and youll learn about some of the methods strings
A String Is a Sequence A string is a sequence of characters. You can access the characters one at a time with the bracket operator:
>>> fruit = banana
>>> letter = fruit[1]
The second statement selects character number 1 from fruit and assigns it to
letter .
The expression in brackets is called an index. The index indicates which character in
the sequence you want (hence the name).
But you might not get what you expect: >>> letter
For most people, the first letter of banana is b, not a. But for computer scientists,
the index is an offset from the beginning of the string, and the offset of the first letter
is zero.
>>> letter = fruit[0]
>>> letter

So b is the 0th letter (“zero-eth”) of banana, a is the 1th letter (“one-eth”), and n is
the 2th letter (“two-eth”).
As an index, you can use an expression that contains variables and operators: >>> i = 1
>>> fruit[i]
>>> fruit[i+1]
But the value of the index has to be an integer. Otherwise you get:
>>> letter = fruit[1.5]
TypeError: string indices must be integers
len is a built-in function that returns the number of characters in a string:
>>> fruit = banana
>>> len(fruit)
To get the last letter of a string, you might be tempted to try something like this:
>>> length = len(fruit)
>>> last = fruit[length]
IndexError: string index out of range
The reason for the IndexError is that there is no letter in banana with the index 6.
Since we started counting at zero, the six letters are numbered 0 to 5. To get the last
character, you have to subtract 1 from length:
>>> last = fruit[length-1]
>>> last
Or you can use negative indices, which count backward from the end of the string. The expression fruit[-1] yields the last letter, fruit[-2] yields the second to last,
and so on.
Traversal with a for Loop A lot of computations involve processing a string one character at a time. Often they
start at the beginning, select each character in turn, do something to it, and continue
until the end. This pattern of processing is called a traversal. One way to write a tra
versal is with a while loop:86 | Chapter 8: Strings

index = 0
while index < len(fruit):
letter = fruit[index]
index = index + 1
This loop traverses the string and displays each letter on a line by itself. The loop con
dition is index < len(fruit) , so when index is equal to the length of the string, the
condition is false, and the body of the loop doesnt run. The last character accessed is
the one with the index len(fruit)-1, which is the last character in the string.
As an exercise, write a function that takes a string as an argument and displays the
letters backward, one per line.
Another way to write a traversal is with a for loop:
for letter in fruit:
Each time through the loop, the next character in the string is assigned to the variable letter . The loop continues until no characters are left.
The following example shows how to use concatenation (string addition) and a for
loop to generate an abecedarian series (that is, in alphabetical order). In Robert
McCloskeys book Make Way for Ducklings , the names of the ducklings are Jack, Kack,
Lack, Mack, Nack, Ouack, Pack, and Quack. This loop outputs these names in order:
prefixes = JKLMNOPQ
suffix = ack
for letter in prefixes:
print(letter + suffix)
The output is:
Of course, thats not quite right because “Ouack” and “Quack” are misspelled. As an
exercise, modify the program to fix this error.
String Slices A segment of a string is called a slice. Selecting a slice is similar to selecting a charac
ter: String Slices | 87

>>> s = Monty Python
>>> s[0:5]
>>> s[6:12]
The operator [n:m] returns the part of the string from the “n-eth” character to the
“m-eth” character, including the first but excluding the last. This behavior is counter
intuitive, but it might help to imagine the indices pointing between the characters, as
in Figure 8-1 .
Figure 8-1. Slice indices.
If you omit the first index (before the colon), the slice starts at the beginning of the
string. If you omit the second index, the slice goes to the end of the string:
>>> fruit = banana
>>> fruit[:3]
>>> fruit[3:]
If the first index is greater than or equal to the second the result is an empty string,
represented by two quotation marks:
>>> fruit = banana
>>> fruit[3:3]

An empty string contains no characters and has length 0, but other than that, it is the
same as any other string.
Continuing this example, what do you think fruit[:] means? Try it and see.
Strings Are Immutable
It is tempting to use the [] operator on the left side of an assignment, with the inten
tion of changing a character in a string. For example:
>>> greeting = Hello, world!
>>> greeting[0] = J
TypeError: str object does not support item assignment
88 | Chapter 8: Strings

The “object” in this case is the string and the “item” is the character you tried to
assign. For now, an object is the same thing as a value, but we will refine that defini tion later ( “Objects and Values” on page 114 ).
The reason for the error is that strings are immutable, which means you cant change
an existing string. The best you can do is create a new string that is a variation on the
>>> greeting = Hello, world!
>>> new_greeting = J + greeting[1:]
>>> new_greeting
Jello, world!
This example concatenates a new first letter onto a slice of greeting. It has no effect
on the original string.
What does the following function do? def find(word, letter):
index = 0
while index < len(word):
if word[index] == letter:
return index
index = index + 1
return -1
In a sense, find is the inverse of the [] operator. Instead of taking an index and
extracting the corresponding character, it takes a character and finds the index where
that character appears. If the character is not found, the function returns -1.
This is the first example we have seen of a return statement inside a loop. If
word[index] == letter , the function breaks out of the loop and returns immedi
If the character doesnt appear in the string, the program exits the loop normally and
returns -1.
This pattern of computationtraversing a sequence and returning when we find
what we are looking foris called a search.
As an exercise, modify find so that it has a third parameter: the index in word where
it should start looking.
Looping and Counting The following program counts the number of times the letter a appears in a string: Searching | 89

word = banana
count = 0
for letter in word:
if letter == a:
count = count + 1
This program demonstrates another pattern of computation called a counter. The
variable count is initialized to 0 and then incremented each time an a is found. When
the loop exits, count contains the resultthe total number of as.
As an exercise, encapsulate this code in a function named count, and generalize it so
that it accepts the string and the letter as arguments.
Then rewrite the function so that instead of traversing the string, it uses the three-
parameter version of find from the previous section.
String Methods Strings provide methods that perform a variety of useful operations. A method is
similar to a functionit takes arguments and returns a valuebut the syntax is dif
ferent. For example, the method upper takes a string and returns a new string with all
uppercase letters.
Instead of the function syntax upper(word), it uses the method syntax word.upper():
>>> word = banana
>>> new_word = word.upper()
>>> new_word
This form of dot notation specifies the name of the method, upper, and the name of
the string to apply the method to, word. The empty parentheses indicate that this
method takes no arguments.
A method call is called an invocation; in this case, we would say that we are invoking
upper on word .
As it turns out, there is a string method named find that is remarkably similar to the
function we wrote:
>>> word = banana
>>> index = word.find(a)
>>> index
In this example, we invoke find on word and pass the letter we are looking for as a
parameter. 90 | Chapter 8: Strings

Actually, the find method is more general than our function; it can find substrings,
not just characters:
>>> word.find(na)
By default, find starts at the beginning of the string, but it can take a second argu
ment, the index where it should start:
>>> word.find(na, 3)
This is an example of an optional argument. find can also take a third argument, the
index where it should stop:
>>> name = bob
>>> name.find(b, 1, 2)
This search fails because b does not appear in the index range from 1 to 2, not includ
ing 2. Searching up to, but not including, the second index makes find consistent
with the slice operator.
The in Operator The word in is a boolean operator that takes two strings and returns True if the first
appears as a substring in the second:
>>> a in banana
>>> seed in banana
For example, the following function prints all the letters from word1 that also appear
in word2 :
def in_both(word1, word2):
for letter in word1:
if letter in word2:
With well-chosen variable names, Python sometimes reads like English. You could
read this loop, “for (each) letter in (the first) word, if (the) letter (appears) in (the sec
ond) word, print (the) letter.”
Heres what you get if you compare apples and oranges: >>> in_both(apples, oranges)
s The in Operator | 91

String Comparison
The relational operators work on strings. To see if two strings are equal: if word == banana:
print(All right, bananas.)
Other relational operations are useful for putting words in alphabetical order:
if word < banana:
print(Your word, + word + , comes before banana.)
elif word > banana:
print(Your word, + word + , comes after banana.)
print(All right, bananas.)
Python does not handle uppercase and lowercase letters the same way people do. All the uppercase letters come before all the lowercase letters, so:
Your word, Pineapple, comes before banana.
A common way to address this problem is to convert strings to a standard format,
such as all lowercase, before performing the comparison. Keep that in mind in case
you have to defend yourself against a man armed with a Pineapple.
When you use indices to traverse the values in a sequence, it is tricky to get the begin
ning and end of the traversal right. Here is a function that is supposed to compare
two words and return True if one of the words is the reverse of the other, but it con
tains two errors:
def is_reverse(word1, word2): if len(word1) != len(word2):
return False

i = 0
j = len(word2)
while j > 0:
if word1[i] != word2[j]:
return False
i = i+1
j = j-1
return True
The first if statement checks whether the words are the same length. If not, we can
return False immediately. Otherwise, for the rest of the function, we can assume that
the words are the same length. This is an example of the guardian pattern in “Check
ing Types” on page 69 .92 | Chapter 8: Strings

i and j are indices: i traverses word1 forward while j traverses word2 backward. If we
find two letters that dont match, we can return False immediately. If we get through
the whole loop and all the letters match, we return True.
If we test this function with the words “pots” and “stop”, we expect the return value True , but we get an IndexError:
>>> is_reverse(pots, stop)
File "", line 15, in is_reverse
if word1[i] != word2[j]:
IndexError: string index out of range
For debugging this kind of error, my first move is to print the values of the indices
immediately before the line where the error appears.
while j > 0:
print(i, j) # print here

if word1[i] != word2[j]:
return False
i = i+1
j = j-1
Now when I run the program again, I get more information:
>>> is_reverse(pots, stop)
0 4
IndexError: string index out of range
The first time through the loop, the value of j is 4, which is out of range for the string
pots . The index of the last character is 3, so the initial value for j should be
len(word2)-1 .
If I fix that error and run the program again, I get: >>> is_reverse(pots, stop)
0 3
1 2
2 1
This time we get the right answer, but it looks like the loop only ran three times,
which is suspicious. To get a better idea of what is happening, it is useful to draw a
state diagram. During the first iteration, the frame for is_reverse is shown in
Figure 8-2 . Debugging | 93

Figure 8-2. State diagram.I took some license by arranging the variables in the frame and adding dotted lines to
show that the values of i and j indicate characters in word1 and word2 .
Starting with this diagram, run the program on paper, changing the values of i and j
during each iteration. Find and fix the second error in this function.
Glossary object: Something a variable can refer to. For now, you can use “object” and “value”interchangeably.
sequence: An ordered collection of values where each value is identified by an integer
item: One of the values in a sequence.
index: An integer value used to select an item in a sequence, such as a character in astring. In Python indices start from 0.
slice: A part of a string specified by a range of indices.
empty string: A string with no characters and length 0, represented by two quotation marks.
immutable: The property of a sequence whose items cannot be changed.
traverse: To iterate through the items in a sequence, performing a similar operation oneach.
search: A pattern of traversal that stops when it finds what it is looking for.
94 | Chapter 8: Strings

counter:A variable used to count something, usually initialized to zero and then incremented.
invocation: A statement that calls a method.
optional argument: A function or method argument that is not required.
Exercises Exercise 8-1.
Read the documentation of the string methods at
stdtypes.html#string-methods . You might want to experiment with some of them to
make sure you understand how they work. strip and replace are particularly useful.
The documentation uses a syntax that might be confusing. For example, in find(sub[, start[, end]]) , the brackets indicate optional arguments. So sub is
required, but start is optional, and if you include start, then end is optional.
Exercise 8-2. There is a string method called count that is similar to the function in “Looping and
Counting” on page 89. Read the documentation of this method and write an invoca
tion that counts the number of as in banana .
Exercise 8-3.
A string slice can take a third index that specifies the “step size”; that is, the number of spaces between successive characters. A step size of 2 means every other character; 3
means every third, etc.
>>> fruit = banana
>>> fruit[0:5:2]
A step size of -1 goes through the word backwards, so the slice [::-1] generates a
reversed string.
Use this idiom to write a one-line version of is_palindrome from Exercise 6-3 . Exercises | 95

Exercise 8-4.
The following functions are all intended to check whether a string contains any lower
case letters, but at least some of them are wrong. For each function, describe what the
function actually does (assuming that the parameter is a string).
def any_lowercase1(s):
for c in s:
if c.islower():
return True
return False
def any_lowercase2(s):
for c in s:
if c.islower():
return True
return False
def any_lowercase3(s):
for c in s:
flag = c.islower()
return flag
def any_lowercase4(s):
flag = False
for c in s:
flag = flag or c.islower()
return flag
def any_lowercase5(s):
for c in s:
if not c.islower():
return False
return True
Exercise 8-5.
A Caesar cypher is a weak form of encryption that involves “rotating” each letter by a
fixed number of places. To rotate a letter means to shift it through the alphabet, wrap
ping around to the beginning if necessary, so A rotated by 3 is D and Z rotated by 1 is A.
To rotate a word, rotate each letter by the same amount. For example, “cheer” rotated
by 7 is “jolly” and “melon” rotated by -10 is “cubed”. In the movie 2001: A Space Odys
sey , the ship computer is called HAL, which is IBM rotated by -1.96 | Chapter 8: Strings

Write a function called rotate_word that takes a string and an integer as parameters,
and returns a new string that contains the letters from the original string rotated by the given amount.
You might want to use the built-in function ord, which converts a character to a
numeric code, and chr, which converts numeric codes to characters. Letters of the
alphabet are encoded in alphabetical order, so for example:
>>> ord(c) - ord(a)
Because c is the two-eth letter of the alphabet. But beware: the numeric codes for
uppercase letters are different.
Potentially offensive jokes on the Internet are sometimes encoded in ROT13, which is a Caesar cypher with rotation 13. If you are not easily offended, find and decode
some of them.
Solution: . Exercises | 97

Case Study: Word Play
This chapter presents the second case study, which involves solving word puzzles by searching for words that have certain properties. For example, well find the longest
palindromes in English and search for words whose letters appear in alphabetical
order. And I will present another program development plan: reduction to a previ ously solved problem.
Reading Word Lists For the exercises in this chapter we need a list of English words. There are lots of word lists available on the Web, but the one most suitable for our purpose is one of
the word lists collected and contributed to the public domain by Grady Ward as part
of the Moby lexicon project (see ). It is a list of
113,809 official crosswords; that is, words that are considered valid in crossword puz
zles and other word games. In the Moby collection, the filename is 113809of.fic;
you can download a copy, with the simpler name words.txt, from http://thinkpy .
This file is in plain text, so you can open it with a text editor, but you can also read it from Python. The built-in function open takes the name of the file as a parameter and
returns a
file object you can use to read the file.
>>> fin = open(words.txt)
fin is a common name for a file object used for input. The file object provides several
methods for reading, including readline, which reads characters from the file until it
gets to a newline and returns the result as a string:
>>> fin.readline()

The first word in this particular list is “aa”, which is a kind of lava. The sequence \r\n
represents two whitespace characters, a carriage return and a newline, that separate this word from the next.
The file object keeps track of where it is in the file, so if you call readline again, you
get the next word:
>>> fin.readline()
The next word is “aah”, which is a perfectly legitimate word, so stop looking at me like that. Or, if its the whitespace thats bothering you, we can get rid of it with the string
method strip:
>>> line = fin.readline()
>>> word = line.strip()
>>> word
You can also use a file object as part of a for loop. This program reads words.txt and
prints each word, one per line:
fin = open(words.txt)
for line in fin:
word = line.strip()
Exercises There are solutions to these exercises in the next section. You should at least attempt
each one before you read the solutions.
Exercise 9-1.
Write a program that reads words.txt and prints only the words with more than 20
characters (not counting whitespace).
Exercise 9-2.
In 1939 Ernest Vincent Wright published a 50,000-word novel called Gadsby that
does not contain the letter “e”. Since “e” is the most common letter in English, thats
not easy to do.
In fact, it is difficult to construct a solitary thought without using that most common
symbol. It is slow going at first, but with caution and hours of training you can gradu
ally gain facility.
All right, Ill stop now.100 | Chapter 9: Case Study: Word Play

Write a function called has_no_e that returns True if the given word doesnt have the
letter “e” in it.
Modify your program from the previous section to print only the words that have no “e” and compute the percentage of the words in the list that have no “e”.
Exercise 9-3.
Write a function named avoids that takes a word and a string of forbidden letters,
and that returns True if the word doesnt use any of the forbidden letters.
Modify your program to prompt the user to enter a string of forbidden letters and
then print the number of words that dont contain any of them. Can you find a com
bination of five forbidden letters that excludes the smallest number of words?
Exercise 9-4.
Write a function named uses_only that takes a word and a string of letters, and that
returns True if the word contains only letters in the list. Can you make a sentence
using only the letters acefhlo? Other than “Hoe alfalfa?”
Exercise 9-5.
Write a function named uses_all that takes a word and a string of required letters,
and that returns True if the word uses all the required letters at least once. How many
words are there that use all the vowels aeiou? How about aeiouy?
Exercise 9-6.
Write a function called is_abecedarian that returns True if the letters in a word
appear in alphabetical order (double letters are okay). How many abecedarian words
are there?
All of the exercises in the previous section have something in common; they can be
solved with the search pattern we saw in “Searching” on page 89. The simplest exam
ple is:
def has_no_e(word):
for letter in word:
if letter == e:
return False
return True Search | 101

The for loop traverses the characters in word. If we find the letter “e”, we can immedi
ately return False; otherwise we have to go to the next letter. If we exit the loop nor
mally, that means we didnt find an “e”, so we return True.
You could write this function more concisely using the in operator, but I started with
this version because it demonstrates the logic of the search pattern.
avoids is a more general version of has_no_e but it has the same structure:
def avoids(word, forbidden):
for letter in word:
if letter in forbidden:
return False
return True
We can return False as soon as we find a forbidden letter; if we get to the end of the
loop, we return True.
uses_only is similar except that the sense of the condition is reversed:
def uses_only(word, available):
for letter in word:
if letter not in available:
return False
return True
Instead of a list of forbidden letters, we have a list of available letters. If we find a let
ter in word that is not in available, we can return False.
uses_all is similar except that we reverse the role of the word and the string of let
def uses_all(word, required):
for letter in required:
if letter not in word:
return False
return True
Instead of traversing the letters in word, the loop traverses the required letters. If any
of the required letters do not appear in the word, we can return False.
If you were really thinking like a computer scientist, you would have recognized that uses_all was an instance of a previously solved problem, and you would have writ
def uses_all(word, required):
return uses_only(required, word)
This is an example of a program development plan called reduction to a previously
solved problem , which means that you recognize the problem you are working on as
an instance of a solved problem and apply an existing solution.102 | Chapter 9: Case Study: Word Play

Looping with IndicesI wrote the functions in the previous section with for loops because I only needed the
characters in the strings; I didnt have to do anything with the indices.
For is_abecedarian we have to compare adjacent letters, which is a little tricky with
a for loop:
def is_abecedarian(word):
previous = word[0]
for c in word:
if c < previous:
return False
previous = c
return True
An alternative is to use recursion:
def is_abecedarian(word):
if len(word) <= 1:
return True
if word[0] > word[1]:
return False
return is_abecedarian(word[1:])
Another option is to use a while loop:
def is_abecedarian(word):
i = 0
while i < len(word)-1:
if word[i+1] < word[i]:
return False
i = i+1
return True
The loop starts at i=0 and ends when i=len(word)-1. Each time through the loop, it
compares the ith character (which you can think of as the current character) to the i
+1 th character (which you can think of as the next).
If the next character is less than (alphabetically before) the current one, then we have
discovered a break in the abecedarian trend, and we return False.
If we get to the end of the loop without finding a fault, then the word passes the test.
To convince yourself that the loop ends correctly, consider an example like flossy.
The length of the word is 6, so the last time the loop runs is when i is 4, which is the
index of the second-to-last character. On the last iteration, it compares the second-to-
last character to the last, which is what we want.
Here is a version of is_palindrome (see Exercise 6-3 ) that uses two indices: one starts
at the beginning and goes up; the other starts at the end and goes down. Looping with Indices | 103

def is_palindrome(word):
i = 0
j = len(word)-1
while i if word[i] != word[j]: return False
i = i+1
j = j-1
return True
Or we could reduce to a previously solved problem and write:
def is_palindrome(word):
return is_reverse(word, word)
Using is_reverse from Figure 8-2 .
Testing programs is hard. The functions in this chapter are relatively easy to test because you can check the results by hand. Even so, it is somewhere between difficult
and impossible to choose a set of words that test for all possible errors.
Taking has_no_e as an example, there are two obvious cases to check: words that have
an e should return False, and words that dont should return True. You should have
no trouble coming up with one of each.
Within each case, there are some less obvious subcases. Among the words that have
an “e”, you should test words with an “e” at the beginning, the end, and somewhere in the middle. You should test long words, short words, and very short words, like the
empty string. The empty string is an example of a special case, which is one of the
non-obvious cases where errors often lurk.
In addition to the test cases you generate, you can also test your program with a word list like words.txt . By scanning the output, you might be able to catch errors, but be
careful: you might catch one kind of error (words that should not be included, but
are) and not another (words that should be included, but arent).
In general, testing can help you find bugs, but it is not easy to generate a good set of
test cases, and even if you do, you cant be sure your program is correct. According to a legendary computer scientist:
Program testing can be used to show the presence of bugs, but never to show theirabsence!
Edsger W. Dijkstra 104 | Chapter 9: Case Study: Word Play

file object:A value that represents an open file.
reduction to a previously solved problem: A way of solving a problem by expressing it as an instance of a previously solved
special case: A test case that is atypical or non-obvious (and less likely to be handledcorrectly).
Exercises Exercise 9-7.
This question is based on a Puzzler that was broadcast on the radio program Car Talk
( ):
Give me a word with three consecutive double letters. Ill give you a couple of words that almost qualify, but dont. For example, the word committee, c-o-m-m-i-t-t-e-e. It
would be great except for the i that sneaks in there. Or Mississippi: M-i-s-s-i-s-s-i-p-
p-i. If you could take out those is it would work. But there is a word that has three
consecutive pairs of letters and to the best of my knowledge this may be the only word. Of course there are probably 500 more but I can only think of one. What is the word?
Write a program to find it.
Solution: .
Exercise 9-8.
Heres another Car Talk Puzzler ( ):
“I was driving on the highway the other day and I happened to notice my odometer. Like most odometers, it shows six digits, in whole miles only. So, if my car had 300,000
miles, for example, Id see 3-0-0-0-0-0.
“Now, what I saw that day was very interesting. I noticed that the last 4 digits were pal
indromic; that is, they read the same forward as backward. For example, 5-4-4-5 is a
palindrome, so my odometer could have read 3-1-5-4-4-5.
“One mile later, the last 5 numbers were palindromic. For example, it could have read 3-6-5-4-5-6. One mile after that, the middle 4 out of 6 numbers were palindromic. And
you ready for this? One mile later, all 6 were palindromic!
“The question is, what was on the odometer when I first looked?”
Glossary | 105

Write a Python program that tests all the six-digit numbers and prints any numbers
that satisfy these requirements.
Solution: .
Exercise 9-9.
Heres another Car Talk Puzzler you can solve with a search (
content/puzzlers ):
“Recently I had a visit with my mom and we realized that the two digits that make up
my age when reversed resulted in her age. For example, if shes 73, Im 37. We won dered how often this has happened over the years but we got sidetracked with other
topics and we never came up with an answer.
“When I got home I figured out that the digits of our ages have been reversible six
times so far. I also figured out that if were lucky it would happen again in a few years,
and if were really lucky it would happen one more time after that. In other words, it would have happened 8 times over all. So the question is, how old am I now?”
Write a Python program that searches for solutions to this Puzzler. Hint: you might
find the string method zfill useful.
Solution: .106 | Chapter 9: Case Study: Word Play

This chapter presents one of Pythons most useful built-in types: lists. You will also learn more about objects and what can happen when you have more than one name
for the same object.
A List Is a Sequence Like a string, a list is a sequence of values. In a string, the values are characters; in a
list, they can be any type. The values in a list are called elements or sometimes items.
There are several ways to create a new list; the simplest is to enclose the elements in square brackets ( [ and ]):
[10, 20, 30, 40]
[crunchy frog, ram bladder, lark vomit]
The first example is a list of four integers. The second is a list of three strings. The
elements of a list dont have to be the same type. The following list contains a string, a
float, an integer, and (lo!) another list:
[spam, 2.0, 5, [10, 20]]
A list within another list is nested.
A list that contains no elements is called an empty list; you can create one with empty
brackets, [].
As you might expect, you can assign list values to variables: >>> cheeses = [Cheddar, Edam, Gouda]
>>> numbers = [42, 123]
>>> empty = []
>>> print(cheeses, numbers, empty)
[Cheddar, Edam, Gouda] [42, 123] []

Lists Are MutableThe syntax for accessing the elements of a list is the same as for accessing the charac
ters of a stringthe bracket operator. The expression inside the brackets specifies the
index. Remember that the indices start at 0:
>>> cheeses[0]
Unlike strings, lists are mutable. When the bracket operator appears on the left side of
an assignment, it identifies the element of the list that will be assigned:
>>> numbers = [42, 123]
>>> numbers[1] = 5
>>> numbers
[42, 5]
The one-eth element of numbers, which used to be 123, is now 5.
Figure 10-1 shows the state diagram for cheeses, numbers and empty .
Figure 10-1. State diagram.
Lists are represented by boxes with the word “list” outside and the elements of the list
inside. cheeses refers to a list with three elements indexed 0, 1 and 2. numbers con
tains two elements; the diagram shows that the value of the second element has been
reassigned from 123 to 5. empty refers to a list with no elements.
108 | Chapter 10: Lists

List indices work the same way as string indices:
Any integer expression can be used as an index.

If you try to read or write an element that does not exist, you get an IndexError.

If an index has a negative value, it counts backward from the end of the list.
The in operator also works on lists:
>>> cheeses = [Cheddar, Edam, Gouda]
>>> Edam in cheeses
>>> Brie in cheeses
Traversing a List The most common way to traverse the elements of a list is with a for loop. The syntax
is the same as for strings:
for cheese in cheeses:
This works well if you only need to read the elements of the list. But if you want to
write or update the elements, you need the indices. A common way to do that is to
combine the built-in functions range and len:
for i in range(len(numbers)):
numbers[i] = numbers[i] * 2
This loop traverses the list and updates each element. len returns the number of ele
ments in the list. range returns a list of indices from 0 to n-1, where n is the length of
the list. Each time through the loop, i gets the index of the next element. The assign
ment statement in the body uses i to read the old value of the element and to assign
the new value.
A for loop over an empty list never runs the body:
for x in []:
print(This never happens.)
Although a list can contain another list, the nested list still counts as a single element.
The length of this list is four:
[spam, 1, [Brie, Roquefort, Pol le Veq], [1, 2, 3]]
Traversing a List | 109

List OperationsThe + operator concatenates lists:
>>> a = [1, 2, 3]
>>> b = [4, 5, 6]
>>> c = a + b
>>> c
[1, 2, 3, 4, 5, 6]
The * operator repeats a list a given number of times:
>>> [0] * 4
[0, 0, 0, 0]
>>> [1, 2, 3] * 3
[1, 2, 3, 1, 2, 3, 1, 2, 3]
The first example repeats [0] four times. The second example repeats the list [1, 2,
3] three times.
List Slices
The slice operator also works on lists: >>> t = [a, b, c, d, e, f]>>> t[1:3]
[b, c]
>>> t[:4]
[a, b, c, d]
>>> t[3:]
[d, e, f]
If you omit the first index, the slice starts at the beginning. If you omit the second, the slice goes to the end. So if you omit both, the slice is a copy of the whole list:
>>> t[:]
[a, b, c, d, e, f]
Since lists are mutable, it is often useful to make a copy before performing operations
that modify lists.
A slice operator on the left side of an assignment can update multiple elements: >>> t = [a, b, c, d, e, f]>>> t[1:3] = [x, y]
>>> t
[a, x, y, d, e, f]
110 | Chapter 10: Lists

List MethodsPython provides methods that operate on lists. For example, append adds a new ele
ment to the end of a list:
>>> t = [a, b, c]
>>> t.append(d)
>>> t
[a, b, c, d]
extend takes a list as an argument and appends all of the elements:
>>> t1 = [a, b, c]
>>> t2 = [d, e]
>>> t1.extend(t2)
>>> t1
[a, b, c, d, e]
This example leaves t2 unmodified.
sort arranges the elements of the list from low to high:
>>> t = [d, c, e, b, a] >>> t.sort()
>>> t
[a, b, c, d, e]
Most list methods are void; they modify the list and return None. If you accidentally
write t = t.sort() , you will be disappointed with the result.
Map, Filter and Reduce To add up all the numbers in a list, you can use a loop like this: def add_all(t):
total = 0
for x in t:
total += x
return total
total is initialized to 0. Each time through the loop, x gets one element from the list.
The += operator provides a short way to update a variable. This augmented assign
ment statement ,
total += x
is equivalent to
total = total + x
As the loop runs, total accumulates the sum of the elements; a variable used this way
is sometimes called an accumulator. List Methods | 111

Adding up the elements of a list is such a common operation that Python provides it
as a built-in function, sum:
>>> t = [1, 2, 3]
>>> sum(t)
An operation like this that combines a sequence of elements into a single value is
sometimes called reduce.
Sometimes you want to traverse one list while building another. For example, the fol
lowing function takes a list of strings and returns a new list that contains capitalized
def capitalize_all(t):
res = []
for s in t:
return res
res is initialized with an empty list; each time through the loop, we append the next
element. So res is another kind of accumulator.
An operation like capitalize_all is sometimes called a map because it “maps” a
function (in this case the method capitalize) onto each of the elements in a
Another common operation is to select some of the elements from a list and return a
sublist. For example, the following function takes a list of strings and returns a list
that contains only the uppercase strings:
def only_upper(t):
res = []
for s in t:
if s.isupper():
return res
isupper is a string method that returns True if the string contains only uppercase let
An operation like only_upper is called a
filter because it selects some of the elements
and filters out the others.
Most common list operations can be expressed as a combination of map, filter and
112 | Chapter 10: Lists

Deleting ElementsThere are several ways to delete elements from a list. If you know the index of the
element you want, you can use pop:
>>> t = [a, b, c] >>> x = t.pop(1)
>>> t
[a, c]
>>> x
pop modifies the list and returns the element that was removed. If you dont provide
an index, it deletes and returns the last element.
If you dont need the removed value, you can use the del operator:
>>> t = [a, b, c]
>>> del t[1]
>>> t
[a, c]
If you know the element you want to remove (but not the index), you can use remove:
>>> t = [a, b, c] >>> t.remove(b)
>>> t
[a, c]
The return value from remove is None .
To remove more than one element, you can use del with a slice index:
>>> t = [a, b, c, d, e, f] >>> del t[1:5]
>>> t
[a, f]
As usual, the slice selects all the elements up to but not including the second index.
Lists and Strings
A string is a sequence of characters and a list is a sequence of values, but a list of char
acters is not the same as a string. To convert from a string to a list of characters, you
can use list:
>>> s = spam
>>> t = list(s)
>>> t
[s, p, a, m] Deleting Elements | 113

Because list is the name of a built-in function, you should avoid using it as a vari
able name. I also avoid l because it looks too much like 1. So thats why I use t.
The list function breaks a string into individual letters. If you want to break a string
into words, you can use the split method:
>>> s = pining for the fjords
>>> t = s.split()
>>> t
[pining, for, the, fjords]
An optional argument called a delimiter specifies which characters to use as word
boundaries. The following example uses a hyphen as a delimiter:
>>> s = spam-spam-spam
>>> delimiter = -
>>> t = s.split(delimiter)
>>> t
[spam, spam, spam]
join is the inverse of split. It takes a list of strings and concatenates the elements.
join is a string method, so you have to invoke it on the delimiter and pass the list as a
>>> t = [pining, for, the, fjords]
>>> delimiter =
>>> s = delimiter.join(t)
>>> s
pining for the fjords
In this case the delimiter is a space character, so join puts a space between words. To
concatenate strings without spaces, you can use the empty string, , as a delimiter.
Objects and Values
If we run these assignment statements: a = banana
b = banana
We know that a and b both refer to a string, but we dont know whether they refer to
the same string. There are two possible states, shown in Figure 10-2.
Figure 10-2. State diagram.
114 | Chapter 10: Lists

In one case, a and b refer to two different objects that have the same value. In the
second case, they refer to the same object.
To check whether two variables refer to the same object, you can use the is operator:
>>> a = banana >>> b = banana
>>> a is b
In this example, Python only created one string object, and both a and b refer to it.
But when you create two lists, you get two objects:
>>> a = [1, 2, 3]
>>> b = [1, 2, 3]
>>> a is b
So the state diagram looks like Figure 10-3.
Figure 10-3. State diagram.
In this case we would say that the two lists are equivalent, because they have the same
elements, but not identical, because they are not the same object. If two objects are
identical, they are also equivalent, but if they are equivalent, they are not necessarily
Until now, we have been using “object” and “value” interchangeably, but it is more precise to say that an object has a value. If you evaluate [1, 2, 3], you get a list
object whose value is a sequence of integers. If another list has the same elements, we
say it has the same value, but it is not the same object.
If a refers to an object and you assign b = a, then both variables refer to the same
>>> a = [1, 2, 3]
>>> b = a
>>> b is a
The state diagram looks like Figure 10-4.
Aliasing | 115

Figure 10-4. State diagram.
The association of a variable with an object is called a reference. In this example,
there are two references to the same object.
An object with more than one reference has more than one name, so we say that the object is aliased.
If the aliased object is mutable, changes made with one alias affect the other: >>> b[0] = 42
>>> a
[42, 2, 3]
Although this behavior can be useful, it is error-prone. In general, it is safer to avoid
aliasing when you are working with mutable objects.
For immutable objects like strings, aliasing is not as much of a problem. In this exam ple:
a = banana
b = banana
It almost never makes a difference whether a and b refer to the same string or not.
List Arguments When you pass a list to a function, the function gets a reference to the list. If the function modifies the list, the caller sees the change. For example, delete_head removes
the first element from a list:
def delete_head(t):
del t[0]
Heres how it is used:
>>> letters = [a, b, c]
>>> delete_head(letters)
>>> letters
[b, c]
The parameter t and the variable letters are aliases for the same object. The stack
diagram looks like Figure 10-5.
116 | Chapter 10: Lists

Figure 10-5. Stack diagram.
Since the list is shared by two frames, I drew it between them.
It is important to distinguish between operations that modify lists and operations that
create new lists. For example, the append method modifies a list, but the + operator
creates a new list:
>>> t1 = [1, 2]
>>> t2 = t1.append(3)
>>> t1
[1, 2, 3]
>>> t2
append modifies the list and returns None:
>>> t3 = t1 + [4]
>>> t1
[1, 2, 3]
>>> t3
[1, 2, 3, 4]
>>> t1
The + operator creates a new list and leaves the original list unchanged.
This difference is important when you write functions that are supposed to modify lists. For example, this function does not delete the head of a list:
def bad_delete_head(t):
t = t[1:] # WRONG!
The slice operator creates a new list and the assignment makes t refer to it, but that
doesnt affect the caller.
>>> t4 = [1, 2, 3]
>>> bad_delete_head(t4)
>>> t4
[1, 2, 3]
At the beginning of bad_delete_head, t and t4 refer to the same list. At the end, t
refers to a new list, but t4 still refers to the original, unmodified list.
An alternative is to write a function that creates and returns a new list. For example, tail returns all but the first element of a list:
List Arguments | 117

def tail(t):
return t[1:]
This function leaves the original list unmodified. Heres how it is used:
>>> letters = [a, b, c]
>>> rest = tail(letters)
>>> rest
[b, c]
Careless use of lists (and other mutable objects) can lead to long hours of debugging. Here are some common pitfalls and ways to avoid them: 1.
Most list methods modify the argument and return None. This is the opposite of
the string methods, which return a new string and leave the original alone.
If you are used to writing string code like this: word = word.strip()
It is tempting to write list code like this: t = t.sort() # WRONG!
Because sort returns None, the next operation you perform with t is likely to fail.
Before using list methods and operators, you should read the documentation carefully and then test them in interactive mode.
Pick an idiom and stick with it.
Part of the problem with lists is that there are too many ways to do things. For example, to remove an element from a list, you can use pop, remove , del , or even
a slice assignment.
To add an element, you can use the append method or the + operator. Assuming
that t is a list and x is a list element, these are correct:
t = t + [x]
t += [x]
And these are wrong: t.append([x]) # WRONG!
t = t.append(x) # WRONG!
t + [x] # WRONG!
t = t + x # WRONG!
Try out each of these examples in interactive mode to make sure you understand what they do. Notice that only the last one causes a runtime error; the other three
are legal, but they do the wrong thing.
118 | Chapter 10: Lists

3. Make copies to avoid aliasing.
If you want to use a method like sort that modifies the argument, but you need
to keep the original list as well, you can make a copy:
>>> t = [3, 1, 2]
>>> t2 = t[:]
>>> t2.sort()
>>> t
[3, 1, 2]
>>> t2
[1, 2, 3]
In this example you could also use the built-in function sorted, which returns a
new, sorted list and leaves the original alone:
>>> t2 = sorted(t)
>>> t
[3, 1, 2]
>>> t2
[1, 2, 3]
Glossary list: A sequence of values.
element: One of the values in a list (or other sequence), also called items.
nested list: A list that is an element of another list.
accumulator: A variable used in a loop to add up or accumulate a result.
augmented assignment: A statement that updates the value of a variable using an operator like +=.
reduce: A processing pattern that traverses a sequence and accumulates the elements intoa single result.
map: A processing pattern that traverses a sequence and performs an operation on
each element. Glossary | 119

A processing pattern that traverses a list and selects the elements that satisfysome criterion.
object: Something a variable can refer to. An object has a type and a value.
equivalent: Having the same value.
identical: Being the same object (which implies equivalence).
reference: The association between a variable and its value.
aliasing: A circumstance where two or more variables refer to the same object.
delimiter: A character or string used to indicate where a string should be split.
Exercises You can download solutions to these exercises from from
code/ .
Exercise 10-1.
Write a function called nested_sum that takes a list of lists of integers and adds up the
elements from all of the nested lists. For example:
>>> t = [[1, 2], [3], [4, 5, 6]]
>>> nested_sum(t)
Exercise 10-2.
Write a function called cumsum that takes a list of numbers and returns the cumulative
sum; that is, a new list where the ith element is the sum of the first i+1 elements from
the original list. For example:
>>> t = [1, 2, 3]
>>> cumsum(t)
[1, 3, 6]
120 | Chapter 10: Lists

Exercise 10-3.
Write a function called middle that takes a list and returns a new list that contains all
but the first and last elements. For example:
>>> t = [1, 2, 3, 4]
>>> middle(t)
[2, 3]
Exercise 10-4.
Write a function called chop that takes a list, modifies it by removing the first and last
elements, and returns None. For example:
>>> t = [1, 2, 3, 4]
>>> chop(t)
>>> t
[2, 3]
Exercise 10-5.
Write a function called is_sorted that takes a list as a parameter and returns True if
the list is sorted in ascending order and False otherwise. For example:
>>> is_sorted([1, 2, 2]) True
>>> is_sorted([b, a])
Exercise 10-6.
Two words are anagrams if you can rearrange the letters from one to spell the other.
Write a function called is_anagram that takes two strings and returns True if they are
Exercise 10-7.
Write a function called has_duplicates that takes a list and returns True if there is
any element that appears more than once. It should not modify the original list.
Exercise 10-8.
This exercise pertains to the so-called Birthday Paradox, which you can read about at .
If there are 23 students in your class, what are the chances that two of you have the
same birthday? You can estimate this probability by generating random samples of 23 Exercises | 121

birthdays and checking for matches. Hint: you can generate random birthdays with
the randint function in the random module.
You can download my solution from .
Exercise 10-9.
Write a function that reads the file words.txt and builds a list with one element per
word. Write two versions of this function, one using the append method and the
other using the idiom t = t + [x]. Which one takes longer to run? Why?
Solution: .
Exercise 10-10.
To check whether a word is in the word list, you could use the in operator, but it
would be slow because it searches through the words in order.
Because the words are in alphabetical order, we can speed things up with a bisection
search (also known as binary search), which is similar to what you do when you look a word up in the dictionary. You start in the middle and check to see whether the
word you are looking for comes before the word in the middle of the list. If so, you
search the first half of the list the same way. Otherwise you search the second half.
Either way, you cut the remaining search space in half. If the word list has 113,809 words, it will take about 17 steps to find the word or conclude that its not there.
Write a function called in_bisect that takes a sorted list and a target value and
returns the index of the value in the list if its there, or None if its not.
Or you could read the documentation of the bisect module and use that!
Solution: .
Exercise 10-11.
Two words are a “reverse pair” if each is the reverse of the other. Write a program that
finds all the reverse pairs in the word list.
Solution: .
Exercise 10-12.
Two words “interlock” if taking alternating letters from each forms a new word. For
example, “shoe” and “cold” interlock to form “schooled”.122 | Chapter 10: Lists

Solution: . Credit: This exercise is inspired
by an example at .1.
Write a program that finds all pairs of words that interlock. Hint: dont enumer
ate all pairs!
Can you find any words that are three-way interlocked; that is, every third letter forms a word, starting from the first, second or third?
Exercises | 123

This chapter presents another built-in type called a dictionary. Dictionaries are one of Pythons best features; they are the building blocks of many efficient and elegant algo
A Dictionary Is a Mapping A dictionary is like a list, but more general. In a list, the indices have to be integers;
in a dictionary they can be (almost) any type.
A dictionary contains a collection of indices, which are called keys, and a collection
of values. Each key is associated with a single value. The association of a key and a
value is called a key-value pair or sometimes an item.
In mathematical language, a dictionary represents a mapping from keys to values, so
you can also say that each key “maps to” a value. As an example, well build a dictio
nary that maps from English to Spanish words, so the keys and the values are all
The function dict creates a new dictionary with no items. Because dict is the name
of a built-in function, you should avoid using it as a variable name.
>>> eng2sp = dict()
>>> eng2sp
The squiggly brackets, {}, represent an empty dictionary. To add items to the dictio
nary, you can use square brackets:
>>> eng2sp[one] = uno
This line creates an item that maps from the key one to the value uno. If we print
the dictionary again, we see a key-value pair with a colon between the key and value:

>>> eng2sp
{one: uno}
This output format is also an input format. For example, you can create a new dictio nary with three items:
>>> eng2sp = {one: uno, two: dos, three: tres}
But if you print eng2sp, you might be surprised:
>>> eng2sp
{one: uno, three: tres, two: dos}
The order of the key-value pairs might not be the same. If you type the same example
on your computer, you might get a different result. In general, the order of items in a
dictionary is unpredictable.
But thats not a problem because the elements of a dictionary are never indexed with integer indices. Instead, you use the keys to look up the corresponding values:
>>> eng2sp[two]
The key two always maps to the value dos so the order of the items doesnt mat
If the key isnt in the dictionary, you get an exception: >>> eng2sp[four]
KeyError: four
The len function works on dictionaries; it returns the number of key-value pairs:
>>> len(eng2sp)
The in operator works on dictionaries, too; it tells you whether something appears as
a key in the dictionary (appearing as a value is not good enough).
>>> one in eng2sp
>>> uno in eng2sp
To see whether something appears as a value in a dictionary, you can use the method values , which returns a collection of values, and then use the in operator:
>>> vals = eng2sp.values()
>>> uno in vals
The in operator uses different algorithms for lists and dictionaries. For lists, it
searches the elements of the list in order, as in “Searching” on page 89. As the list gets
longer, the search time gets longer in direct proportion. 126 | Chapter 11: Dictionaries

For dictionaries, Python uses an algorithm called a hashtable that has a remarkable
property: the in operator takes about the same amount of time no matter how many
items are in the dictionary. I explain how thats possible in “Hashtables” on page 251,
but the explanation might not make sense until youve read a few more chapters.
Dictionary as a Collection of Counters Suppose you are given a string and you want to count how many times each letter
appears. There are several ways you could do it:1.
You could create 26 variables, one for each letter of the alphabet. Then you could
traverse the string and, for each character, increment the corresponding counter,
probably using a chained conditional.
You could create a list with 26 elements. Then you could convert each character
to a number (using the built-in function ord), use the number as an index into
the list, and increment the appropriate counter.
You could create a dictionary with characters as keys and counters as the corre sponding values. The first time you see a character, you would add an item to the
dictionary. After that you would increment the value of an existing item.
Each of these options performs the same computation, but each of them implements
that computation in a different way.
An implementation is a way of performing a computation; some implementations
are better than others. For example, an advantage of the dictionary implementation is
that we dont have to know ahead of time which letters appear in the string and we
only have to make room for the letters that do appear.
Here is what the code might look like: def histogram(s):
d = dict()
for c in s:
if c not in d:
d[c] = 1
d[c] += 1
return d
The name of the function is histogram, which is a statistical term for a collection of
counters (or frequencies).
The first line of the function creates an empty dictionary. The for loop traverses the
string. Each time through the loop, if the character c is not in the dictionary, we cre
ate a new item with key c and the initial value 1 (since we have seen this letter once).
If c is already in the dictionary we increment d[c].
Dictionary as a Collection of Counters | 127

Heres how it works:>>> h = histogram(brontosaurus)>>> h
{a: 1, b: 1, o: 2, n: 1, s: 2, r: 2, u: 2, t: 1}
The histogram indicates that the letters a and b appear once; o appears twice,
and so on.
Dictionaries have a method called get that takes a key and a default value. If the key
appears in the dictionary, get returns the corresponding value; otherwise it returns
the default value. For example:
>>> h = histogram(a)
>>> h
{a: 1}
>>> h.get(a, 0)
>>> h.get(b, 0)
As an exercise, use get to write histogram more concisely. You should be able to
eliminate the if statement.
Looping and Dictionaries If you use a dictionary in a for statement, it traverses the keys of the dictionary. For
example, print_hist prints each key and the corresponding value:
def print_hist(h): for c in h:
print(c, h[c])
Heres what the output looks like:
>>> h = histogram(parrot)
>>> print_hist(h)
a 1
p 1
r 2
t 1
o 1
Again, the keys are in no particular order. To traverse the keys in sorted order, you
can use the built-in function sorted:128 | Chapter 11: Dictionaries

>>> for key in sorted(h):
... print(key, h[key])
a 1
o 1
p 1
r 2
t 1
Reverse Lookup
Given a dictionary d and a key k, it is easy to find the corresponding value v = d[k].
This operation is called a lookup.
But what if you have v and you want to find k? You have two problems: first, there
might be more than one key that maps to the value v. Depending on the application,
you might be able to pick one, or you might have to make a list that contains all of them. Second, there is no simple syntax to do a reverse lookup; you have to search.
Here is a function that takes a value and returns the first key that maps to that value: def reverse_lookup(d, v):
for k in d:
if d[k] == v:
return k
raise LookupError()
This function is yet another example of the search pattern, but it uses a feature we
havent seen before: raise. The raise statement causes an exception; in this case it
causes a LookupError , which is a built-in exception used to indicate that a lookup
operation failed.
If we get to the end of the loop, that means v doesnt appear in the dictionary as a
value, so we raise an exception.
Here is an example of a successful reverse lookup: >>> h = histogram(parrot)
>>> k = reverse_lookup(h, 2)
>>> k
And an unsuccessful one:
>>> k = reverse_lookup(h, 3)
Traceback (most recent call last):
File "", line 1, in
File "", line 5, in reverse_lookup LookupError
The effect when you raise an exception is the same as when Python raises one: it prints a traceback and an error message. Reverse Lookup | 129

The raise statement can take a detailed error message as an optional argument. For
>>> raise LookupError(value does not appear in the dictionary)
Traceback (most recent call last): File "", line 1, in ?
LookupError: value does not appear in the dictionary
A reverse lookup is much slower than a forward lookup; if you have to do it often, or
if the dictionary gets big, the performance of your program will suffer.
Dictionaries and Lists Lists can appear as values in a dictionary. For example, if you are given a dictionary
that maps from letters to frequencies, you might want to invert it; that is, create a dic tionary that maps from frequencies to letters. Since there might be several letters with
the same frequency, each value in the inverted dictionary should be a list of letters.
Here is a function that inverts a dictionary: def invert_dict(d):
inverse = dict()
for key in d:
val = d[key]
if val not in inverse:
inverse[val] = [key]
return inverse
Each time through the loop, key gets a key from d and val gets the corresponding
value. If val is not in inverse, that means we havent seen it before, so we create a
new item and initialize it with a singleton (a list that contains a single element).
Otherwise we have seen this value before, so we append the corresponding key to the list.
Here is an example: >>> hist = histogram(parrot)
>>> hist
{a: 1, p: 1, r: 2, t: 1, o: 1}
>>> inverse = invert_dict(hist)
>>> inverse
{1: [a, p, t, o], 2: [r]}
Figure 11-1 is a state diagram showing hist and inverse . A dictionary is represented
as a box with the type dict above it and the key-value pairs inside. If the values are
integers, floats or strings, I draw them inside the box, but I usually draw lists outside the box, just to keep the diagram simple.130 | Chapter 11: Dictionaries

Figure 11-1. State diagram.
Lists can be values in a dictionary, as this example shows, but they cannot be keys.
Heres what happens if you try:
>>> t = [1, 2, 3]
>>> d = dict()
>>> d[t] = oops
Traceback (most recent call last):
File "", line 1, in ?
TypeError: list objects are unhashable
I mentioned earlier that a dictionary is implemented using a hashtable and that
means that the keys have to be hashable.
A hash is a function that takes a value (of any kind) and returns an integer. Dictionar
ies use these integers, called hash values, to store and look up key-value pairs.
This system works fine if the keys are immutable. But if the keys are mutable, like
lists, bad things happen. For example, when you create a key-value pair, Python hashes the key and stores it in the corresponding location. If you modify the key and
then hash it again, it would go to a different location. In that case you might have two
entries for the same key, or you might not be able to find a key. Either way, the dictio nary wouldnt work correctly.
Thats why keys have to be hashable, and why mutable types like lists arent. The sim
plest way to get around this limitation is to use tuples, which we will see in the next
Since dictionaries are mutable, they cant be used as keys, but they can be used as
Memos If you played with the fibonacci function from “One More Example” on page 68,
you might have noticed that the bigger the argument you provide, the longer the
function takes to run. Furthermore, the runtime increases quickly.
Memos | 131

To understand why, consider Figure 11-2, which shows the call graph for fibonacci
with n=4.
Figure 11-2. Call graph.
A call graph shows a set of function frames, with lines connecting each frame to the frames of the functions it calls. At the top of the graph, fibonacci with n=4 calls fibo
nacci with n=3 and n=2. In turn, fibonacci with n=3 calls fibonacci with n=2 and
n=1 . And so on.
Count how many times fibonacci(0) and fibonacci(1) are called. This is an ineffi
cient solution to the problem, and it gets worse as the argument gets bigger.
One solution is to keep track of values that have already been computed by storing
them in a dictionary. A previously computed value that is stored for later use is called
a memo . Here is a “memoized” version of fibonacci:
known = {0:0, 1:1}
def fibonacci(n):
if n in known:
return known[n]
res = fibonacci(n-1) + fibonacci(n-2)
known[n] = res
return res
known is a dictionary that keeps track of the Fibonacci numbers we already know. It
starts with two items: 0 maps to 0 and 1 maps to 1.
132 | Chapter 11: Dictionaries

Whenever fibonacci is called, it checks known. If the result is already there, it can
return immediately. Otherwise it has to compute the new value, add it to the dictio nary, and return it.
If you run this version of fibonacci and compare it with the original, you will find
that it is much faster.
Global Variables
In the previous example, known is created outside the function, so it belongs to the
special frame called __main__. Variables in __main__ are sometimes called global
because they can be accessed from any function. Unlike local variables, which disap pear when their function ends, global variables persist from one function call to the
It is common to use global variables for
ags ; that is, boolean variables that indicate
(“flag”) whether a condition is true. For example, some programs use a flag named
verbose to control the level of detail in the output:
verbose = True
def example1():
if verbose:
print(Running example1)
If you try to reassign a global variable, you might be surprised. The following exam
ple is supposed to keep track of whether the function has been called:
been_called = False
def example2():
been_called = True # WRONG
But if you run it you will see that the value of been_called doesnt change. The prob
lem is that example2 creates a new local variable named been_called. The local vari
able goes away when the function ends, and has no effect on the global variable.
To reassign a global variable inside a function you have to declare the global variable
before you use it:
been_called = False
def example2():
global been_called
been_called = True
The global statement tells the interpreter something like, “In this function, when I
say been_called , I mean the global variable; dont create a local one.”
Global Variables | 133

Heres an example that tries to update a global variable:count = 0
def example3():
count = count + 1 # WRONG
If you run it you get:
UnboundLocalError: local variable count referenced before assignment
Python assumes that count is local, and under that assumption you are reading it
before writing it. The solution, again, is to declare count global:
def example3():
global count
count += 1
If a global variable refers to a mutable value, you can modify the value without
declaring the variable:
known = {0:0, 1:1}
def example4():
known[2] = 1
So you can add, remove and replace elements of a global list or dictionary, but if you
want to reassign the variable, you have to declare it:
def example5():
global known
known = dict()
Global variables can be useful, but if you have a lot of them, and you modify them
frequently, they can make programs hard to debug.
As you work with bigger datasets it can become unwieldy to debug by printing and
checking the output by hand. Here are some suggestions for debugging large datasets:
Scale down the input: If possible, reduce the size of the dataset. For example if the program reads a text
file, start with just the first 10 lines, or with the smallest example you can find. You can either edit the files themselves, or (better) modify the program so it
reads only the first n lines.
If there is an error, you can reduce n to the smallest value that manifests the error,
and then increase it gradually as you find and correct errors.134 | Chapter 11: Dictionaries

Check summaries and types:Instead of printing and checking the entire dataset, consider printing summariesof the data: for example, the number of items in a dictionary or the total of a list
of numbers.
A common cause of runtime errors is a value that is not the right type. For debugging this kind of error, it is often enough to print the type of a value.
Write self-checks: Sometimes you can write code to check for errors automatically. For example, ifyou are computing the average of a list of numbers, you could check that the
result is not greater than the largest element in the list or less than the smallest.
This is called a “sanity check” because it detects results that are “insane”.
Another kind of check compares the results of two different computations to see
if they are consistent. This is called a “consistency check”.
Format the output: Formatting debugging output can make it easier to spot an error. We saw an
example in “Debugging” on page 70 . The pprint module provides a pprint func
tion that displays built-in types in a more human-readable format ( pprint stands
for “pretty print”).
Again, time you spend building scaffolding can reduce the time you spend debugging.
Glossary mapping: A relationship in which each element of one set corresponds to an element ofanother set.
dictionary: A mapping from keys to their corresponding values.
key-value pair: The representation of the mapping from a key to a value.
item: In a dictionary, another name for a key-value pair.
key: An object that appears in a dictionary as the first part of a key-value pair.
value: An object that appears in a dictionary as the second part of a key-value pair. Thisis more specific than our previous use of the word “value”. Glossary | 135

implementation:A way of performing a computation.
hashtable: The algorithm used to implement Python dictionaries.
hash function: A function used by a hashtable to compute the location for a key.
hashable: A type that has a hash function. Immutable types like integers, floats and stringsare hashable; mutable types like lists and dictionaries are not.
lookup: A dictionary operation that takes a key and finds the corresponding value.
reverse lookup: A dictionary operation that takes a value and finds one or more keys that map to
raise statement: A statement that (deliberately) raises an exception.
singleton: A list (or other sequence) with a single element.
call graph: A diagram that shows every frame created during the execution of a program,with an arrow from each caller to each callee.
memo: A computed value stored to avoid unnecessary future computation.
global variable: A variable defined outside a function. Global variables can be accessed from anyfunction.
global statement: A statement that declares a variable name global.
A boolean variable used to indicate whether a condition is true.
declaration: A statement like global that tells the interpreter something about a variable.
136 | Chapter 11: Dictionaries

ExercisesExercise 11-1.
Write a function that reads the words in words.txt and stores them as keys in a dic
tionary. It doesnt matter what the values are. Then you can use the in operator as a
fast way to check whether a string is in the dictionary.
If you did Exercise 10-10 , you can compare the speed of this implementation with the
list in operator and the bisection search.
Exercise 11-2.
Read the documentation of the dictionary method setdefault and use it to write a
more concise version of invert_dict.
Solution: .
Exercise 11-3.
Memoize the Ackermann function from Exercise 6-2 and see if memoization makes it
possible to evaluate the function with bigger arguments. Hint: no.
Solution: .
Exercise 11-4.
If you did Exercise 10-7 , you already have a function named has_duplicates that
takes a list as a parameter and returns True if there is any object that appears more
than once in the list.
Use a dictionary to write a faster, simpler version of has_duplicates.
Solution: .
Exercise 11-5.
Two words are “rotate pairs” if you can rotate one of them and get the other (see
rotate_word in Exercise 8-5 ).
Write a program that reads a wordlist and finds all the rotate pairs.
Solution: . Exercises | 137

Exercise 11-6.
Heres another Puzzler from Car Talk ( ):
This was sent in by a fellow named Dan OLeary. He came upon a common one-
syllable, five-letter word recently that has the following unique property. When you
remove the first letter, the remaining letters form a homophone of the original word, that is a word that sounds exactly the same. Replace the first letter, that is, put it back
and remove the second letter, and the result is yet another homophone of the original
word. And the question is, whats the word?
Now Im going to give you an example that doesnt work. Lets look at the five-letter word, wrack. W-R-A-C-K, you know like to wrack with pain. If I remove the first let
ter, I am left with a four-letter word, R-A-C-K. As in, Holy cow, did you see the rack
on that buck! It must have been a nine-pointer! Its a perfect homophone. If you put
the w back, and remove the r, instead, youre left with the word, wack, which is a real word, its just not a homophone of the other two words.
But there is, however, at least one word that Dan and we know of, which will yield two
homophones if you remove either of the first two letters to make two, new four-letter
words. The question is, whats the word?
You can use the dictionary from Exercise 11-1 to check whether a string is in the
word list.
To check whether two words are homophones, you can use the CMU Pronouncing
Dictionary. You can download it from
or from and you can also download http://thinkpy , which provides a function named read_dictionary
that reads the pronouncing dictionary and returns a Python dictionary that maps
from each word to a string that describes its primary pronunciation.
Write a program that lists all the words that solve the Puzzler. Solution: .138 | Chapter 11: Dictionaries

CHAPTER 12Tuples
This chapter presents one more built-in type, the tuple, and then shows how lists, dic tionaries, and tuples work together. I also present a useful feature for variable-length argument lists: the gather and scatter operators.
One note: there is no consensus on how to pronounce “tuple”. Some people say “tuh-
ple”, which rhymes with “supple”. But in the context of programming, most people say
“too-ple”, which rhymes with “quadruple”.
Tuples Are Immutable
A tuple is a sequence of values. The values can be any type, and they are indexed by integers, so in that respect tuples are a lot like lists. The important difference is that
tuples are immutable.
Syntactically, a tuple is a comma-separated list of values: >>> t = a, b, c, d, e
Although it is not necessary, it is common to enclose tuples in parentheses:
>>> t = (a, b, c, d, e)
To create a tuple with a single element, you have to include a final comma:
>>> t1 = a,
>>> type(t1)

A value in parentheses is not a tuple:
>>> t2 = (a)
>>> type(t2)


Another way to create a tuple is the built-in function tuple. With no argument, it
creates an empty tuple:
>>> t = tuple()
>>> t
If the argument is a sequence (string, list or tuple), the result is a tuple with the ele ments of the sequence:
>>> t = tuple(lupins)
>>> t
(l, u, p, i, n, s)
Because tuple is the name of a built-in function, you should avoid using it as a vari
able name.
Most list operators also work on tuples. The bracket operator indexes an element: >>> t = (a, b, c, d, e)
>>> t[0]
And the slice operator selects a range of elements:
>>> t[1:3]
(b, c)
But if you try to modify one of the elements of the tuple, you get an error:
>>> t[0] = A
TypeError: object doesnt support item assignment
Because tuples are immutable, you cant modify the elements. But you can replace
one tuple with another:
>>> t = (A,) + t[1:]
>>> t
(A, b, c, d, e)
This statement makes a new tuple and then makes t refer to it.
The relational operators work with tuples and other sequences; Python starts by com
paring the first element from each sequence. If they are equal, it goes on to the next elements, and so on, until it finds elements that differ. Subsequent elements are not
considered (even if they are really big).
>>> (0, 1, 2) < (0, 3, 4)
>>> (0, 1, 2000000) < (0, 3, 4)
True140 | Chapter 12: Tuples

Tuple Assignment
It is often useful to swap the values of two variables. With conventional assignments, you have to use a temporary variable. For example, to swap a and b:
>>> temp = a
>>> a = b
>>> b = temp
This solution is cumbersome; tuple assignment is more elegant:
>>> a, b = b, a
The left side is a tuple of variables; the right side is a tuple of expressions. Each value
is assigned to its respective variable. All the expressions on the right side are evalu
ated before any of the assignments.
The number of variables on the left and the number of values on the right have to be the same:
>>> a, b = 1, 2, 3
ValueError: too many values to unpack
More generally, the right side can be any kind of sequence (string, list or tuple). For
example, to split an email address into a user name and a domain, you could write:
>>> addr =
>>> uname, domain = addr.split(@)
The return value from split is a list with two elements; the first element is assigned
to uname , the second to domain:
>>> uname
>>> domain
Tuples as Return Values
Strictly speaking, a function can only return one value, but if the value is a tuple, the effect is the same as returning multiple values. For example, if you want to divide two
integers and compute the quotient and remainder, it is inefficient to compute x/y and
then x%y. It is better to compute them both at the same time.
The built-in function divmod takes two arguments and returns a tuple of two values:
the quotient and remainder. You can store the result as a tuple:
>>> t = divmod(7, 3)
>>> t
(2, 1)
Or use tuple assignment to store the elements separately: Tuple Assignment | 141

>>> quot, rem = divmod(7, 3)
>>> quot
>>> rem
Here is an example of a function that returns a tuple:
def min_max(t):
return min(t), max(t)
max and min are built-in functions that find the largest and smallest elements of a
sequence. min_max computes both and returns a tuple of two values.
Variable-Length Argument Tuples
Functions can take a variable number of arguments. A parameter name that begins
with * gathers arguments into a tuple. For example, printall takes any number of
arguments and prints them:
def printall(*args):
The gather parameter can have any name you like, but args is conventional. Heres
how the function works:
>>> printall(1, 2.0, 3)
(1, 2.0, 3)
The complement of gather is scatter. If you have a sequence of values and you want
to pass it to a function as multiple arguments, you can use the * operator. For exam
ple, divmod takes exactly two arguments; it doesnt work with a tuple:
>>> t = (7, 3)
>>> divmod(t)
TypeError: divmod expected 2 arguments, got 1
But if you scatter the tuple, it works:
>>> divmod(*t)
(2, 1)
Many of the built-in functions use variable-length argument tuples. For example, max
and min can take any number of arguments:
>>> max(1, 2, 3)
But sum does not:
>>> sum(1, 2, 3)
TypeError: sum expected at most 2 arguments, got 3 142 | Chapter 12: Tuples

As an exercise, write a function called sumall that takes any number of arguments
and returns their sum.
Lists and Tuples
zip is a built-in function that takes two or more sequences and returns a list of tuples
where each tuple contains one element from each sequence. The name of the func
tion refers to a zipper, which joins and interleaves two rows of teeth.
This example zips a string and a list: >>> s = abc
>>> t = [0, 1, 2]
>>> zip(s, t)

The result is a zip object that knows how to iterate through the pairs. The most com
mon use of zip is in a for loop:
>>> for pair in zip(s, t):
... print(pair)
(a, 0)
(b, 1)
(c, 2)
A zip object is a kind of iterator, which is any object that iterates through a sequence.
Iterators are similar to lists in some ways, but unlike lists, you cant use an index to select an element from an iterator.
If you want to use list operators and methods, you can use a zip object to make a list: >>> list(zip(s, t))
[(a, 0), (b, 1), (c, 2)]
The result is a list of tuples; in this example, each tuple contains a character from the string and the corresponding element from the list.
If the sequences are not the same length, the result has the length of the shorter one: >>> list(zip(Anne, Elk))
[(A, E), (n, l), (n, k)]
You can use tuple assignment in a for loop to traverse a list of tuples:
t = [(a, 0), (b, 1), (c, 2)] for letter, number in t:
print(number, letter)
Each time through the loop, Python selects the next tuple in the list and assigns the
elements to letter and number . The output of this loop is: Lists and Tuples | 143

0 a
1 b
2 c
If you combine zip, for and tuple assignment, you get a useful idiom for traversing
two (or more) sequences at the same time. For example, has_match takes two sequen
ces, t1 and t2, and returns True if there is an index i such that t1[i] == t2[i] :
def has_match(t1, t2): for x, y in zip(t1, t2):
if x == y:
return True
return False
If you need to traverse the elements of a sequence and their indices, you can use the
built-in function enumerate:
for index, element in enumerate(abc): print(index, element)
The result from enumerate is an enumerate object, which iterates a sequence of pairs;
each pair contains an index (starting from 0) and an element from the given
sequence. In this example, the output is
0 a
1 b
2 c
Dictionaries and Tuples Dictionaries have a method called items that returns a sequence of tuples, where each
tuple is a key-value pair:
>>> d = {a:0, b:1, c:2}
>>> t = d.items()
>>> t
dict_items([(c, 2), (a, 0), (b, 1)])
The result is a dict_items object, which is an iterator that iterates the key-value pairs.
You can use it in a for loop like this:
>>> for key, value in d.items():
... print(key, value)
c 2
a 0
b 1
As you should expect from a dictionary, the items are in no particular order. 144 | Chapter 12: Tuples

Going in the other direction, you can use a list of tuples to initialize a new dictionary:>>> t = [(a, 0), (c, 2), (b, 1)]
>>> d = dict(t)
>>> d
{a: 0, c: 2, b: 1}
Combining dict with zip yields a concise way to create a dictionary:
>>> d = dict(zip(abc, range(3)))
>>> d
{a: 0, c: 2, b: 1}
The dictionary method update also takes a list of tuples and adds them, as key-value
pairs, to an existing dictionary.
It is common to use tuples as keys in dictionaries (primarily because you cant use
lists). For example, a telephone directory might map from last-name, first-name pairs
to telephone numbers. Assuming that we have defined last, first and number , we
could write:
directory[last, first] = number
The expression in brackets is a tuple. We could use tuple assignment to traverse this
for last, first in directory:
print(first, last, directory[last,first])
This loop traverses the keys in directory, which are tuples. It assigns the elements of
each tuple to last and first , then prints the name and corresponding telephone
There are two ways to represent tuples in a state diagram. The more detailed version
shows the indices and elements just as they appear in a list. For example, the tuple
(Cleese, John) would appear as in Figure 12-1.
Figure 12-1. State diagram.
But in a larger diagram you might want to leave out the details. For example, a dia
gram of the telephone directory might appear as in Figure 12-2.
Dictionaries and Tuples | 145

Figure 12-2. State diagram.
Here the tuples are shown using Python syntax as a graphical shorthand. The tele
phone number in the diagram is the complaints line for the BBC, so please dont call it.
Sequences of Sequences I have focused on lists of tuples, but almost all of the examples in this chapter also
work with lists of lists, tuples of tuples, and tuples of lists. To avoid enumerating the
possible combinations, it is sometimes easier to talk about sequences of sequences.
In many contexts, the different kinds of sequences (strings, lists and tuples) can be used interchangeably. So how should you choose one over the others?
To start with the obvious, strings are more limited than other sequences because the
elements have to be characters. They are also immutable. If you need the ability to
change the characters in a string (as opposed to creating a new string), you might
want to use a list of characters instead.
Lists are more common than tuples, mostly because they are mutable. But there are a few cases where you might prefer tuples:
In some contexts, like a return statement, it is syntactically simpler to create a
tuple than a list.
If you want to use a sequence as a dictionary key, you have to use an immutable
type like a tuple or string.
If you are passing a sequence as an argument to a function, using tuples reducesthe potential for unexpected behavior due to aliasing.
Because tuples are immutable, they dont provide methods like sort and reverse ,
which modify existing lists. But Python provides the built-in function sorted, which
takes any sequence and returns a new list with the same elements in sorted order, and
146 | Chapter 12: Tuples

reversed, which takes a sequence and returns an iterator that traverses the list in
reverse order.
Lists, dictionaries and tuples are examples of data structures; in this chapter we are
starting to see compound data structures, like lists of tuples, or dictionaries that con tain tuples as keys and lists as values. Compound data structures are useful, but they
are prone to what I call shape errors; that is, errors caused when a data structure has
the wrong type, size, or structure. For example, if you are expecting a list with one integer and I give you a plain old integer (not in a list), it wont work.
To help debug these kinds of errors, I have written a module called structshape that
provides a function, also called structshape, that takes any kind of data structure as
an argument and returns a string that summarizes its shape. You can download it
from .
Heres the result for a simple list: >>> from structshape import structshape
>>> t = [1, 2, 3]
>>> structshape(t)
list of 3 int
A fancier program might write “list of 3 int s”, but it was easier not to deal with plu
rals. Heres a list of lists:
>>> t2 = [[1,2], [3,4], [5,6]]
>>> structshape(t2)
list of 3 list of 2 int
If the elements of the list are not the same type, structshape groups them, in order,
by type:
>>> t3 = [1, 2, 3, 4.0, 5, 6, [7], [8], 9]>>> structshape(t3)
list of (3 int, float, 2 str, 2 list of int, int)
Heres a list of tuples:
>>> s = abc
>>> lt = list(zip(t, s))
>>> structshape(lt)
list of 3 tuple of (int, str)
And heres a dictionary with three items that map integers to strings:
>>> d = dict(lt)
>>> structshape(d)
dict of 3 int->str
If you are having trouble keeping track of your data structures, structshape can help. Debugging | 147

Glossarytuple: An immutable sequence of elements.
tuple assignment: An assignment with a sequence on the right side and a tuple of variables on the
left. The right side is evaluated and then its elements are assigned to the variables on the left.
gather: The operation of assembling a variable-length argument tuple.
scatter: The operation of treating a sequence as a list of arguments.
zip object: The result of calling a built-in function zip; an object that iterates through a
sequence of tuples.
iterator: An object that can iterate through a sequence, but which does not provide list
operators and methods.
data structure: A collection of related values, often organized in lists, dictionaries, tuples, etc.
shape error: An error caused because a value has the wrong shape; that is, the wrong type orsize.
Exercise 12-1.
Write a function called most_frequent that takes a string and prints the letters in
decreasing order of frequency. Find text samples from several different languages and
see how letter frequency varies between languages. Compare your results with the
tables at .
Solution: .
Exercise 12-2.
More anagrams!148 | Chapter 12: Tuples

1. Write a program that reads a word list from a file (see “Reading Word Lists” on
page 99 ) and prints all the sets of words that are anagrams.
Here is an example of what the output might look like: [deltas, desalt, lasted, salted, slated, staled]
[retainers, ternaries]
[generating, greatening]
[resmelts, smelters, termless]
Hint: you might want to build a dictionary that maps from a collection of letters
to a list of words that can be spelled with those letters. The question is, how can you represent the collection of letters in a way that can be used as a key? 2.
Modify the previous program so that it prints the longest list of anagrams first,
followed by the second longest, and so on.
In Scrabble, a “bingo” is when you play all seven tiles in your rack, along with a letter on the board, to form an eight-letter word. What collection of eight letters
forms the most possible bingos? Hint: there are seven.
Solution: .
Exercise 12-3.
Two words form a “metathesis pair” if you can transform one into the other by swap
ping two letters; for example, “converse” and “conserve”. Write a program that finds all of the metathesis pairs in the dictionary. Hint: dont test all pairs of words, and
dont test all possible swaps.
Solution: . Credit: This exercise is inspired
by an example at .
Exercise 12-4.
Heres another Car Talk Puzzler ( ):
What is the longest English word, that remains a valid English word, as you remove its
letters one at a time?
Now, letters can be removed from either end, or the middle, but you cant rearrange
any of the letters. Every time you drop a letter, you wind up with another English word. If you do that, youre eventually going to wind up with one letter and that too is
going to be an English wordone thats found in the dictionary. I want to know whats
the longest word and how many letters does it have?
Im going to give you a little modest example: Sprite. Ok? You start off with sprite, you take a letter off, one from the interior of the word, take the r away, and were left with
the word spite, then we take the e off the end, were left with spit, we take the s off,
were left with pit, it, and I.
Exercises | 149

Write a program to find all words that can be reduced in this way, and then find the
longest one.
This exercise is a little more challenging than most, so here are some suggestions:1.
You might want to write a function that takes a word and computes a list of all the words that can be formed by removing one letter. These are the “children” of
the word.
Recursively, a word is reducible if any of its children are reducible. As a base case, you can consider the empty string reducible.
The wordlist I provided, words.txt, doesnt contain single letter words. So you
might want to add “I”, “a”, and the empty string.
To improve the performance of your program, you might want to memoize the
words that are known to be reducible.
Solution: .
150 | Chapter 12: Tuples

Case Study: Data Structure Selection
At this point you have learned about Pythons core data structures, and you have seen some of the algorithms that use them. If you would like to know more about algo
rithms, this might be a good time to read Chapter 21. But you dont have to read it
before you go on; you can read it whenever you are interested.
This chapter presents a case study with exercises that let you think about choosing data structures and practice using them.
Word Frequency Analysis As usual, you should at least attempt the exercises before you read my solutions.
Exercise 13-1.
Write a program that reads a file, breaks each line into words, strips whitespace and
punctuation from the words, and converts them to lowercase.
Hint: The string module provides a string named whitespace, which contains space,
tab, newline, etc., and punctuation which contains the punctuation characters. Lets
see if we can make Python swear:
>>> import string
>>> string.punctuation
Also, you might consider using the string methods strip, replace and translate .

Exercise 13-2.
Go to Project Gutenberg ( and download your favorite out-of-
copyright book in plain text format.
Modify your program from the previous exercise to read the book you downloaded, skip over the header information at the beginning of the file, and process the rest of
the words as before.
Then modify the program to count the total number of words in the book, and the
number of times each word is used.
Print the number of different words used in the book. Compare different books by
different authors, written in different eras. Which author uses the most extensive vocabulary?
Exercise 13-3.
Modify the program from the previous exercise to print the 20 most frequently used
words in the book.
Exercise 13-4.
Modify the previous program to read a word list (see “Reading Word Lists” on page
99 ) and then print all the words in the book that are not in the word list. How many
of them are typos? How many of them are common words that should be in the word
list, and how many of them are really obscure?
Random Numbers Given the same inputs, most computer programs generate the same outputs every
time, so they are said to be deterministic. Determinism is usually a good thing, since
we expect the same calculation to yield the same result. For some applications, though, we want the computer to be unpredictable. Games are an obvious example,
but there are more.
Making a program truly nondeterministic turns out to be difficult, but there are ways
to make it at least seem nondeterministic. One of them is to use algorithms that gen erate pseudorandom numbers. Pseudorandom numbers are not truly random
because they are generated by a deterministic computation, but just by looking at the
numbers it is all but impossible to distinguish them from random.
The random module provides functions that generate pseudorandom numbers (which
I will simply call “random” from here on).152 | Chapter 13: Case Study: Data Structure Selection

The function random returns a random float between 0.0 and 1.0 (including 0.0 but
not 1.0). Each time you call random, you get the next number in a long series. To see a
sample, run this loop:
import random
for i in range(10):
x = random.random()
The function randint takes parameters low and high and returns an integer between
low and high (including both):
>>> random.randint(5, 10)
>>> random.randint(5, 10)
To choose an element from a sequence at random, you can use choice:
>>> t = [1, 2, 3]
>>> random.choice(t)
>>> random.choice(t)
The random module also provides functions to generate random values from continu
ous distributions including Gaussian, exponential, gamma, and a few more.
Exercise 13-5.
Write a function named choose_from_hist that takes a histogram as defined in “Dic
tionary as a Collection of Counters” on page 127 and returns a random value from
the histogram, chosen with probability in proportion to frequency. For example, for
this histogram:
>>> t = [a, a, b]
>>> hist = histogram(t)
>>> hist
{a: 2, b: 1}
your function should return a with probability 2/3 and b with probability 1/3.
Word Histogram
You should attempt the previous exercises before you go on. You can download my solution from . You will also need .
Here is a program that reads a file and builds a histogram of the words in the file: Word Histogram | 153

import string
def process_file(filename):
hist = dict()
fp = open(filename)
for line in fp:
process_line(line, hist) return hist
def process_line(line, hist):
line = line.replace(-, )

for word in line.split():
word = word.strip(string.punctuation + string.whitespace) word = word.lower()
hist[word] = hist.get(word, 0) + 1
hist = process_file(emma.txt)
This program reads emma.txt, which contains the text of Emma by Jane Austen.
process_file loops through the lines of the file, passing them one at a time to pro
cess_line . The histogram hist is being used as an accumulator.
process_line uses the string method replace to replace hyphens with spaces before
using split to break the line into a list of strings. It traverses the list of words and
uses strip and lower to remove punctuation and convert to lowercase. (It is short
hand to say that strings are “converted”; remember that strings are immutable, so
methods like strip and lower return new strings.)
Finally, process_line updates the histogram by creating a new item or incrementing
an existing one.
To count the total number of words in the file, we can add up the frequencies in the histogram:
def total_words(hist):
return sum(hist.values())
The number of different words is just the number of items in the dictionary:
def different_words(hist):
return len(hist)
Here is some code to print the results:
print(Total number of words:, total_words(hist))print(Number of different words:, different_words(hist))
And the results:
Total number of words: 161080
Number of different words: 7214 154 | Chapter 13: Case Study: Data Structure Selection

Most Common Words
To find the most common words, we can make a list of tuples, where each tuple con
tains a word and its frequency, and sort it.
The following function takes a histogram and returns a list of word-frequency tuples: def most_common(hist):
t = []
for key, value in hist.items(): t.append((value, key))
return t
In each tuple, the frequency appears first, so the resulting list is sorted by frequency.
Here is a loop that prints the 10 most common words:
t = most_common(hist)
print(The most common words are:)
for freq, word in t[:10]:
print(word, freq, sep=\t)
I use the keyword argument sep to tell print to use a tab character as a “separator”,
rather than a space, so the second column is lined up. Here are the results from Emma :
The most common words are:
to 5242
the 5205
and 4897
of 4295
i 3191
a 3130
it 2529
her 2483
was 2400
she 2364
This code can be simplified using the key parameter of the sort function. If you are
curious, you can read about it at .
Optional Parameters
We have seen built-in functions and methods that take optional arguments. It is pos sible to write programmer-defined functions with optional arguments, too. For example, here is a function that prints the most common words in a histogram: Most Common Words | 155

def print_most_common(hist, num=10):
t = most_common(hist)
print(The most common words are:)
for freq, word in t[:num]:
print(word, freq, sep=\t)
The first parameter is required; the second is optional. The default value of num is 10.
If you only provide one argument: print_most_common(hist)
num gets the default value. If you provide two arguments:
print_most_common(hist, 20)
num gets the value of the argument instead. In other words, the optional argument
overrides the default value.
If a function has both required and optional parameters, all the required parameters have to come first, followed by the optional ones.
Dictionary Subtraction Finding the words from the book that are not in the word list from words.txt is a
problem you might recognize as set subtraction; that is, we want to find all the words
from one set (the words in the book) that are not in the other (the words in the list).
subtract takes dictionaries d1 and d2 and returns a new dictionary that contains all
the keys from d1 that are not in d2. Since we dont really care about the values, we set
them all to None:
def subtract(d1, d2):
res = dict()
for key in d1:
if key not in d2:
res[key] = None
return res
To find the words in the book that are not in words.txt, we can use process_file to
build a histogram for words.txt, and then subtract:
words = process_file(words.txt) diff = subtract(hist, words)
print("Words in the book that arent in the word list:")
for word in diff:
print(word, end= ) 156 | Chapter 13: Case Study: Data Structure Selection

Here are some of the results from Emma:
Words in the book that arent in the word list: rencontre janes blanche woodhouses disingenuousness friends venice apartment ...
Some of these words are names and possessives. Others, like “rencontre”, are no longer in common use. But a few are common words that should really be in the list!
Exercise 13-6.
Python provides a data structure called set that provides many common set opera
tions. You can read about them in “Sets” on page 226, or read the documentation at .
Write a program that uses set subtraction to find words in the book that are not in the
word list.
Solution: .
Random Words To choose a random word from the histogram, the simplest algorithm is to build a list with multiple copies of each word, according to the observed frequency, and then
choose from the list:
def random_word(h):
t = []
for word, freq in h.items():
t.extend([word] * freq)
return random.choice(t)
The expression [word] * freq creates a list with freq copies of the string word. The
extend method is similar to append except that the argument is a sequence.
This algorithm works, but it is not very efficient; each time you choose a random
word, it rebuilds the list, which is as big as the original book. An obvious improve ment is to build the list once and then make multiple selections, but the list is still big.
An alternative is:1.
Use keys to get a list of the words in the book.
Build a list that contains the cumulative sum of the word frequencies (see Exer
cise 10-2 ). The last item in this list is the total number of words in the book, n.
Random Words | 157

3. Choose a random number from 1 to n. Use a bisection search (See Exercise
10-10 ) to find the index where the random number would be inserted in the
cumulative sum. 4.
Use the index to find the corresponding word in the word list.
Exercise 13-7.
Write a program that uses this algorithm to choose a random word from the book. Solution: .
Markov Analysis If you choose words from the book at random, you can get a sense of the vocabulary,
but you probably wont get a sentence:
this the small regard harriet which knightleys it most things
A series of random words seldom makes sense because there is no relationship
between successive words. For example, in a real sentence you would expect an article
like “the” to be followed by an adjective or a noun, and probably not a verb or adverb.
One way to measure these kinds of relationships is Markov analysis, which character izes, for a given sequence of words, the probability of the words that might come
next. For example, the song “Eric, the Half a Bee” begins:
Half a bee, philosophically,Must, ipso facto, half not be.
But half the bee has got to be
Vis a vis, its entity. Dyou see?
But can a bee be said to be
Or not to be an entire bee When half the bee is not a bee
Due to some ancient injury?
In this text, the phrase “half the” is always followed by the word “bee”, but the phrase
“the bee” might be followed by either “has” or “is”.
The result of Markov analysis is a mapping from each prefix (like “half the” and “the
bee”) to all possible suffixes (like “has” and “is”).
Given this mapping, you can generate a random text by starting with any prefix and choosing at random from the possible suffixes. Next, you can combine the end of the
prefix and the new suffix to form the next prefix, and repeat.
158 | Chapter 13: Case Study: Data Structure Selection

For example, if you start with the prefix “Half a”, then the next word has to be “bee”,
because the prefix only appears once in the text. The next prefix is “a bee”, so the next suffix might be “philosophically”, “be” or “due”.
In this example the length of the prefix is always two, but you can do Markov analysis
with any prefix length.
Exercise 13-8.
Markov analysis:1.
Write a program to read a text from a file and perform Markov analysis. The result should be a dictionary that maps from prefixes to a collection of possible
suffixes. The collection might be a list, tuple, or dictionary; it is up to you to make an appropriate choice. You can test your program with prefix length 2, but you should write the program in a way that makes it easy to try other lengths.
Add a function to the previous program to generate random text based on theMarkov analysis. Here is an example from Emma with prefix length 2:
He was very clever, be it sweetness or be angry, ashamed or only amused, at such a
stroke. She had never thought of Hannah till you were never meant for me?” “I cannot make speeches, Emma:” he soon cut it all himself.
For this example, I left the punctuation attached to the words. The result is
almost syntactically correct, but not quite. Semantically, it almost makes sense, but not quite.
What happens if you increase the prefix length? Does the random text make
more sense?
Once your program is working, you might want to try a mash-up: if you combine text from two or more books, the random text you generate will blend the
vocabulary and phrases from the sources in interesting ways.
Credit: This case study is based on an example from Kernighan and Pike,
e Practice
of Programming , Addison-Wesley, 1999.
You should attempt this exercise before you go on; then you can can download my
solution from . You will also need http://think .
Data Structures
Using Markov analysis to generate random text is fun, but there is also a point to this exercise: data structure selection. In your solution to the previous exercises, you had
to choose:
Data Structures | 159

How to represent the prefixes.
How to represent the collection of possible suffixes.

How to represent the mapping from each prefix to the collection of possible suffixes.
The last one is easy: a dictionary is the obvious choice for a mapping from keys to
corresponding values.
For the prefixes, the most obvious options are string, list of strings, or tuple of strings. For the suffixes, one option is a list; another is a histogram (dictionary).
How should you choose? The first step is to think about the operations you will need
to implement for each data structure. For the prefixes, we need to be able to remove
words from the beginning and add to the end. For example, if the current prefix is “Half a”, and the next word is “bee”, you need to be able to form the next prefix, “a
Your first choice might be a list, since it is easy to add and remove elements, but we
also need to be able to use the prefixes as keys in a dictionary, so that rules out lists. With tuples, you cant append or remove, but you can use the addition operator to
form a new tuple:
def shift(prefix, word):
return prefix[1:] + (word,)
shift takes a tuple of words, prefix, and a string, word, and forms a new tuple that
has all the words in prefix except the first, and word added to the end.
For the collection of suffixes, the operations we need to perform include adding a
new suffix (or increasing the frequency of an existing one), and choosing a random
Adding a new suffix is equally easy for the list implementation or the histogram.
Choosing a random element from a list is easy; choosing from a histogram is harder to do efficiently (see Exercise 13-7).
So far we have been talking mostly about ease of implementation, but there are other
factors to consider in choosing data structures. One is runtime. Sometimes there is a
theoretical reason to expect one data structure to be faster than other; for example, I
mentioned that the in operator is faster for dictionaries than for lists, at least when
the number of elements is large.
But often you dont know ahead of time which implementation will be faster. One
option is to implement both of them and see which is better. This approach is called
benchmarking . A practical alternative is to choose the data structure that is easiest to
implement, and then see if it is fast enough for the intended application. If so, there is
160 | Chapter 13: Case Study: Data Structure Selection

no need to go on. If not, there are tools, like the profile module, that can identify the
places in a program that take the most time.
The other factor to consider is storage space. For example, using a histogram for the
collection of suffixes might take less space because you only have to store each word
once, no matter how many times it appears in the text. In some cases, saving space
can also make your program run faster, and in the extreme, your program might not run at all if you run out of memory. But for many applications, space is a secondary
consideration after runtime.
One final thought: in this discussion, I have implied that we should use one data
structure for both analysis and generation. But since these are separate phases, it
would also be possible to use one structure for analysis and then convert to another
structure for generation. This would be a net win if the time saved during generation
exceeded the time spent in conversion.
Debugging When you are debugging a program, and especially if you are working on a hard bug,
there are five things to try:
Reading: Examine your code, read it back to yourself, and check that it says what you
meant to say.
Running: Experiment by making changes and running different versions. Often if you display the right thing at the right place in the program, the problem becomes obvi
ous, but sometimes you have to build scaffolding.
Ruminating: Take some time to think! What kind of error is it: syntax, runtime, or semantic?What information can you get from the error messages, or from the output of the
program? What kind of error could cause the problem youre seeing? What did you change last, before the problem appeared?
Rubberducking: If you explain the problem to someone else, you sometimes find the answerbefore you finish asking the question. Often you dont need the other person; you
could just talk to a rubber duck. And thats the origin of the well-known strategy called rubber duck debugging . I am not making this up; see https://en.wikipe . Debugging | 161

Retreating:At some point, the best thing to do is back off and undo recent changes until youget back to a program that works and that you understand. Then you can start
Beginning programmers sometimes get stuck on one of these activities and forget the
others. Each activity comes with its own failure mode.
For example, reading your code might help if the problem is a typographical error, but not if the problem is a conceptual misunderstanding. If you dont understand
what your program does, you can read it 100 times and never see the error, because
the error is in your head.
Running experiments can help, especially if you run small, simple tests. But if you run experiments without thinking or reading your code, you might fall into a pattern I
call “random walk programming”, which is the process of making random changes
until the program does the right thing. Needless to say, random walk programming
can take a long time.
You have to take time to think. Debugging is like an experimental science. You should have at least one hypothesis about what the problem is. If there are two or more pos
sibilities, try to think of a test that would eliminate one of them.
But even the best debugging techniques will fail if there are too many errors, or if the
code you are trying to fix is too big and complicated. Sometimes the best option is to retreat, simplifying the program until you get to something that works and that you
Beginning programmers are often reluctant to retreat because they cant stand to
delete a line of code (even if its wrong). If it makes you feel better, copy your program into another file before you start stripping it down. Then you can copy the pieces
back one at a time.
Finding a hard bug requires reading, running, ruminating, and sometimes retreating. If you get stuck on one of these activities, try the others.
Glossary deterministic: Pertaining to a program that does the same thing each time it runs, given the
same inputs.
pseudorandom: Pertaining to a sequence of numbers that appears to be random, but is generatedby a deterministic program.162 | Chapter 13: Case Study: Data Structure Selection

default value:The value given to an optional parameter if no argument is provided.
override: To replace a default value with an argument.
benchmarking: The process of choosing between data structures by implementing alternativesand testing them on a sample of the possible inputs.
rubber duck debugging: Debugging by explaining your problem to an inanimate object such as a rubber
duck. Articulating the problem can help you solve it, even if the rubber duck doesnt know Python.
Exercises Exercise 13-9.
The “rank” of a word is its position in a list of words sorted by frequency: the most
common word has rank 1, the second most common has rank 2, etc.
Zipf s law describes a relationship between the ranks and frequencies of words in nat
ural languages ( s_law ). Specifically, it predicts that
the frequency, f, of the word with rank r is:
f = cr
where s and c are parameters that depend on the language and the text. If you take the
logarithm of both sides of this equation, you get:
log f= log c slog r
So if you plot log f versus log r, you should get a straight line with slope -s and inter
cept log c.
Write a program that reads a text from a file, counts word frequencies, and prints one
line for each word, in descending order of frequency, with log f and log r. Use the
graphing program of your choice to plot the results and check whether they form a
straight line. Can you estimate the value of s?
Solution: . To run my solution, you need the
plotting module matplotlib. If you installed Anaconda, you already have matplot
lib ; otherwise you might have to install it.
Exercises | 163

This chapter introduces the idea of “persistent” programs that keep data in perma nent storage, and shows how to use different kinds of permanent storage, like files
and databases.
Persistence Most of the programs we have seen so far are transient in the sense that they run for a
short time and produce some output, but when they end, their data disappears. If you run the program again, it starts with a clean slate.
Other programs are persistent: they run for a long time (or all the time); they keep at
least some of their data in permanent storage (a hard drive, for example); and if they
shut down and restart, they pick up where they left off.
Examples of persistent programs are operating systems, which run pretty much
whenever a computer is on, and web servers, which run all the time, waiting for requests to come in on the network.
One of the simplest ways for programs to maintain their data is by reading and writ
ing text files. We have already seen programs that read text files; in this chapter we
will see programs that write them.
An alternative is to store the state of the program in a database. In this chapter I will present a simple database and a module, pickle, that makes it easy to store program

Reading and WritingA text file is a sequence of characters stored on a permanent medium like a hard
drive, flash memory, or CD-ROM. We saw how to open and read a file in “Reading
Word Lists” on page 99 .
To write a file, you have to open it with mode w as a second parameter:
>>> fout = open(output.txt, w)
If the file already exists, opening it in write mode clears out the old data and starts fresh, so be careful! If the file doesnt exist, a new one is created.
open returns a file object that provides methods for working with the file. The write
method puts data into the file:
>>> line1 = "This heres the wattle,\n">>> fout.write(line1)
The return value is the number of characters that were written. The file object keeps
track of where it is, so if you call write again, it adds the new data to the end of the
>>> line2 = "the emblem of our land.\n">>> fout.write(line2)
When you are done writing, you should close the file:
>>> fout.close()
If you dont close the file, it gets closed for you when the program ends.
Format Operator The argument of write has to be a string, so if we want to put other values in a file,
we have to convert them to strings. The easiest way to do that is with str:
>>> x = 52
>>> fout.write(str(x))
An alternative is to use the format operator, %. When applied to integers, % is the
modulus operator. But when the first operand is a string, % is the format operator.
The first operand is the format string, which contains one or more format sequen
ces , which specify how the second operand is formatted. The result is a string.
For example, the format sequence %d means that the second operand should be for
matted as a decimal integer:166 | Chapter 14: Files

>>> camels = 42
>>> %d % camels
The result is the string 42, which is not to be confused with the integer value 42.
A format sequence can appear anywhere in the string, so you can embed a value in a
>>> I have spotted %d camels. % camelsI have spotted 42 camels.
If there is more than one format sequence in the string, the second argument has to be a tuple. Each format sequence is matched with an element of the tuple, in order.
The following example uses %d to format an integer, %g to format a floating-point
number, and %s to format a string:
>>> In %d years I have spotted %g %s. % (3, 0.1, camels) In 3 years I have spotted 0.1 camels.
The number of elements in the tuple has to match the number of format sequences in
the string. Also, the types of the elements have to match the format sequences:
>>> %d %d %d % (1, 2)
TypeError: not enough arguments for format string
>>> %d % dollars
TypeError: %d format: a number is required, not str
In the first example, there arent enough elements; in the second, the element is the
wrong type.
For more information on the format operator, see
stdtypes.html#printf-style-string-formatting . A more powerful alternative is the string
format method, which you can read about at
stdtypes.html#str.format .
Filenames and Paths Files are organized into directories (also called “folders”). Every running program
has a “current directory”, which is the default directory for most operations. For
example, when you open a file for reading, Python looks for it in the current direc
The os module provides functions for working with files and directories (“os” stands
for “operating system”). os.getcwd returns the name of the current directory:
>>> import os
>>> cwd = os.getcwd()
>>> cwd
/home/dinsdale Filenames and Paths | 167

cwd stands for “current working directory”. The result in this example is /home/dins
dale , which is the home directory of a user named dinsdale.
A string like /home/dinsdale that identifies a file or directory is called a path.
A simple filename, like memo.txt, is also considered a path, but it is a relative path
because it relates to the current directory. If the current directory is /home/dinsdale,
the filename memo.txt would refer to /home/dinsdale/memo.txt .
A path that begins with / does not depend on the current directory; it is called an
absolute path . To find the absolute path to a file, you can use os.path.abspath:
>>> os.path.abspath(memo.txt)
os.path provides other functions for working with filenames and paths. For example,
os.path.exists checks whether a file or directory exists:
>>> os.path.exists(memo.txt)
If it exists, os.path.isdir checks whether its a directory:
>>> os.path.isdir(memo.txt) False
>>> os.path.isdir(/home/dinsdale) True
Similarly, os.path.isfile checks whether its a file.
os.listdir returns a list of the files (and other directories) in the given directory:
>>> os.listdir(cwd)
[music, photos, memo.txt]
To demonstrate these functions, the following example “walks” through a directory, prints the names of all the files, and calls itself recursively on all the directories:
def walk(dirname):
for name in os.listdir(dirname):
path = os.path.join(dirname, name)
if os.path.isfile(path):
os.path.join takes a directory and a filename and joins them into a complete path.
The os module provides a function called walk that is similar to this one but more
versatile. As an exercise, read the documentation and use it to print the names of the
files in a given directory and its subdirectories. You can download my solution from .168 | Chapter 14: Files

Catching ExceptionsA lot of things can go wrong when you try to read and write files. If you try to open a
file that doesnt exist, you get an IOError:
>>> fin = open(bad_file)
IOError: [Errno 2] No such file or directory: bad_file
If you dont have permission to access a file:
>>> fout = open(/etc/passwd, w)
PermissionError: [Errno 13] Permission denied: /etc/passwd
And if you try to open a directory for reading, you get
>>> fin = open(/home)
IsADirectoryError: [Errno 21] Is a directory: /home
To avoid these errors, you could use functions like os.path.exists and
os.path.isfile , but it would take a lot of time and code to check all the possibilities
(if “ Errno 21 ” is any indication, there are at least 21 things that can go wrong).
It is better to go ahead and tryand deal with problems if they happenwhich is
exactly what the try statement does. The syntax is similar to an if...else statement:
try: fin = open(bad_file)
print(Something went wrong.)
Python starts by executing the try clause. If all goes well, it skips the except clause
and proceeds. If an exception occurs, it jumps out of the try clause and runs the
except clause.
Handling an exception with a try statement is called catching an exception. In this
example, the except clause prints an error message that is not very helpful. In gen
eral, catching an exception gives you a chance to fix the problem, or try again, or at
least end the program gracefully.
Databases A database is a file that is organized for storing data. Many databases are organized
like a dictionary in the sense that they map from keys to values. The biggest differ
ence between a database and a dictionary is that the database is on disk (or other per manent storage), so it persists after the program ends.
The module dbm provides an interface for creating and updating database files. As an
example, Ill create a database that contains captions for image files.
Opening a database is similar to opening other files: Catching Exceptions | 169

>>> import dbm
>>> db =, c)
The mode c means that the database should be created if it doesnt already exist.
The result is a database object that can be used (for most operations) like a dictionary.
When you create a new item, dbm updates the database file:
>>> db[cleese.png] = Photo of John Cleese.
When you access one of the items, dbm reads the file:
>>> db[cleese.png] bPhoto of John Cleese.
The result is a bytes object, which is why it begins with b. A bytes object is similar to
a string in many ways. When you get farther into Python, the difference becomes important, but for now we can ignore it.
If you make another assignment to an existing key, dbm replaces the old value:
>>> db[cleese.png] = Photo of John Cleese doing a silly walk. >>> db[cleese.png]
bPhoto of John Cleese doing a silly walk.
Some dictionary methods, like keys and items , dont work with database objects. But
iteration with a for loop works:
for key in db:
print(key, db[key])
As with other files, you should close the database when you are done:
>>> db.close()
A limitation of dbm is that the keys and values have to be strings or bytes. If you try to
use any other type, you get an error.
The pickle module can help. It translates almost any type of object into a string suit
able for storage in a database, and then translates strings back into objects.
pickle.dumps takes an object as a parameter and returns a string representation
( dumps is short for “dump string”):
>>> import pickle
>>> t = [1, 2, 3]
>>> pickle.dumps(t)
b\x80\x03]q\x00(K\x01K\x02K\x03e. 170 | Chapter 14: Files

1popen is deprecated now, which means we are supposed to stop using it and start using the subprocess mod
ule. But for simple cases, I find subprocess more complicated than necessary. So I am going to keep using
popen until they take it away.The format isnt obvious to human readers; it is meant to be easy for pickle to inter
pret. pickle.loads (“load string”) reconstitutes the object:
>>> t1 = [1, 2, 3]
>>> s = pickle.dumps(t1)
>>> t2 = pickle.loads(s)
>>> t2
[1, 2, 3]
Although the new object has the same value as the old, it is not (in general) the same
>>> t1 == t2
>>> t1 is t2
In other words, pickling and then unpickling has the same effect as copying the
You can use pickle to store non-strings in a database. In fact, this combination is so
common that it has been encapsulated in a module called shelve.
Pipes Most operating systems provide a command-line interface, also known as a shell.
Shells usually provide commands to navigate the file system and launch applications.
For example, in Unix you can change directories with cd, display the contents of a
directory with ls, and launch a web browser by typing (for example) firefox.
Any program that you can launch from the shell can also be launched from Python
using a pipe object , which represents a running program.
For example, the Unix command ls -l normally displays the contents of the current
directory in long format. You can launch ls with os.popen 1
>>> cmd = ls -l
>>> fp = os.popen(cmd)
The argument is a string that contains a shell command. The return value is an object
that behaves like an open file. You can read the output from the ls process one line at
a time with readline or get the whole thing at once with read:
>>> res = Pipes | 171

When you are done, you close the pipe like a file:>>> stat = fp.close()
>>> print(stat)
The return value is the final status of the ls process; None means that it ended nor
mally (with no errors).
For example, most Unix systems provide a command called md5sum that reads the
contents of a file and computes a “checksum”. You can read about MD5 at http:// . This command provides an efficient way to check
whether two files have the same contents. The probability that different contents yield
the same checksum is very small (that is, unlikely to happen before the universe col lapses).
You can use a pipe to run md5sum from Python and get the result:
>>> filename = book.tex
>>> cmd = md5sum + filename
>>> fp = os.popen(cmd)
>>> res =
>>> stat = fp.close()
>>> print(res)
1e0033f0ed0656636de0d75144ba32e0 book.tex
>>> print(stat)
Writing Modules Any file that contains Python code can be imported as a module. For example, sup
pose you have a file named with the following code:
def linecount(filename):
count = 0
for line in open(filename): count += 1
return count
If you run this program, it reads itself and prints the number of lines in the file, which
is 7. You can also import it like this:
>>> import wc
Now you have a module object wc:
>>> wc
172 | Chapter 14: Files

The module object provides linecount:
>>> wc.linecount(
So thats how you write modules in Python.
The only problem with this example is that when you import the module it runs the test code at the bottom. Normally when you import a module, it defines new func
tions but it doesnt run them.
Programs that will be imported as modules often use the following idiom: if __name__ == __main__:
__name__ is a built-in variable that is set when the program starts. If the program is
running as a script, __name__ has the value __main__; in that case, the test code
runs. Otherwise, if the module is being imported, the test code is skipped.
As an exercise, type this example into a file named and run it as a script. Then
run the Python interpreter and import wc. What is the value of __name__ when the
module is being imported?
Warning: If you import a module that has already been imported, Python does noth
ing. It does not re-read the file, even if it has changed.
If you want to reload a module, you can use the built-in function reload, but it can
be tricky, so the safest thing to do is restart the interpreter and then import the mod
ule again.
Debugging When you are reading and writing files, you might run into problems with white
space. These errors can be hard to debug because spaces, tabs and newlines are nor mally invisible:
>>> s = 1 2\t 3\n 4
>>> print(s)
1 2 3
The built-in function repr can help. It takes any object as an argument and returns a
string representation of the object. For strings, it represents whitespace characters
with backslash sequences:
>>> print(repr(s))
1 2\t 3\n 4
This can be helpful for debugging. Debugging | 173

One other problem you might run into is that different systems use different charac
ters to indicate the end of a line. Some systems use a newline, represented \n. Others
use a return character, represented \r. Some use both. If you move files between dif
ferent systems, these inconsistencies can cause problems.
For most systems, there are applications to convert from one format to another. You
can find them (and read more about this issue) at
Newline . Or, of course, you could write one yourself.
Glossary persistent: Pertaining to a program that runs indefinitely and keeps at least some of its datain permanent storage.
format operator: An operator, %, that takes a format string and a tuple and generates a string that
includes the elements of the tuple formatted as specified by the format string.
format string: A string, used with the format operator, that contains format sequences.
format sequence: A sequence of characters in a format string, like %d, that specifies how a value
should be formatted.
A sequence of characters stored in permanent storage like a hard drive.
directory: A named collection of files, also called a folder.
path: A string that identifies a file.
relative path: A path that starts from the current directory.
absolute path: A path that starts from the topmost directory in the file system.
catch: To prevent an exception from terminating a program by using the try and
except statements.
174 | Chapter 14: Files

database:A file whose contents are organized like a dictionary with keys that correspond tovalues.
bytes object: An object similar to a string.
shell: A program that allows users to type commands and then executes them by start
ing other programs.
pipe object: An object that represents a running program, allowing a Python program to runcommands and read the results.
Exercises Exercise 14-1.
Write a function called sed that takes as arguments a pattern string, a replacement
string, and two filenames; it should read the first file and write the contents into the
second file (creating it if necessary). If the pattern string appears anywhere in the file,
it should be replaced with the replacement string.
If an error occurs while opening, reading, writing or closing files, your program should catch the exception, print an error message, and exit.
Solution: .
Exercise 14-2.
If you download my solution to Exercise 12-2 from , youll see that it creates a dictionary that maps from a sorted string
of letters to the list of words that can be spelled with those letters. For example,
opst maps to the list [opts, post, pots, spot, stop, tops] .
Write a module that imports anagram_sets and provides two new functions:
store_anagrams should store the anagram dictionary in a “shelf ”; read_anagrams
should look up a word and return a list of its anagrams.
Solution: Exercises | 175

Exercise 14-3.
In a large collection of MP3 files, there may be more than one copy of the same song,
stored in different directories or with different filenames. The goal of this exercise is
to search for duplicates.1.
Write a program that searches a directory and all of its subdirectories, recursively, and returns a list of complete paths for all files with a given suffix
(like .mp3). Hint: os.path provides several useful functions for manipulating file-
and path names.
To recognize duplicates, you can use md5sum to compute a “checksum” for each
files. If two files have the same checksum, they probably have the same contents.
To double-check, you can use the Unix command diff.
Solution: .
176 | Chapter 14: Files

Classes and Objects
At this point you know how to use functions to organize code and built-in types to organize data. The next step is to learn “object-oriented programming”, which uses
programmer-defined types to organize both code and data. Object-oriented program ming is a big topic; it will take a few chapters to get there.
Code examples from this chapter are available from ; solutions to the exercises are available from .
Programmer-Defined Types
We have used many of Pythons built-in types; now we are going to define a new type.
As an example, we will create a type called Point that represents a point in two-
dimensional space.
In mathematical notation, points are often written in parentheses with a comma sepa
rating the coordinates. For example, (0,0) represents the origin, and (x,y) represents
the point x units to the right and y units up from the origin.
There are several ways we might represent points in Python:

We could store the coordinates separately in two variables, x and y.

We could store the coordinates as elements in a list or tuple.

We could create a new type to represent points as objects.
Creating a new type is more complicated than the other options, but it has advantages
that will be apparent soon.
A programmer-defined type is also called a class. A class definition looks like this:

class Point:
"""Represents a point in 2-D space."""
The header indicates that the new class is called Point. The body is a docstring that
explains what the class is for. You can define variables and methods inside a class defi
nition, but we will get back to that later.
Defining a class named Point creates a class object :
>>> Point

Because Point is defined at the top level, its “full name” is __main__.Point.
The class object is like a factory for creating objects. To create a Point, you call Point
as if it were a function:
>>> blank = Point()
>>> blank
<__main__.Point object at 0xb7e9d3ac>
The return value is a reference to a Point object, which we assign to blank.
Creating a new object is called instantiation, and the object is an instance of the
When you print an instance, Python tells you what class it belongs to and where it is
stored in memory (the prefix 0x means that the following number is in hexadecimal).
Every object is an instance of some class, so “object” and “instance” are interchangea ble. But in this chapter I use “instance” to indicate that I am talking about a
programmer-defined type.
Attributes You can assign values to an instance using dot notation: >>> blank.x = 3.0
>>> blank.y = 4.0
This syntax is similar to the syntax for selecting a variable from a module, such as math.pi or string.whitespace . In this case, though, we are assigning values to
named elements of an object. These elements are called attributes.
As a noun, “AT-trib-ute” is pronounced with emphasis on the first syllable, as
opposed to “a-TRIB-ute”, which is a verb.
The following diagram shows the result of these assignments. A state diagram that shows an object and its attributes is called an object diagram; see Figure 15-1 . 178 | Chapter 15: Classes and Objects

Figure 15-1. Object diagram.
The variable blank refers to a Point object, which contains two attributes. Each
attribute refers to a floating-point number.
You can read the value of an attribute using the same syntax: >>> blank.y
>>> x = blank.x
>>> x
The expression blank.x means, “Go to the object blank refers to and get the value of
x .” In the example, we assign that value to a variable named x. There is no conflict
between the variable x and the attribute x.
You can use dot notation as part of any expression. For example: >>> (%g, %g) % (blank.x, blank.y)
(3.0, 4.0)
>>> distance = math.sqrt(blank.x**2 + blank.y**2) >>> distance
You can pass an instance as an argument in the usual way. For example:
def print_point(p):
print((%g, %g) % (p.x, p.y))
print_point takes a point as an argument and displays it in mathematical notation.
To invoke it, you can pass blank as an argument:
>>> print_point(blank)
(3.0, 4.0)
Inside the function, p is an alias for blank, so if the function modifies p, blank
As an exercise, write a function called distance_between_points that takes two
Points as arguments and returns the distance between them.
Sometimes it is obvious what the attributes of an object should be, but other times
you have to make decisions. For example, imagine you are designing a class to repre
Rectangles | 179

sent rectangles. What attributes would you use to specify the location and size of a
rectangle? You can ignore angle; to keep things simple, assume that the rectangle is either vertical or horizontal.
There are at least two possibilities:
You could specify one corner of the rectangle (or the center), the width, and the height.

You could specify two opposing corners.
At this point it is hard to say whether either is better than the other, so well imple
ment the first one, just as an example.
Here is the class definition: class Rectangle:
"""Represents a rectangle.
attributes: width, height, corner.
The docstring lists the attributes: width and height are numbers; corner is a Point
object that specifies the lower-left corner.
To represent a rectangle, you have to instantiate a Rectangle object and assign values
to the attributes:
box = Rectangle()
box.width = 100.0
box.height = 200.0
box.corner = Point()
box.corner.x = 0.0
box.corner.y = 0.0
The expression box.corner.x means, “Go to the object box refers to and select the
attribute named corner; then go to that object and select the attribute named x.”
Figure 15-2 shows the state of this object. An object that is an attribute of another
object is embedded .
Figure 15-2. Object diagram.
180 | Chapter 15: Classes and Objects

Instances as Return ValuesFunctions can return instances. For example, find_center takes a Rectangle as an
argument and returns a Point that contains the coordinates of the center of the
Rectangle :
def find_center(rect):
p = Point()
p.x = rect.corner.x + rect.width/2
p.y = rect.corner.y + rect.height/2
return p
Here is an example that passes box as an argument and assigns the resulting Point to
center :
>>> center = find_center(box)
>>> print_point(center)
(50, 100)
Objects Are Mutable
You can change the state of an object by making an assignment to one of its attributes. For example, to change the size of a rectangle without changing its posi
tion, you can modify the values of width and height :
box.width = box.width + 50
box.height = box.height + 100
You can also write functions that modify objects. For example, grow_rectangle takes
a Rectangle object and two numbers, dwidth and dheight , and adds the numbers to
the width and height of the rectangle:
def grow_rectangle(rect, dwidth, dheight):
rect.width += dwidth
rect.height += dheight
Here is an example that demonstrates the effect:
>>> box.width, box.height
(150.0, 300.0)
>>> grow_rectangle(box, 50, 100)
>>> box.width, box.height
(200.0, 400.0)
Inside the function, rect is an alias for box, so when the function modifies rect, box
As an exercise, write a function named move_rectangle that takes a Rectangle and
two numbers named dx and dy. It should change the location of the rectangle by
adding dx to the x coordinate of corner and adding dy to the y coordinate of corner. Instances as Return Values | 181

CopyingAliasing can make a program difficult to read because changes in one place might
have unexpected effects in another place. It is hard to keep track of all the variables
that might refer to a given object.
Copying an object is often an alternative to aliasing. The copy module contains a
function called copy that can duplicate any object:
>>> p1 = Point()
>>> p1.x = 3.0
>>> p1.y = 4.0
>>> import copy
>>> p2 = copy.copy(p1)
p1 and p2 contain the same data, but they are not the same Point:
>>> print_point(p1)
(3, 4)
>>> print_point(p2)
(3, 4)
>>> p1 is p2
>>> p1 == p2
The is operator indicates that p1 and p2 are not the same object, which is what we
expected. But you might have expected == to yield True because these points contain
the same data. In that case, you will be disappointed to learn that for instances, the
default behavior of the == operator is the same as the is operator; it checks object
identity, not object equivalence. Thats because for programmer-defined types,
Python doesnt know what should be considered equivalent. At least, not yet.
If you use copy.copy to duplicate a Rectangle, you will find that it copies the Rectan
gle object but not the embedded Point:
>>> box2 = copy.copy(box)
>>> box2 is box
>>> box2.corner is box.corner
Figure 15-3 shows what the object diagram looks like. This operation is called a shal
low copy because it copies the object and any references it contains, but not the
embedded objects.182 | Chapter 15: Classes and Objects

Figure 15-3. Object diagram.
For most applications, this is not what you want. In this example, invoking grow_rec
tangle on one of the Rectangles would not affect the other, but invoking move_rec
tangle on either would affect both! This behavior is confusing and error-prone.
Fortunately, the copy module provides a method named deepcopy that copies not
only the object but also the objects it refers to, and the objects they refer to, and so on.
You will not be surprised to learn that this operation is called a deep copy.
>>> box3 = copy.deepcopy(box)
>>> box3 is box
>>> box3.corner is box.corner
box3 and box are completely separate objects.
As an exercise, write a version of move_rectangle that creates and returns a new Rec
tangle instead of modifying the old one.
When you start working with objects, you are likely to encounter some new excep
tions. If you try to access an attribute that doesnt exist, you get an AttributeError:
>>> p = Point() >>> p.x = 3
>>> p.y = 4
>>> p.z
AttributeError: Point instance has no attribute z
If you are not sure what type an object is, you can ask:
>>> type(p)

You can also use isinstance to check whether an object is an instance of a class:
>>> isinstance(p, Point)
If you are not sure whether an object has a particular attribute, you can use the built-
in function hasattr:
Debugging | 183

>>> hasattr(p, x)
>>> hasattr(p, z)
The first argument can be any object; the second argument is a string that contains
the name of the attribute.
You can also use a try statement to see if the object has the attributes you need:
x = p.x
except AttributeError:
x = 0
This approach can make it easier to write functions that work with different types;
more on that topic is coming up in “Polymorphism” on page 202.
class: A programmer-defined type. A class definition creates a new class object.
class object: An object that contains information about a programmer-defined type. The class
object can be used to create instances of the type.
instance: An object that belongs to a class.
instantiate: To create a new object.
attribute: One of the named values associated with an object.
embedded object: An object that is stored as an attribute of another object.
shallow copy: To copy the contents of an object, including any references to embedded objects;
implemented by the copy function in the copy module.
deep copy: To copy the contents of an object as well as any embedded objects, and any
objects embedded in them, and so on; implemented by the deepcopy function in
the copy module. 184 | Chapter 15: Classes and Objects

object diagram:A diagram that shows objects, their attributes, and the values of the attributes.
Exercises Exercise 15-1.
Write a definition for a class named Circle with attributes center and radius , where
center is a Point object and radius is a number.
Instantiate a Circle object that represents a circle with its center at 150, 100
radius 75.
Write a function named point_in_circle that takes a Circle and a Point and returns
True if the Point lies in or on the boundary of the circle.
Write a function named rect_in_circle that takes a Circle and a Rectangle and
returns True if the Rectangle lies entirely in or on the boundary of the circle.
Write a function named rect_circle_overlap that takes a Circle and a Rectangle
and returns True if any of the corners of the Rectangle fall inside the circle. Or as a
more challenging version, return True if any part of the Rectangle falls inside the circle.
Solution: .
Exercise 15-2.
Write a function called draw_rect that takes a Turtle object and a Rectangle and uses
the Turtle to draw the Rectangle. See Chapter 4 for examples using Turtle objects.
Write a function called draw_circle that takes a Turtle and a Circle and draws the
Solution: .
Exercises | 185

Classes and Functions
Now that we know how to create new types, the next step is to write functions that take programmer-defined objects as parameters and return them as results. In this
chapter I also present “functional programming style” and two new program develop
ment plans.
Code examples from this chapter are available from . Solutions to the exercises are at .
Time As another example of a programmer-defined type, well define a class called Time
that records the time of day. The class definition looks like this:
class Time:
"""Represents the time of day.
attributes: hour, minute, second """
We can create a new Time object and assign attributes for hours, minutes, and
time = Time()
time.hour = 11
time.minute = 59
time.second = 30
The state diagram for the Time object looks like Figure 16-1.

As an exercise, write a function called print_time that takes a Time object and prints
it in the form hour:minute:second . Hint: the format sequence %.2d prints an inte
ger using at least two digits, including a leading zero if necessary.
Write a boolean function called is_after that takes two Time objects, t1 and t2, and
returns True if t1 follows t2 chronologically and False otherwise. Challenge: dont
use an if statement.
Figure 16-1. Object diagram.
Pure Functions In the next few sections, well write two functions that add time values. They demon
strate two kinds of functions: pure functions and modifiers. They also demonstrate a
development plan Ill call prototype and patch, which is a way of tackling a complex
problem by starting with a simple prototype and incrementally dealing with the com
Here is a simple prototype of add_time:
def add_time(t1, t2):
sum = Time()
sum.hour = t1.hour + t2.hour
sum.minute = t1.minute + t2.minute
sum.second = t1.second + t2.second
return sum
The function creates a new Time object, initializes its attributes, and returns a refer
ence to the new object. This is called a pure function because it does not modify any
of the objects passed to it as arguments and it has no effect, like displaying a value or
getting user input, other than returning a value.
To test this function, Ill create two Time objects: start contains the start time of a
movie, like Monty Python and the Holy Grail , and duration contains the runtime of
the movie, which is 1 hour 35 minutes.
add_time figures out when the movie will be done:
>>> start = Time()
>>> start.hour = 9
>>> start.minute = 45
>>> start.second = 0
188 | Chapter 16: Classes and Functions

>>> duration = Time()
>>> duration.hour = 1
>>> duration.minute = 35
>>> duration.second = 0
>>> done = add_time(start, duration)
>>> print_time(done)
The result, 10:80:00, might not be what you were hoping for. The problem is that
this function does not deal with cases where the number of seconds or minutes adds
up to more than sixty. When that happens, we have to “carry” the extra seconds into
the minute column or the extra minutes into the hour column.
Heres an improved version: def add_time(t1, t2):
sum = Time()
sum.hour = t1.hour + t2.hour
sum.minute = t1.minute + t2.minute
sum.second = t1.second + t2.second
if sum.second >= 60:
sum.second -= 60
sum.minute += 1
if sum.minute >= 60:
sum.minute -= 60
sum.hour += 1
return sum
Although this function is correct, it is starting to get big. We will see a shorter alterna
tive later.
Sometimes it is useful for a function to modify the objects it gets as parameters. In
that case, the changes are visible to the caller. Functions that work this way are called
modifiers .
increment , which adds a given number of seconds to a Time object, can be written
naturally as a modifier. Here is a rough draft:
def increment(time, seconds):
time.second += seconds
if time.second >= 60:
time.second -= 60
time.minute += 1
Modifiers | 189

if time.minute >= 60:
time.minute -= 60
time.hour += 1
The first line performs the basic operation; the remainder deals with the special cases
we saw before.
Is this function correct? What happens if seconds is much greater than 60?
In that case, it is not enough to carry once; we have to keep doing it until time.sec
ond is less than 60. One solution is to replace the if statements with while state
ments. That would make the function correct, but not very efficient. As an exercise,
write a correct version of increment that doesnt contain any loops.
Anything that can be done with modifiers can also be done with pure functions. In fact, some programming languages only allow pure functions. There is some evidence
that programs that use pure functions are faster to develop and less error-prone than
programs that use modifiers. But modifiers are convenient at times, and functional
programs tend to be less efficient.
In general, I recommend that you write pure functions whenever it is reasonable and resort to modifiers only if there is a compelling advantage. This approach might be
called a functional programming style .
As an exercise, write a “pure” version of increment that creates and returns a new
Time object rather than modifying the parameter.
Prototyping versus Planning
The development plan I am demonstrating is called “prototype and patch”. For each function, I wrote a prototype that performed the basic calculation and then tested it,
patching errors along the way.
This approach can be effective, especially if you dont yet have a deep understanding
of the problem. But incremental corrections can generate code that is unnecessarily
complicated (since it deals with many special cases) and unreliable (since it is hard to know if you have found all the errors).
An alternative is designed development , in which high-level insight into the problem
can make the programming much easier. In this case, the insight is that a Time object
is really a three-digit number in base 60 (see
mal .)! The second attribute is the “ones column”, the minute attribute is the “sixties
column”, and the hour attribute is the “thirty-six hundreds column”.
When we wrote add_time and increment , we were effectively doing addition in base
60, which is why we had to carry from one column to the next.
190 | Chapter 16: Classes and Functions

This observation suggests another approach to the whole problemwe can convert
Time objects to integers and take advantage of the fact that the computer knows how to do integer arithmetic.
Here is a function that converts Times to integers: def time_to_int(time):
minutes = time.hour * 60 + time.minute
seconds = minutes * 60 + time.second return seconds
And here is a function that converts an integer to a Time (recall that divmod divides
the first argument by the second and returns the quotient and remainder as a tuple):
def int_to_time(seconds):
time = Time()
minutes, time.second = divmod(seconds, 60)
time.hour, time.minute = divmod(minutes, 60)
return time
You might have to think a bit, and run some tests, to convince yourself that these
functions are correct. One way to test them is to check that
time_to_int(int_to_time(x)) == x for many values of x. This is an example of a
consistency check.
Once you are convinced they are correct, you can use them to rewrite add_time:
def add_time(t1, t2):
seconds = time_to_int(t1) + time_to_int(t2) return int_to_time(seconds)
This version is shorter than the original, and easier to verify. As an exercise, rewrite
increment using time_to_int and int_to_time .
In some ways, converting from base 60 to base 10 and back is harder than just dealing with times. Base conversion is more abstract; our intuition for dealing with time val
ues is better.
But if we have the insight to treat times as base 60 numbers and make the investment
of writing the conversion functions ( time_to_int and int_to_time ), we get a pro
gram that is shorter, easier to read and debug, and more reliable.
It is also easier to add features later. For example, imagine subtracting two Times to find the duration between them. The naive approach would be to implement subtrac
tion with borrowing. Using the conversion functions would be easier and more likely
to be correct.
Ironically, sometimes making a problem harder (or more general) makes it easier (because there are fewer special cases and fewer opportunities for error). Prototyping versus Planning | 191

A Time object is well-formed if the values of minute and second are between 0 and 60
(including 0 but not 60) and if hour is positive. hour and minute should be integral
values, but we might allow second to have a fraction part.
Requirements like these are called invariants because they should always be true. To
put it a different way, if they are not true, something has gone wrong.
Writing code to check invariants can help detect errors and find their causes. For
example, you might have a function like valid_time that takes a Time object and
returns False if it violates an invariant:
def valid_time(time):
if time.hour < 0 or time.minute < 0 or time.second < 0:
return False
if time.minute >= 60 or time.second >= 60:
return False
return True
At the beginning of each function you could check the arguments to make sure they are valid:
def add_time(t1, t2):
if not valid_time(t1) or not valid_time(t2):
raise ValueError(invalid Time object in add_time)
seconds = time_to_int(t1) + time_to_int(t2) return int_to_time(seconds)
Or you could use an assert statement, which checks a given invariant and raises an
exception if it fails:
def add_time(t1, t2):
assert valid_time(t1) and valid_time(t2) seconds = time_to_int(t1) + time_to_int(t2) return int_to_time(seconds)
assert statements are useful because they distinguish code that deals with normal
conditions from code that checks for errors.
prototype and patch: A development plan that involves writing a rough draft of a program, testing, andcorrecting errors as they are found.
designed development: A development plan that involves high-level insight into the problem and moreplanning than incremental development or prototype development.192 | Chapter 16: Classes and Functions

pure function:A function that does not modify any of the objects it receives as arguments. Mostpure functions are fruitful.
A function that changes one or more of the objects it receives as arguments. Most
modifiers are void; that is, they return None.
functional programming style: A style of program design in which the majority of functions are pure.
invariant: A condition that should always be true during the execution of a program.
assert statement: A statement that check a condition and raises an exception if it fails.
Code examples from this chapter are available from ; solutions to the exercises are available from .
Exercise 16-1.
Write a function called mul_time that takes a Time object and a number and returns a
new Time object that contains the product of the original Time and the number.
Then use mul_time to write a function that takes a Time object that represents the
finishing time in a race, and a number that represents the distance, and returns a
Time object that represents the average pace (time per mile).
Exercise 16-2.
The datetime module provides time objects that are similar to the Time objects in
this chapter, but they provide a rich set of methods and operators. Read the docu
mentation at .
Use the datetime module to write a program that gets the current date and prints
the day of the week.
Write a program that takes a birthday as input and prints the users age and the
number of days, hours, minutes and seconds until their next birthday.
Exercises | 193

3. For two people born on different days, there is a day when one is twice as old asthe other. Thats their Double Day. Write a program that takes two birthdays and
computes their Double Day. 4.
For a little more challenge, write the more general version that computes the day when one person is n times older than the other.
Solution: .
194 | Chapter 16: Classes and Functions

Classes and Methods
Although we are using some of Pythons object-oriented features, the programs from the last two chapters are not really object-oriented because they dont represent the
relationships between programmer-defined types and the functions that operate on
them. The next step is to transform those functions into methods that make the rela tionships explicit.
Code examples from this chapter are available from , and solutions to the exercises are in .
Object-Oriented Features Python is an object-oriented programming language , which means that it provides
features that support object-oriented programming, which has these defining charac teristics:

Programs include class and method definitions.

Most of the computation is expressed in terms of operations on objects.

Objects often represent things in the real world, and methods often correspondto the ways things in the real world interact.
For example, the Time class defined in Chapter 16 corresponds to the way people
record the time of day, and the functions we defined correspond to the kinds of
things people do with times. Similarly, the Point and Rectangle classes in Chapter 15
correspond to the mathematical concepts of a point and a rectangle.
So far, we have not taken advantage of the features Python provides to support
object-oriented programming. These features are not strictly necessary; most of them

provide alternative syntax for things we have already done. But in many cases, the
alternative is more concise and more accurately conveys the structure of the program.
For example, in there is no obvious connection between the class definition
and the function definitions that follow. With some examination, it is apparent that
every function takes at least one Time object as an argument.
This observation is the motivation for methods; a method is a function that is associ
ated with a particular class. We have seen methods for strings, lists, dictionaries and
tuples. In this chapter, we will define methods for programmer-defined types.
Methods are semantically the same as functions, but there are two syntactic differ ences:
Methods are defined inside a class definition in order to make the relationshipbetween the class and the method explicit.

The syntax for invoking a method is different from the syntax for calling a function.
In the next few sections, we will take the functions from the previous two chapters
and transform them into methods. This transformation is purely mechanical; you can
do it by following a sequence of steps. If you are comfortable converting from one
form to another, you will be able to choose the best form for whatever you are doing.
Printing Objects
In Chapter 16 , we defined a class named Time and in “Time” on page 187 , you wrote a
function named print_time:
class Time:
"""Represents the time of day."""
def print_time(time):
print(%.2d:%.2d:%.2d % (time.hour, time.minute, time.second))
To call this function, you have to pass a Time object as an argument:
>>> start = Time() >>> start.hour = 9
>>> start.minute = 45
>>> start.second = 00
>>> print_time(start)
To make print_time a method, all we have to do is move the function definition
inside the class definition. Notice the change in indentation.
196 | Chapter 17: Classes and Methods

class Time:
def print_time(time): print(%.2d:%.2d:%.2d % (time.hour, time.minute, time.second))
Now there are two ways to call print_time. The first (and less common) way is to use
function syntax:
>>> Time.print_time(start)
In this use of dot notation, Time is the name of the class, and print_time is the name
of the method. start is passed as a parameter.
The second (and more concise) way is to use method syntax: >>> start.print_time()
In this use of dot notation, print_time is the name of the method (again), and start
is the object the method is invoked on, which is called the subject. Just as the subject
of a sentence is what the sentence is about, the subject of a method invocation is what
the method is about.
Inside the method, the subject is assigned to the first parameter, so in this case start
is assigned to time.
By convention, the first parameter of a method is called self, so it would be more
common to write print_time like this:
class Time:
def print_time(self):
print(%.2d:%.2d:%.2d % (self.hour, self.minute, self.second))
The reason for this convention is an implicit metaphor:
The syntax for a function call, print_time(start), suggests that the function is
the active agent. It says something like, “Hey print_time! Heres an object for
you to print.”

In object-oriented programming, the objects are the active agents. A method
invocation like start.print_time() says “Hey start! Please print yourself.”
This change in perspective might be more polite, but it is not obvious that it is useful.
In the examples we have seen so far, it may not be. But sometimes shifting responsi
bility from the functions onto the objects makes it possible to write more versatile
functions (or methods), and makes it easier to maintain and reuse code.
As an exercise, rewrite time_to_int (from “Prototyping versus Planning” on page
190 ) as a method. You might be tempted to rewrite int_to_time as a method, too,
but that doesnt really make sense because there would be no object to invoke it on.
Printing Objects | 197

Another Example
Heres a version of increment (from “Modifiers” on page 189 ) rewritten as a method:
# inside class Time:
def increment(self, seconds): seconds += self.time_to_int() return int_to_time(seconds)
This version assumes that time_to_int is written as a method. Also, note that it is a
pure function, not a modifier.
Heres how you would invoke increment:
>>> start.print_time() 09:45:00
>>> end = start.increment(1337)
>>> end.print_time()
The subject, start, gets assigned to the first parameter, self. The argument, 1337,
gets assigned to the second parameter, seconds.
This mechanism can be confusing, especially if you make an error. For example, if
you invoke increment with two arguments, you get:
>>> end = start.increment(1337, 460)
TypeError: increment() takes 2 positional arguments but 3 were given
The error message is initially confusing, because there are only two arguments in parentheses. But the subject is also considered an argument, so all together thats
By the way, a positional argument is an argument that doesnt have a parameter
name; that is, it is not a keyword argument. In this function call:
sketch(parrot, cage, dead=True)
parrot and cage are positional, and dead is a keyword argument.
A More Complicated Example
Rewriting is_after (from “Time” on page 187) is slightly more complicated because
it takes two Time objects as parameters. In this case it is conventional to name the
first parameter self and the second parameter other:
# inside class Time:
def is_after(self, other):
return self.time_to_int() > other.time_to_int()198 | Chapter 17: Classes and Methods

To use this method, you have to invoke it on one object and pass the other as an
>>> end.is_after(start)
One nice thing about this syntax is that it almost reads like English: “end is after
The init Method
The init method (short for “initialization”) is a special method that gets invoked when
an object is instantiated. Its full name is __init__ (two underscore characters, fol
lowed by init, and then two more underscores). An init method for the Time class
might look like this:
# inside class Time:
def __init__(self, hour=0, minute=0, second=0):
self.hour = hour
self.minute = minute
self.second = second
It is common for the parameters of __init__ to have the same names as the
attributes. The statement
self.hour = hour
stores the value of the parameter hour as an attribute of self.
The parameters are optional, so if you call Time with no arguments, you get the
default values:
>>> time = Time()
>>> time.print_time()
If you provide one argument, it overrides hour:
>>> time = Time (9)
>>> time.print_time()
If you provide two arguments, they override hour and minute :
>>> time = Time(9, 45)
>>> time.print_time()
And if you provide three arguments, they override all three default values.
As an exercise, write an init method for the Point class that takes x and y as optional
parameters and assigns them to the corresponding attributes. The init Method | 199

The __str__ Method
__str__ is a special method, like __init__, that is supposed to return a string repre
sentation of an object.
For example, here is a str method for Time objects:
# inside class Time:
def __str__(self):
return %.2d:%.2d:%.2d % (self.hour, self.minute, self.second)
When you print an object, Python invokes the str method:
>>> time = Time(9, 45)
>>> print(time)
When I write a new class, I almost always start by writing __init__, which makes it
easier to instantiate objects, and __str__, which is useful for debugging.
As an exercise, write a str method for the Point class. Create a Point object and print
Operator Overloading
By defining other special methods, you can specify the behavior of operators on
programmer-defined types. For example, if you define a method named __add__ for
the Time class, you can use the + operator on Time objects.
Here is what the definition might look like: # inside class Time:
def __add__(self, other):
seconds = self.time_to_int() + other.time_to_int() return int_to_time(seconds)
And here is how you could use it:
>>> start = Time(9, 45)
>>> duration = Time(1, 35)
>>> print(start + duration)
When you apply the + operator to Time objects, Python invokes __add__. When you
print the result, Python invokes __str__. So there is a lot happening behind the
Changing the behavior of an operator so that it works with programmer-defined types is called operator overloading . For every operator in Python there is a corre200 | Chapter 17: Classes and Methods

sponding special method, like __add__. For more details, see
reference/datamodel.html#specialnames .
As an exercise, write an add method for the Point class.
Type-Based Dispatch In the previous section we added two Time objects, but you also might want to add
an integer to a Time object. The following is a version of __add__ that checks the type
of other and invokes either add_time or increment :
# inside class Time:
def __add__(self, other):
if isinstance(other, Time):
return self.add_time(other)
return self.increment(other)
def add_time(self, other): seconds = self.time_to_int() + other.time_to_int() return int_to_time(seconds)
def increment(self, seconds): seconds += self.time_to_int() return int_to_time(seconds)
The built-in function isinstance takes a value and a class object, and returns True if
the value is an instance of the class.
If other is a Time object, __add__ invokes add_time . Otherwise it assumes that the
parameter is a number and invokes increment. This operation is called a type-based
dispatch because it dispatches the computation to different methods based on the
type of the arguments.
Here are examples that use the + operator with different types:
>>> start = Time(9, 45) >>> duration = Time(1, 35)
>>> print(start + duration)
>>> print(start + 1337)
Unfortunately, this implementation of addition is not commutative. If the integer is
the first operand, you get
>>> print(1337 + start)
TypeError: unsupported operand type(s) for +: int and instance Type-Based Dispatch | 201

The problem is, instead of asking the Time object to add an integer, Python is asking
an integer to add a Time object, and it doesnt know how. But there is a clever solu
tion for this problem: the special method __radd__, which stands for “right-side add”.
This method is invoked when a Time object appears on the right side of the + opera
tor. Heres the definition:
# inside class Time:
def __radd__(self, other): return self.__add__(other)
And heres how its used:
>>> print(1337 + start)
As an exercise, write an add method for Points that works with either a Point object
or a tuple:
If the second operand is a Point, the method should return a new Point whose x
coordinate is the sum of the x coordinates of the operands, and likewise for the y

If the second operand is a tuple, the method should add the first element of the tuple to the x coordinate and the second element to the y coordinate, and return
a new Point with the result.
Type-based dispatch is useful when it is necessary, but (fortunately) it is not always necessary. Often you can avoid it by writing functions that work correctly for argu
ments with different types.
Many of the functions we wrote for strings also work for other sequence types. For
example, in “Dictionary as a Collection of Counters” on page 127 we used histogram
to count the number of times each letter appears in a word:
def histogram(s):
d = dict()
for c in s:
if c not in d:
d[c] = 1
d[c] = d[c]+1
return d
This function also works for lists, tuples, and even dictionaries, as long as the ele
ments of s are hashable, so they can be used as keys in d:
202 | Chapter 17: Classes and Methods

>>> t = [spam, egg, spam, spam, bacon, spam]>>> histogram(t)
{bacon: 1, egg: 1, spam: 4}
Functions that work with several types are called polymorphic. Polymorphism can
facilitate code reuse. For example, the built-in function sum, which adds the elements
of a sequence, works as long as the elements of the sequence support addition.
Since Time objects provide an add method, they work with sum:
>>> t1 = Time(7, 43)
>>> t2 = Time(7, 41)
>>> t3 = Time(7, 37)
>>> total = sum([t1, t2, t3])
>>> print(total)
In general, if all of the operations inside a function work with a given type, the func
tion works with that type.
The best kind of polymorphism is the unintentional kind, where you discover that a
function you already wrote can be applied to a type you never planned for.
Interface and Implementation One of the goals of object-oriented design is to make software more maintainable,
which means that you can keep the program working when other parts of the system change, and modify the program to meet new requirements.
A design principle that helps achieve that goal is to keep interfaces separate from
implementations. For objects, that means that the methods a class provides should
not depend on how the attributes are represented.
For example, in this chapter we developed a class that represents a time of day. Meth
ods provided by this class include time_to_int, is_after , and add_time .
We could implement those methods in several ways. The details of the implementa tion depend on how we represent time. In this chapter, the attributes of a Time object
are hour , minute , and second .
As an alternative, we could replace these attributes with a single integer representing
the number of seconds since midnight. This implementation would make some
methods, like is_after, easier to write, but it makes other methods harder.
After you deploy a new class, you might discover a better implementation. If other
parts of the program are using your class, it might be time-consuming and error-
prone to change the interface. Interface and Implementation | 203

But if you designed the interface carefully, you can change the implementation
without changing the interface, which means that other parts of the program dont have to change.
Debugging It is legal to add attributes to objects at any point in the execution of a program, but if
you have objects with the same type that dont have the same attributes, it is easy to
make mistakes. It is considered a good idea to initialize all of an objects attributes in the init method.
If you are not sure whether an object has a particular attribute, you can use the built-
in function hasattr (see “Debugging” on page 183 ).
Another way to access attributes is the built-in function vars, which takes an object
and returns a dictionary that maps from attribute names (as strings) to their values:
>>> p = Point(3, 4)
>>> vars(p)
{y: 4, x: 3}
For purposes of debugging, you might find it useful to keep this function handy:
def print_attributes(obj):
for attr in vars(obj):
print(attr, getattr(obj, attr))
print_attributes traverses the dictionary and prints each attribute name and its
corresponding value.
The built-in function getattr takes an object and an attribute name (as a string) and
returns the attributes value.
Glossary object-oriented language: A language that provides features, such as programmer-defined types and methods, that facilitate object-oriented programming.
object-oriented programming: A style of programming in which data and the operations that manipulate it are
organized into classes and methods.
method: A function that is defined inside a class definition and is invoked on instances ofthat class.204 | Chapter 17: Classes and Methods

subject:The object a method is invoked on.
positional argument: An argument that does not include a parameter name, so it is not a keywordargument.
operator overloading: Changing the behavior of an operator like + so it works with a programmer-
defined type.
type-based dispatch: A programming pattern that checks the type of an operand and invokes differentfunctions for different types.
polymorphic: Pertaining to a function that can work with more than one type.
information hiding: The principle that the interface provided by an object should not depend on itsimplementation, in particular the representation of its attributes.
Exercises Exercise 17-1.
Download the code from this chapter from
Change the attributes of Time to be a single integer representing seconds since mid
night. Then modify the methods (and the function int_to_time) to work with the
new implementation. You should not have to modify the test code in main. When you
are done, the output should be the same as before.
Exercise 17-2.
This exercise is a cautionary tale about one of the most common, and difficult to find,
errors in Python. Write a definition for a class named Kangaroo with the following
An __init__ method that initializes an attribute named pouch_contents to an
empty list.
A method named put_in_pouch that takes an object of any type and adds it to
pouch_contents .
Exercises | 205

3. A __str__ method that returns a string representation of the Kangaroo object
and the contents of the pouch.
Test your code by creating two Kangaroo objects, assigning them to variables named
kanga and roo, and then adding roo to the contents of kangas pouch.
Download . It contains a solution to the
previous problem with one big, nasty bug. Find and fix the bug.
If you get stuck, you can download ,
which explains the problem and demonstrates a solution. 206 | Chapter 17: Classes and Methods

The language feature most often associated with object-oriented programming is inheritance . Inheritance is the ability to define a new class that is a modified version
of an existing class. In this chapter I demonstrate inheritance using classes that repre
sent playing cards, decks of cards, and poker hands.
If you dont play poker, you can read about it at,
but you dont have to; Ill tell you what you need to know for the exercises.
Code examples from this chapter are available from .
Card Objects
There are 52 cards in a deck, each of which belongs to 1 of 4 suits and 1 of 13 ranks. The suits are Spades, Hearts, Diamonds, and Clubs (in descending order in bridge).
The ranks are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. Depending on the
game that you are playing, an Ace may be higher than King or lower than 2.
If we want to define a new object to represent a playing card, it is obvious what the
attributes should be: rank and suit. It is not as obvious what type the attributes
should be. One possibility is to use strings containing words like Spade for suits
and Queen for ranks. One problem with this implementation is that it would not be
easy to compare cards to see which had a higher rank or suit.
An alternative is to use integers to encode the ranks and suits. In this context,
“encode” means that we are going to define a mapping between numbers and suits, or between numbers and ranks. This kind of encoding is not meant to be a secret (that
would be “encryption”).

For example, this table shows the suits and the corresponding integer codes:Spades
This code makes it easy to compare cards; because higher suits map to higher num
bers, we can compare suits by comparing their codes.
The mapping for ranks is fairly obvious; each of the numerical ranks maps to the cor responding integer, and for face cards:
I am using the , symbol to make it clear that these mappings are not part of the
Python program. They are part of the program design, but they dont appear explic itly in the code.
The class definition for Card looks like this:
class Card:
"""Represents a standard playing card."""
def __init__(self, suit=0, rank=2): self.suit = suit
self.rank = rank
As usual, the init method takes an optional parameter for each attribute. The default
card is the 2 of Clubs.
To create a Card, you call Card with the suit and rank of the card you want:
queen_of_diamonds = Card(1, 12)
Class Attributes
In order to print Card objects in a way that people can easily read, we need a mapping from the integer codes to the corresponding ranks and suits. A natural way to do that
is with lists of strings. We assign these lists to class attributes:
208 | Chapter 18: Inheritance

# inside class Card:
suit_names = [Clubs, Diamonds, Hearts, Spades] rank_names = [None, Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King]
def __str__(self):
return %s of %s % (Card.rank_names[self.rank],
Variables like suit_names and rank_names , which are defined inside a class but out
side of any method, are called class attributes because they are associated with the
class object Card.
This term distinguishes them from variables like suit and rank, which are called
instance attributes because they are associated with a particular instance.
Both kinds of attribute are accessed using dot notation. For example, in __str__,
self is a Card object, and self.rank is its rank. Similarly, Card is a class object, and
Card.rank_names is a list of strings associated with the class.
Every card has its own suit and rank, but there is only one copy of suit_names and
rank_names .
Putting it all together, the expression Card.rank_names[self.rank] means “use the
attribute rank from the object self as an index into the list rank_names from the
class Card, and select the appropriate string.”
The first element of rank_names is None because there is no card with rank zero. By
including None as a place-keeper, we get a mapping with the nice property that the
index 2 maps to the string 2, and so on. To avoid this tweak, we could have used a
dictionary instead of a list.
With the methods we have so far, we can create and print cards: >>> card1 = Card(2, 11)
>>> print(card1)
Jack of Hearts
Figure 18-1 is a diagram of the Card class object and one Card instance. Card is a class
object; its type is type. card1 is an instance of Card, so its type is Card. To save space,
I didnt draw the contents of suit_names and rank_names . Class Attributes | 209

Figure 18-1. Object diagram.
Comparing Cards For built-in types, there are relational operators ( <, >, == , etc.) that compare values
and determine when one is greater than, less than, or equal to another. For
programmer-defined types, we can override the behavior of the built-in operators by
providing a method named __lt__, which stands for “less than”.
__lt__ takes two parameters, self and other , and True if self is strictly less than
other .
The correct ordering for cards is not obvious. For example, which is better, the 3 of
Clubs or the 2 of Diamonds? One has a higher rank, but the other has a higher suit. In order to compare cards, you have to decide whether rank or suit is more impor
The answer might depend on what game you are playing, but to keep things simple,
well make the arbitrary choice that suit is more important, so all of the Spades out rank all of the Diamonds, and so on.
With that decided, we can write __lt__:
# inside class Card:
def __lt__(self, other):
# check the suits
if self.suit < other.suit: return True
if self.suit > other.suit: return False
# suits are the same... check ranks
return self.rank < other.rank
You can write this more concisely using tuple comparison:
210 | Chapter 18: Inheritance

# inside class Card:
def __lt__(self, other):
t1 = self.suit, self.rank
t2 = other.suit, other.rank
return t1 < t2
As an exercise, write an __lt__ method for Time objects. You can use tuple compari
son, but you also might consider comparing integers.
Decks Now that we have Cards, the next step is to define Decks. Since a deck is made up of
cards, it is natural for each Deck to contain a list of cards as an attribute.
The following is a class definition for Deck. The init method creates the attribute
cards and generates the standard set of 52 cards:
class Deck:
def __init__(self): = []
for suit in range(4):
for rank in range(1, 14):
card = Card(suit, rank)
The easiest way to populate the deck is with a nested loop. The outer loop enumerates the suits from 0 to 3. The inner loop enumerates the ranks from 1 to 13. Each itera
tion creates a new Card with the current suit and rank, and appends it to
Printing the Deck
Here is a __str__ method for Deck:
#inside class Deck:
def __str__(self):
res = []
for card in
return \n.join(res)
This method demonstrates an efficient way to accumulate a large string: building a
list of strings and then using the string method join. The built-in function str
invokes the __str__ method on each card and returns the string representation.
Since we invoke join on a newline character, the cards are separated by newlines.
Heres what the result looks like: Decks | 211

>>> deck = Deck()
>>> print(deck)
Ace of Clubs
2 of Clubs
3 of Clubs
10 of Spades
Jack of Spades
Queen of Spades
King of Spades
Even though the result appears on 52 lines, it is one long string that contains new lines.
Add, Remove,
Shuffle and Sort
To deal cards, we would like a method that removes a card from the deck and returns
it. The list method pop provides a convenient way to do that:
#inside class Deck:
def pop_card(self):
Since pop removes the last card in the list, we are dealing from the bottom of the
To add a card, we can use the list method append:
#inside class Deck:
def add_card(self, card):
A method like this that uses another method without doing much work is sometimes called a veneer. The metaphor comes from woodworking, where a veneer is a thin
layer of good quality wood glued to the surface of a cheaper piece of wood to improve
the appearance.
In this case add_card is a “thin” method that expresses a list operation in terms
appropriate for decks. It improves the appearance, or interface, of the implementa
As another example, we can write a Deck method named shuffle using the function
shuffle from the random module:
# inside class Deck:

def shuffle(self):
212 | Chapter 18: Inheritance

Dont forget to import random.
As an exercise, write a Deck method named sort that uses the list method sort to
sort the cards in a Deck. sort uses the __lt__ method we defined to determine the
Inheritance Inheritance is the ability to define a new class that is a modified version of an existing
class. As an example, lets say we want a class to represent a “hand”, that is, the cards
held by one player. A hand is similar to a deck: both are made up of a collection of
cards, and both require operations like adding and removing cards.
A hand is also different from a deck; there are operations we want for hands that
dont make sense for a deck. For example, in poker we might compare two hands to
see which one wins. In bridge, we might compute a score for a hand in order to make
a bid.
This relationship between classessimilar, but differentlends itself to inheritance. To define a new class that inherits from an existing class, you put the name of the
existing class in parentheses:
class Hand(Deck):
"""Represents a hand of playing cards."""
This definition indicates that Hand inherits from Deck; that means we can use meth
ods like pop_card and add_card for Hands as well as Decks.
When a new class inherits from an existing one, the existing one is called the parent
and the new class is called the child.
In this example, Hand inherits __init__ from Deck, but it doesnt really do what we
want: instead of populating the hand with 52 new cards, the init method for Hands
should initialize cards with an empty list.
If we provide an init method in the Hand class, it overrides the one in the Deck class:
# inside class Hand:
def __init__(self, label=): = []
self.label = label
When you create a Hand, Python invokes this init method, not the one in Deck.
>>> hand = Hand(new hand) >>>
>>> hand.label
new hand Inheritance | 213

The other methods are inherited from Deck, so we can use pop_card and add_card to
deal a card:
>>> deck = Deck()
>>> card = deck.pop_card()
>>> hand.add_card(card)
>>> print(hand)
King of Spades
A natural next step is to encapsulate this code in a method called move_cards:
#inside class Deck:
def move_cards(self, hand, num): for i in range(num):
move_cards takes two arguments, a Hand object and the number of cards to deal. It
modifies both self and hand, and returns None.
In some games, cards are moved from one hand to another, or from a hand back to the deck. You can use move_cards for any of these operations: self can be either a
Deck or a Hand, and hand, despite the name, can also be a Deck.
Inheritance is a useful feature. Some programs that would be repetitive without
inheritance can be written more elegantly with it. Inheritance can facilitate code
reuse, since you can customize the behavior of parent classes without having to mod ify them. In some cases, the inheritance structure reflects the natural structure of the
problem, which makes the design easier to understand.
On the other hand, inheritance can make programs difficult to read. When a method
is invoked, it is sometimes not clear where to find its definition. The relevant code may be spread across several modules. Also, many of the things that can be done
using inheritance can be done as well or better without it.
Class Diagrams So far we have seen stack diagrams, which show the state of a program, and object
diagrams, which show the attributes of an object and their values. These diagrams represent a snapshot in the execution of a program, so they change as the program
They are also highly detailed; for some purposes, too detailed. A class diagram is a
more abstract representation of the structure of a program. Instead of showing indi
vidual objects, it shows classes and the relationships between them.
There are several kinds of relationship between classes:214 | Chapter 18: Inheritance

Objects in one class might contain references to objects in another class. Forexample, each Rectangle contains a reference to a Point, and each Deck contains
references to many Cards. This kind of relationship is called HAS-A, as in, “a
Rectangle has a Point.”
One class might inherit from another. This relationship is called IS-A, as in, “a
Hand is a kind of a Deck.”

One class might depend on another in the sense that objects in one class take objects in the second class as parameters, or use objects in the second class as
part of a computation. This kind of relationship is called a dependency.
A class diagram is a graphical representation of these relationships. For example,
Figure 18-2 shows the relationships between Card, Deck and Hand.
Figure 18-2. Class diagram.
The arrow with a hollow triangle head represents an IS-A relationship; in this case it indicates that Hand inherits from Deck.
The standard arrowhead represents a HAS-A relationship; in this case a Deck has ref
erences to Card objects.
The star ( *) near the arrowhead is a multiplicity; it indicates how many Cards a Deck
has. A multiplicity can be a simple number like 52, a range like 5..7 or a star, which
indicates that a Deck can have any number of Cards.
There are no dependencies in this diagram. They would normally be shown with a dashed arrow. Or if there are a lot of dependencies, they are sometimes omitted.
A more detailed diagram might show that a Deck actually contains a list of Cards, but
built-in types like list and dict are usually not included in class diagrams.
Data Encapsulation The previous chapters demonstrate a development plan we might call “object-
oriented design”. We identified objects we neededlike Point, Rectangle and Time
and defined classes to represent them. In each case there is an obvious correspond
Data Encapsulation | 215

ence between the object and some entity in the real world (or at least a mathematical
But sometimes it is less obvious what objects you need and how they should interact. In that case you need a different development plan. In the same way that we discov
ered function interfaces by encapsulation and generalization, we can discover class
interfaces by data encapsulation .
Markov analysis, from “Markov Analysis” on page 158, provides a good example. If
you download my code from , youll see that
it uses two global variables suffix_map and prefix that are read and written from
several functions.
suffix_map = {}
prefix = ()
Because these variables are global, we can only run one analysis at a time. If we read
two texts, their prefixes and suffixes would be added to the same data structures
(which makes for some interesting generated text).
To run multiple analyses, and keep them separate, we can encapsulate the state of each analysis in an object. Heres what that looks like:
class Markov:
def __init__(self):
self.suffix_map = {}
self.prefix = ()
Next, we transform the functions into methods. For example, heres process_word:
def process_word(self, word, order=2):
if len(self.prefix) < order:
self.prefix += (word,)
except KeyError:
# if there is no entry for this prefix, make one self.suffix_map[self.prefix] = [word]
self.prefix = shift(self.prefix, word)
Transforming a program like thischanging the design without changing the behav
ioris another example of refactoring (see “Refactoring” on page 41).
This example suggests a development plan for designing objects and methods:1.
Start by writing functions that read and write global variables (when necessary).
216 | Chapter 18: Inheritance

2. Once you get the program working, look for associations between global variables and the functions that use them. 3.
Encapsulate related variables as attributes of an object.
Transform the associated functions into methods of the new class.
As an exercise, download my Markov code from , and follow the steps described above to encapsulate the global variables as
attributes of a new class called Markov.
Solution: (note the capital M).
Inheritance can make debugging difficult because when you invoke a method on an
object, it might be hard to figure out which method will be invoked.
Suppose you are writing a function that works with Hand objects. You would like it to
work with all kinds of Hands, like PokerHands, BridgeHands, etc. If you invoke a
method like shuffle, you might get the one defined in Deck, but if any of the sub
classes override this method, youll get that version instead. This behavior is usually a
good thing, but it can be confusing.
Any time you are unsure about the flow of execution through your program, the sim
plest solution is to add print statements at the beginning of the relevant methods. If
Deck.shuffle prints a message that says something like Running Deck.shuffle, then
as the program runs it traces the flow of execution.
As an alternative, you could use this function, which takes an object and a method name (as a string) and returns the class that provides the definition of the method:
def find_defining_class(obj, meth_name):
for ty in type(obj).mro():
if meth_name in ty.__dict__:
return ty
Heres an example:
>>> hand = Hand()
>>> find_defining_class(hand, shuffle)

So the shuffle method for this Hand is the one in Deck.
find_defining_class uses the mro method to get the list of class objects (types) that
will be searched for methods. “MRO” stands for “method resolution order”, which is
the sequence of classes Python searches to “resolve” a method name.
Debugging | 217

Heres a design suggestion: when you override a method, the interface of the new
method should be the same as the old. It should take the same parameters, return the same type, and obey the same preconditions and postconditions. If you follow this
rule, you will find that any function designed to work with an instance of a parent
class, like a Deck, will also work with instances of child classes like a Hand and Poker
If you violate this rule, which is called the “Liskov substitution principle”, your code
will collapse like (sorry) a house of cards.
Glossary encode: To represent one set of values using another set of values by constructing a map
ping between them.
class attribute: An attribute associated with a class object. Class attributes are defined inside aclass definition but outside any method.
instance attribute: An attribute associated with an instance of a class.
veneer: A method or function that provides a different interface to another functionwithout doing much computation.
inheritance: The ability to define a new class that is a modified version of a previously defined
parent class: The class from which a child class inherits.
child class: A new class created by inheriting from an existing class; also called a “subclass”.
IS-A relationship: A relationship between a child class and its parent class.
HAS-A relationship: A relationship between two classes where instances of one class contain references to instances of the other.
dependency: A relationship between two classes where instances of one class use instances of
the other class, but do not store them as attributes.218 | Chapter 18: Inheritance

class diagram:A diagram that shows the classes in a program and the relationships betweenthem.
multiplicity: A notation in a class diagram that shows, for a HAS-A relationship, how many
references there are to instances of another class.
data encapsulation: A program development plan that involves a prototype using global variables anda final version that makes the global variables into instance attributes.
Exercises Exercise 18-1.
For the following program, draw a UML class diagram that shows these classes and
the relationships among them.
class PingPongParent:
class Ping(PingPongParent):
def __init__(self, pong):
self.pong = pong
class Pong(PingPongParent):
def __init__(self, pings=None):
if pings is None:
self.pings = []
self.pings = pings
def add_ping(self, ping):
pong = Pong()
ping = Ping(pong)
Exercise 18-2.
Write a Deck method called deal_hands that takes two parameters: the number of
hands and the number of cards per hand. It should create the appropriate number of
Hand objects, deal the appropriate number of cards per hand, and return a list of
Hands. Exercises | 219

Exercise 18-3.
The following are the possible hands in poker, in increasing order of value and
decreasing order of probability:
pair: Two cards with the same rank.
two pair: Two pairs of cards with the same rank.
three of a kind: Three cards with the same rank.
straight: Five cards with ranks in sequence (aces can be high or low, so Ace-2-3-4-5 is a
straight and so is 10-Jack-Queen-King-Ace , but Queen-King-Ace-2-3 is not.)
Five cards with the same suit.
full house: Three cards with one rank, two cards with another.
four of a kind: Four cards with the same rank.
Five cards in sequence (as defined above) and with the same suit.
The goal of these exercises is to estimate the probability of drawing these various
Download the following files from :
A complete version of the Card, Deck and Hand classes in this chapter. :
An incomplete implementation of a class that represents a poker hand, and
some code that tests it.
If you run , it deals seven 7-card poker hands and checks to see if
any of them contains a flush. Read this code carefully before you go on.
Add methods to named has_pair , has_twopair , etc. that return
True or False according to whether or not the hand meets the relevant criteria.
Your code should work correctly for “hands” that contain any number of cards (although 5 and 7 are the most common sizes).
220 | Chapter 18: Inheritance

4. Write a method named classify that figures out the highest-value classification
for a hand and sets the label attribute accordingly. For example, a 7-card hand
might contain a flush and a pair; it should be labeled “flush”. 5.
When you are convinced that your classification methods are working, the next step is to estimate the probabilities of the various hands. Write a function in that shuffles a deck of cards, divides it into hands, classifies the
hands, and counts the number of times various classifications appear.
Print a table of the classifications and their probabilities. Run your program with larger and larger numbers of hands until the output values converge to a reason
able degree of accuracy. Compare your results to the values at http://en.wikipe .
Solution: .
Exercises | 221

The Goodies
One of my goals for this book has been to teach you as little Python as possible. When there were two ways to do something, I picked one and avoided mentioning
the other. Or sometimes I put the second one into an exercise.
Now I want to go back for some of the good bits that got left behind. Python provides
a number of features that are not really necessaryyou can write good code without thembut with them you can sometimes write code thats more concise, readable or
efficient, and sometimes all three.
Conditional Expressions
We saw conditional statements in “Conditional Execution” on page 49. Conditional
statements are often used to choose one of two values; for example:
if x > 0:
y = math.log(x)
y = float(nan)
This statement checks whether x is positive. If so, it computes math.log. If not,
math.log would raise a ValueError. To avoid stopping the program, we generate a
“NaN”, which is a special floating-point value that represents “Not a Number”.
We can write this statement more concisely using a conditional expression:
y = math.log(x) if x > 0 else float(nan)
You can almost read this line like English: “ y gets log- x if x is greater than 0; other
wise it gets NaN”.
Recursive functions can sometimes be rewritten using conditional expressions. For
example, here is a recursive version of factorial:

def factorial(n):
if n == 0:
return 1
return n * factorial(n-1)
We can rewrite it like this:
def factorial(n):
return 1 if n == 0 else n * factorial(n-1)
Another use of conditional expressions is handling optional arguments. For example,
here is the init method from GoodKangaroo (see Exercise 17-2 ):
def __init__(self, name, contents=None): = name
if contents == None:
contents = []
self.pouch_contents = contents
We can rewrite this one like this:
def __init__(self, name, contents=None): = name
self.pouch_contents = [] if contents == None else contents
In general, you can replace a conditional statement with a conditional expression if
both branches contain simple expressions that are either returned or assigned to the same variable.
List Comprehensions
In “Map, Filter and Reduce” on page 111 we saw the map and filter patterns. For
example, this function takes a list of strings, maps the string method capitalize to
the elements, and returns a new list of strings:
def capitalize_all(t):
res = []
for s in t:
res.append(s.capitalize()) return res
We can write this more concisely using a list comprehension:
def capitalize_all(t):
return [s.capitalize() for s in t]
The bracket operators indicate that we are constructing a new list. The expression
inside the brackets specifies the elements of the list, and the for clause indicates what
sequence we are traversing.
The syntax of a list comprehension is a little awkward because the loop variable, s in
this example, appears in the expression before we get to the definition. 224 | Chapter 19: The Goodies

List comprehensions can also be used for filtering. For example, this function selects
only the elements of t that are uppercase, and returns a new list:
def only_upper(t):
res = []
for s in t:
if s.isupper():
return res
We can rewrite it using a list comprehension:
def only_upper(t):
return [s for s in t if s.isupper()]
List comprehensions are concise and easy to read, at least for simple expressions. And
they are usually faster than the equivalent for loops, sometimes much faster. So if you
are mad at me for not mentioning them earlier, I understand.
But, in my defense, list comprehensions are harder to debug because you cant put a print statement inside the loop. I suggest that you use them only if the computation is
simple enough that you are likely to get it right the first time. And for beginners that
means never.
Generator Expressions Generator expressions are similar to list comprehensions, but with parentheses
instead of square brackets:
>>> g = (x**2 for x in range(5))
>>> g
at 0x7f4c45a786c0>
The result is a generator object that knows how to iterate through a sequence of val ues. But unlike a list comprehension, it does not compute the values all at once; it
waits to be asked. The built-in function next gets the next value from the generator:
>>> next(g)
>>> next(g)
When you get to the end of the sequence, next raises a StopIteration exception.
You can also use a for loop to iterate through the values:
>>> for val in g:
... print(val)
16 Generator Expressions | 225

The generator object keeps track of where it is in the sequence, so the for loop picks
up where next left off. Once the generator is exhausted, it continues to raise
StopException :
>>> next(g)
Generator expressions are often used with functions like sum, max , and min:
>>> sum(x**2 for x in range(5))
any and all Python provides a built-in function, any, that takes a sequence of boolean values and
returns True if any of the values are True. It works on lists:
>>> any([False, False, True])
But it is often used with generator expressions:
>>> any(letter == t for letter in monty)True
That example isnt very useful because it does the same thing as the in operator. But
we could use any to rewrite some of the search functions we wrote in “Search” on
page 101 . For example, we could write avoids like this:
def avoids(word, forbidden):
return not any(letter in forbidden for letter in word)
The function almost reads like English: “ word avoids forbidden if there are not any
forbidden letters in word.”
Using any with a generator expression is efficient because it stops immediately if it
finds a True value, so it doesnt have to evaluate the whole sequence.
Python provides another built-in function, all, that returns True if every element of
the sequence is True. As an exercise, use all to rewrite uses_all from “Search” on
page 101 .
Sets In “Dictionary Subtraction” on page 156 I use dictionaries to find the words that
appear in a document but not in a word list. The function I wrote takes d1, which
contains the words from the document as keys, and d2, which contains the list of
words. It returns a dictionary that contains the keys from d1 that are not in d2:226 | Chapter 19: The Goodies

def subtract(d1, d2):
res = dict()
for key in d1:
if key not in d2:
res[key] = None
return res
In all of these dictionaries, the values are None because we never use them. As a result,
we waste some storage space.
Python provides another built-in type, called a set, that behaves like a collection of
dictionary keys with no values. Adding elements to a set is fast; so is checking mem
bership. And sets provide methods and operators to compute common set opera
For example, set subtraction is available as a method called difference or as an oper
ator, -. So we can rewrite subtract like this:
def subtract(d1, d2):
return set(d1) - set(d2)
The result is a set instead of a dictionary, but for operations like iteration, the behav ior is the same.
Some of the exercises in this book can be done concisely and efficiently with sets. For
example, here is a solution to has_duplicates, from Exercise 10-7 , that uses a dictio
def has_duplicates(t):
d = {}
for x in t:
if x in d:
return True
d[x] = True
return False
When an element appears for the first time, it is added to the dictionary. If the same
element appears again, the function returns True.
Using sets, we can write the same function like this: def has_duplicates(t):
return len(set(t)) < len(t)
An element can only appear in a set once, so if an element in t appears more than
once, the set will be smaller than t. If there are no duplicates, the set will be the same
size as t.
We can also use sets to do some of the exercises in Chapter 9. For example, heres a
version of uses_only with a loop: Sets | 227

def uses_only(word, available):
for letter in word:
if letter not in available:
return False
return True
uses_only checks whether all letters in word are in available . We can rewrite it like
def uses_only(word, available): return set(word) <= set(available)
The <= operator checks whether one set is a subset or another, including the possibil
ity that they are equal, which is true if all the letters in word appear in available .
As an exercise, rewrite avoids using sets.
Counters A Counter is like a set, except that if an element appears more than once, the Counter
keeps track of how many times it appears. If you are familiar with the mathematical
idea of a multiset, a Counter is a natural way to represent a multiset.
Counter is defined in a standard module called collections, so you have to import
it. You can initialize a Counter with a string, list, or anything else that supports itera tion:
>>> from collections import Counter
>>> count = Counter(parrot)
>>> count
Counter({r: 2, t: 1, o: 1, p: 1, a: 1})
Counters behave like dictionaries in many ways; they map from each key to the num
ber of times it appears. As in dictionaries, the keys have to be hashable.
Unlike dictionaries, Counters dont raise an exception if you access an element that doesnt appear. Instead, they return 0:
>>> count[d]
We can use Counters to rewrite is_anagram from Exercise 10-6 :
def is_anagram(word1, word2):
return Counter(word1) == Counter(word2)
If two words are anagrams, they contain the same letters with the same counts, so their Counters are equivalent.
Counters provide methods and operators to perform set-like operations, including
addition, subtraction, union and intersection. And they provide an often-useful 228 | Chapter 19: The Goodies

method, most_common , which returns a list of value-frequency pairs, sorted from most
common to least:
>>> count = Counter(parrot)
>>> for val, freq in count.most_common(3):
... print(val, freq)
r 2
p 1
a 1
The collections module also provides defaultdict, which is like a dictionary
except that if you access a key that doesnt exist, it can generate a new value on the fly.
When you create a defaultdict, you provide a function thats used to create new val
ues. A function used to create objects is sometimes called a factory. The built-in
functions that create lists, sets, and other types can be used as factories:
>>> from collections import defaultdict
>>> d = defaultdict(list)
Notice that the argument is list, which is a class object, not list(), which is a new
list. The function you provide doesnt get called unless you access a key that doesnt exist:
>>> t = d[new key]
>>> t
The new list, which were calling t, is also added to the dictionary. So if we modify t,
the change appears in d:
>>> t.append(new value)
>>> d
defaultdict(, {new key: [new value]})
If you are making a dictionary of lists, you can often write simpler code using
defaultdict . In my solution to Exercise 12-2, which you can get from http://thinkpy , I make a dictionary that maps from a sorted string
of letters to the list of words that can be spelled with those letters. For example,
opst maps to the list [opts, post, pots, spot, stop, tops] .
Heres the original code: def all_anagrams(filename):
d = {}
for line in open(filename):
word = line.strip().lower()
t = signature(word)
if t not in d: defaultdict | 229

d[t] = [word]
d[t].append(word) return d
This can be simplified using setdefault, which you might have used in Exercise
11-2 :
def all_anagrams(filename):
d = {}
for line in open(filename):
word = line.strip().lower() t = signature(word)
d.setdefault(t, []).append(word)
return d
This solution has the drawback that it makes a new list every time, regardless of
whether it is needed. For lists, thats no big deal, but if the factory function is compli
cated, it might be.
We can avoid this problem and simplify the code using a defaultdict:
def all_anagrams(filename):
d = defaultdict(list)
for line in open(filename):
word = line.strip().lower() t = signature(word)
return d
My solution to Exercise 18-3, which you can download from
code/ , uses setdefault in the function has_straightflush . This
solution has the drawback of creating a Hand object every time through the loop,
whether it is needed or not. As an exercise, rewrite it using a defaultdict.
Named Tuples Many simple objects are basically collections of related values. For example, the Point
object defined in Chapter 15 contains two numbers, x and y. When you define a class
like this, you usually start with an init method and a str method:
class Point:
def __init__(self, x=0, y=0):
self.x = x
self.y = y
def __str__(self):
return (%g, %g) % (self.x, self.y) 230 | Chapter 19: The Goodies

This is a lot of code to convey a small amount of information. Python provides a
more concise way to say the same thing:
from collections import namedtuple
Point = namedtuple(Point, [x, y])
The first argument is the name of the class you want to create. The second is a list of the attributes Point objects should have, as strings. The return value from namedtuple
is a class object:
>>> Point

Point automatically provides methods like __init__ and __str__ so you dont have
to write them.
To create a Point object, you use the Point class as a function: >>> p = Point(1, 2)
>>> p
Point(x=1, y=2)
The init method assigns the arguments to attributes using the names you provided.
The str method prints a representation of the Point object and its attributes.
You can access the elements of the named tuple by name: >>> p.x, p.y
(1, 2)
But you can also treat a named tuple as a tuple:
>>> p[0], p[1]
(1, 2)
>>> x, y = p
>>> x, y
(1, 2)
Named tuples provide a quick way to define simple classes. The drawback is that sim
ple classes dont always stay simple. You might decide later that you want to add methods to a named tuple. In that case, you could define a new class that inherits
from the named tuple:
class Pointier(Point):
# add more methods here
Or you could switch to a conventional class definition. Named Tuples | 231

Gathering Keyword Args
In “Variable-Length Argument Tuples” on page 142, we saw how to write a function
that gathers its arguments into a tuple:
def printall(*args):
You can call this function with any number of positional arguments (that is, argu
ments that dont have keywords):
>>> printall(1, 2.0, 3)
(1, 2.0, 3)
But the * operator doesnt gather keyword arguments:
>>> printall(1, 2.0, third=3)
TypeError: printall() got an unexpected keyword argument third
To gather keyword arguments, you can use the ** operator:
def printall(*args, **kwargs):
print(args, kwargs)
You can call the keyword gathering parameter anything you want, but kwargs is a
common choice. The result is a dictionary that maps keywords to values:
>>> printall(1, 2.0, third=3)
(1, 2.0) {third: 3}
If you have a dictionary of keywords and values, you can use the scatter operator, **,
to call a function:
>>> d = dict(x=1, y=2)
>>> Point(**d)
Point(x=1, y=2)
Without the scatter operator, the function would treat d as a single positional argu
ment, so it would assign d to x and complain because theres nothing to assign to y:
>>> d = dict(x=1, y=2) >>> Point(d)
Traceback (most recent call last): File "", line 1, in
TypeError: __new__() missing 1 required positional argument: y
When you are working with functions that have a large number of parameters, it is
often useful to create and pass around dictionaries that specify frequently used options.
232 | Chapter 19: The Goodies

Glossaryconditional expression: An expression that has one of two values, depending on a condition.
list comprehension: An expression with a for loop in square brackets that yields a new list.
generator expression: An expression with a for loop in parentheses that yields a generator object.
multiset: A mathematical entity that represents a mapping between the elements of a set
and the number of times they appear.
factory: A function, usually passed as a parameter, used to create objects.
Exercises Exercise 19-1.
The following is a function that computes the binomial coefficient recursively: def binomial_coeff(n, k):
"""Compute the binomial coefficient "n choose k".
n: number of trials
k: number of successes
returns: int
if k == 0:
return 1
if n == 0:
return 0
res = binomial_coeff(n-1, k) + binomial_coeff(n-1, k-1) return res
Rewrite the body of the function using nested conditional expressions.
One note: this function is not very efficient because it ends up computing the same
values over and over. You could make it more efficient by memoizing (see “Memos”
on page 131 ). But you will find that its harder to memoize if you write it using condi
tional expressions. Glossary | 233

When you are debugging, you should distinguish among different kinds of errors in order to track them down more quickly:

Syntax errors are discovered by the interpreter when it is translating the sourcecode into byte code. They indicate that there is something wrong with the struc
ture of the program. Example: Omitting the colon at the end of a def statement
generates the somewhat redundant message SyntaxError: invalid syntax.

Runtime errors are produced by the interpreter if something goes wrong while
the program is running. Most runtime error messages include information about
where the error occurred and what functions were executing. Example: An infin
ite recursion eventually causes the runtime error maximum recursion depth
exceeded .

Semantic errors are problems with a program that runs without producing error
messages but doesnt do the right thing. Example: An expression may not be eval
uated in the order you expect, yielding an incorrect result.
The first step in debugging is to figure out which kind of error you are dealing with.
Although the following sections are organized by error type, some techniques are applicable in more than one situation.
Syntax Errors
Syntax errors are usually easy to fix once you figure out what they are. Unfortunately, the error messages are often not helpful. The most common messages are SyntaxEr
ror: invalid syntax and SyntaxError: invalid token , neither of which is very

On the other hand, the message does tell you where in the program the problem
occurred. Actually, it tells you where Python noticed a problem, which is not neces sarily where the error is. Sometimes the error is prior to the location of the error mes
sage, often on the preceding line.
If you are building the program incrementally, you should have a good idea about
where the error is. It will be in the last line you added.
If you are copying code from a book, start by comparing your code to the books code
very carefully. Check every character. At the same time, remember that the book might be wrong, so if you see something that looks like a syntax error, it might be.
Here are some ways to avoid the most common syntax errors:1.
Make sure you are not using a Python keyword for a variable name.
Check that you have a colon at the end of the header of every compound state ment, including for, while , if , and def statements.
Make sure that any strings in the code have matching quotation marks. Make
sure that all quotation marks are straight quotes, not curly quotes.
If you have multiline strings with triple quotes (single or double), make sure you
have terminated the string properly. An unterminated string may cause an inva
lid token error at the end of your program, or it may treat the following part of
the program as a string until it comes to the next string. In the second case, it
might not produce an error message at all!
An unclosed opening operator (, { , or [makes Python continue with the next
line as part of the current statement. Generally, an error occurs almost immedi
ately in the next line.
Check for the classic = instead of == inside a conditional.
Check the indentation to make sure it lines up the way it is supposed to. Python
can handle space and tabs, but if you mix them it can cause problems. The best
way to avoid this problem is to use a text editor that knows about Python and generates consistent indentation.
If you have non-ASCII characters in the code (including strings and comments),that might cause a problem, although Python 3 usually handles non-ASCII char
acters. Be careful if you paste in text from a web page or other source.
If nothing works, move on to the next section...
236 | Chapter 20: Debugging

I keep making changes and it makes no difference.
If the interpreter says there is an error and you dont see it, that might be because you
and the interpreter are not looking at the same code. Check your programming envi
ronment to make sure that the program you are editing is the one Python is trying to
If you are not sure, try putting an obvious and deliberate syntax error at the begin ning of the program. Now run it again. If the interpreter doesnt find the new error,
you are not running the new code.
There are a few likely culprits:

You edited the file and forgot to save the changes before running it again. Some programming environments do this for you, but some dont.

You changed the name of the file, but you are still running the old name.

Something in your development environment is configured incorrectly.

If you are writing a module and using import, make sure you dont give your
module the same name as one of the standard Python modules.

If you are using import to read a module, remember that you have to restart the
interpreter or use reload to read a modified file. If you import the module again,
it doesnt do anything.
If you get stuck and you cant figure out what is going on, one approach is to start
again with a new program like “Hello, World!”, and make sure you can get a known
program to run. Then gradually add the pieces of the original program to the new
Runtime Errors Once your program is syntactically correct, Python can read it and at least start run
ning it. What could possibly go wrong?
My program does absolutely nothing.
This problem is most common when your file consists of functions and classes but
does not actually invoke a function to start execution. This may be intentional if you only plan to import this module to supply classes and functions.
If it is not intentional, make sure there is a function call in the program, and make
sure the flow of execution reaches it (see “Flow of execution” below).
Runtime Errors | 237

My program hangs.If a program stops and seems to be doing nothing, it is “hanging”. Often that means
that it is caught in an infinite loop or infinite recursion.
If there is a particular loop that you suspect is the problem, add a print state
ment immediately before the loop that says “entering the loop” and another
immediately after that says “exiting the loop”.
Run the program. If you get the first message and not the second, youve got an infinite loop. Go to the “Infinite loop” section below.

Most of the time, an infinite recursion will cause the program to run for a while
and then produce a “RuntimeError: Maximum recursion depth exceeded” error.
If that happens, go to the “Infinite recursion” section below.
If you are not getting this error but you suspect there is a problem with a recur sive method or function, you can still use the techniques in the “Infinite recur
sion” section.

If neither of those steps works, start testing other loops and other recursive func
tions and methods.

If that doesnt work, then it is possible that you dont understand the flow of execution in your program. Go to the “Flow of execution” section below.
Infinite loop
If you think you have an infinite loop and you think you know what loop is causing
the problem, add a print statement at the end of the loop that prints the values of the
variables in the condition and the value of the condition.
For example: while x > 0 and y < 0 :
# do something to x
# do something to y
print(x: , x)
print(y: , y)
print("condition: ", (x > 0 and y < 0))
Now when you run the program, you will see three lines of output for each time
through the loop. The last time through the loop, the condition should be False. If
the loop keeps going, you will be able to see the values of x and y, and you might
figure out why they are not being updated correctly.
238 | Chapter 20: Debugging

Infinite recursion
Most of the time, infinite recursion causes the program to run for a while and then
produce a Maximum recursion depth exceeded error.
If you suspect that a function is causing an infinite recursion, make sure that there is
a base case. There should be some condition that causes the function to return
without making a recursive invocation. If not, you need to rethink the algorithm and identify a base case.
If there is a base case but the program doesnt seem to be reaching it, add a print
statement at the beginning of the function that prints the parameters. Now when you
run the program, you will see a few lines of output every time the function is invoked, and you will see the parameter values. If the parameters are not moving toward the
base case, you will get some ideas about why not.
Flow of execution
If you are not sure how the flow of execution is moving through your program, add
print statements to the beginning of each function with a message like “entering
function foo”, where foo is the name of the function.
Now when you run the program, it will print a trace of each function as it is invoked.
When I run the program I get an exception. If something goes wrong during runtime, Python prints a message that includes the
name of the exception, the line of the program where the problem occurred, and a traceback.
The traceback identifies the function that is currently running, and then the function
that called it, and then the function that called that, and so on. In other words, it
traces the sequence of function calls that got you to where you are, including the line number in your file where each call occurred.
The first step is to examine the place in the program where the error occurred and see
if you can figure out what happened. These are some of the most common runtime
NameError: You are trying to use a variable that doesnt exist in the current environment.Check if the name is spelled right, or at least consistently. And remember that
local variables are local; you cannot refer to them from outside the function where they are defined. Runtime Errors | 239

TypeError:There are several possible causes:
You are trying to use a value improperly. Example: indexing a string, list, or
tuple with something other than an integer.

There is a mismatch between the items in a format string and the items
passed for conversion. This can happen if either the number of items does not match or an invalid conversion is called for.

You are passing the wrong number of arguments to a function. For methods,
look at the method definition and check that the first parameter is self.
Then look at the method invocation; make sure you are invoking the method
on an object with the right type and providing the other arguments correctly.
KeyError: You are trying to access an element of a dictionary using a key that the dictionary
does not contain. If the keys are strings, remember that capitalization matters.
AttributeError: You are trying to access an attribute or method that does not exist. Check the
spelling! You can use the built-in function vars to list the attributes that do exist.
If an AttributeError indicates that an object has NoneType, that means that it is
None . So the problem is not the attribute name, but the object.
The reason the object is none might be that you forgot to return a value from a function; if you get to the end of a function without hitting a return statement, it
returns None. Another common cause is using the result from a list method, like
sort , that returns None.
IndexError: The index you are using to access a list, string, or tuple is greater than its length
minus one. Immediately before the site of the error, add a print statement to dis
play the value of the index and the length of the array. Is the array the right size?
Is the index the right value?
The Python debugger ( pdb) is useful for tracking down exceptions because it allows
you to examine the state of the program immediately before the error. You can read
about pdb at .
I added so many print statements I get inundated with output.
One of the problems with using print statements for debugging is that you can end
up buried in output. There are two ways to proceed: simplify the output or simplify
the program.
240 | Chapter 20: Debugging

To simplify the output, you can remove or comment out print statements that arent
helping, or combine them, or format the output so it is easier to understand.
To simplify the program, there are several things you can do. First, scale down the problem the program is working on. For example, if you are searching a list, search a
small list. If the program takes input from the user, give it the simplest input that
causes the problem.
Second, clean up the program. Remove dead code and reorganize the program to
make it as easy to read as possible. For example, if you suspect that the problem is in a deeply nested part of the program, try rewriting that part with simpler structure. If you suspect a large function, try splitting it into smaller functions and testing them
Often the process of finding the minimal test case leads you to the bug. If you find
that a program works in one situation but not in another, that gives you a clue about what is going on.
Similarly, rewriting a piece of code can help you find subtle bugs. If you make a
change that you think shouldnt affect the program, and it does, that can tip you off.
Semantic Errors In some ways, semantic errors are the hardest to debug, because the interpreter pro
vides no information about what is wrong. Only you know what the program is sup posed to do.
The first step is to make a connection between the program text and the behavior you
are seeing. You need a hypothesis about what the program is actually doing. One of
the things that makes that hard is that computers run so fast.
You will often wish that you could slow the program down to human speed, and with some debuggers you can. But the time it takes to insert a few well-placed print state
ments is often short compared to setting up the debugger, inserting and removing
breakpoints, and “stepping” the program to where the error is occurring.
My program doesnt work. You should ask yourself these questions:
Is there something the program was supposed to do but which doesnt seem to be happening? Find the section of the code that performs that function and make
sure it is executing when you think it should.

Is something happening that shouldnt? Find code in your program that performs that function and see if it is executing when it shouldnt.
Semantic Errors | 241

Is a section of code producing an effect that is not what you expected? Make surethat you understand the code in question, especially if it involves functions or
methods in other Python modules. Read the documentation for the functions
you call. Try them out by writing simple test cases and checking the results.
In order to program, you need a mental model of how programs work. If you write a
program that doesnt do what you expect, often the problem is not in the program; its
in your mental model.
The best way to correct your mental model is to break the program into its compo nents (usually the functions and methods) and test each component independently.
Once you find the discrepancy between your model and reality, you can solve the
Of course, you should be building and testing components as you develop the pro
gram. If you encounter a problem, there should be only a small amount of new code that is not known to be correct.
Ive got a big hairy expression and it doesnt do what I expect.
Writing complex expressions is fine as long as they are readable, but they can be hard
to debug. It is often a good idea to break a complex expression into a series of assign ments to temporary variables.
For example: self.hands[i].addCard(self.hands[self.findNeighbor(i)].popCard())
This can be rewritten as:
neighbor = self.findNeighbor(i)
pickedCard = self.hands[neighbor].popCard() self.hands[i].addCard(pickedCard)
The explicit version is easier to read because the variable names provide additional
documentation, and it is easier to debug because you can check the types of the inter
mediate variables and display their values.
Another problem that can occur with big expressions is that the order of evaluation may not be what you expect. For example, if you are translating the expression
into Python, you might write:
y = x / 2 * math.pi
That is not correct because multiplication and division have the same precedence and are evaluated from left to right. So this expression computes
x / 2
A good way to debug expressions is to add parentheses to make the order of evalua
tion explicit:
242 | Chapter 20: Debugging

y = x / (2 * math.pi)
Whenever you are not sure of the order of evaluation, use parentheses. Not only will
the program be correct (in the sense of doing what you intended), it will also be more readable for other people who havent memorized the order of operations.
Ive got a function that doesnt return what I expect. If you have a return statement with a complex expression, you dont have a chance to
print the result before returning. Again, you can use a temporary variable. For exam
ple, instead of:
return self.hands[i].removeMatches()
you could write:
count = self.hands[i].removeMatches()
return count
Now you have the opportunity to display the value of count before returning.
Im really, really stuck and I need help. First, try getting away from the computer for a few minutes. Computers emit waves
that affect the brain, causing these symptoms:
Frustration and rage.

Superstitious beliefs (“the computer hates me”) and magical thinking (“the pro
gram only works when I wear my hat backward”).

Random walk programming (the attempt to program by writing every possibleprogram and choosing the one that does the right thing).
If you find yourself suffering from any of these symptoms, get up and go for a walk.
When you are calm, think about the program. What is it doing? What are some pos
sible causes of that behavior? When was the last time you had a working program,
and what did you do next?
Sometimes it just takes time to find a bug. I often find bugs when I am away from the computer and let my mind wander. Some of the best places to find bugs are on trains,
in the shower, and in bed just before you fall asleep.
No, I really need help.
It happens. Even the best programmers occasionally get stuck. Sometimes you work
on a program so long that you cant see the error. You need a fresh pair of eyes.
Before you bring someone else in, make sure you are prepared. Your program should
be as simple as possible, and you should be working on the smallest input that causes
Semantic Errors | 243

the error. You should have print statements in the appropriate places (and the output
they produce should be comprehensible). You should understand the problem well
enough to describe it concisely.
When you bring someone in to help, be sure to give them the information they need:
If there is an error message, what is it and what part of the program does it indicate?

What was the last thing you did before this error occurred? What were the lastlines of code that you wrote, or what is the new test case that fails?

What have you tried so far, and what have you learned?
When you find the bug, take a second to think about what you could have done to
find it faster. Next time you see something similar, you will be able to find the bug more quickly.
Remember, the goal is not just to make the program work. The goal is to learn how to
make the program work.
244 | Chapter 20: Debugging

1But if you get a question like this in an interview, I think a better answer is, “The fastest way to sort a million
integers is to use whatever sort function is provided by the language Im using. Its performance is good
enough for the vast majority of applications, but if it turned out that my application was too slow, I would use a profiler to see where the time was being spent. If it looked like a faster sort algorithm would have a signifi
cant effect on performance, then I would look around for a good implementation of radix sort.”
Analysis of Algorithms
This appendix is an edited excerpt from
ink Complexity , by Allen B. Downey, also
published by OReilly Media (2012). When you are done with this book, you might want to move on to that one.
Analysis of algorithms is a branch of computer science that studies the performance
of algorithms, especially their runtime and space requirements. See http://en.wikipe .
The practical goal of algorithm analysis is to predict the performance of different algorithms in order to guide design decisions.
During the 2008 United States presidential campaign, candidate Barack Obama was
asked to perform an impromptu analysis when he visited Google. Chief executive
Eric Schmidt jokingly asked him for “the most efficient way to sort a million 32-bit integers.” Obama had apparently been tipped off, because he quickly replied, “I think
the bubble sort would be the wrong way to go.” See
This is true: bubble sort is conceptually simple but slow for large datasets. The answer
Schmidt was probably looking for is “radix sort” (
Radix_sort ).1
The goal of algorithm analysis is to make meaningful comparisons between algo
rithms, but there are some problems:

The relative performance of the algorithms might depend on characteristics ofthe hardware, so one algorithm might be faster on Machine A, another on
Machine B. The general solution to this problem is to specify a machine model
and analyze the number of steps, or operations, an algorithm requires under a given model.
Relative performance might depend on the details of the dataset. For example, some sorting algorithms run faster if the data are already partially sorted; other
algorithms run slower in this case. A common way to avoid this problem is to analyze the worst-case scenario. It is sometimes useful to analyze average-case
performance, but thats usually harder, and it might not be obvious what set of
cases to average over.

Relative performance also depends on the size of the problem. A sorting algo rithm that is fast for small lists might be slow for long lists. The usual solution to
this problem is to express runtime (or number of operations) as a function of problem size, and group functions into categories depending on how quicklythey grow as problem size increases.
The good thing about this kind of comparison is that it lends itself to simple classifi cation of algorithms. For example, if I know that the runtime of Algorithm A tends to
be proportional to the size of the input, n, and Algorithm B tends to be proportional
to n2
, then I expect A to be faster than B, at least for large values of n.
This kind of analysis comes with some caveats, but well get to that later.
Order of Growth Suppose you have analyzed two algorithms and expressed their runtimes in terms of
the size of the input: Algorithm A takes 100n+1 steps to solve a problem with size n;
Algorithm B takes
n 2
+ n+ 1
The following table shows the runtime of these algorithms for different problem
Input sizeRuntime of Algorithm ARuntime of Algorithm B101 00111110010 00110 1011 000100 0011 001 00110 0001 000 001
> 10 10
n=10 , Algorithm A looks pretty bad; it takes almost 10 times longer than Algo
rithm B. But for n=100 they are about the same, and for larger values A is much
246 | Chapter 21: Analysis of Algorithms

The fundamental reason is that for large values of n, any function that contains an n2
term will grow faster than a function whose leading term is n. The leading term is the
term with the highest exponent.
For Algorithm A, the leading term has a large coefficient, 100, which is why B does better than A for small n. But regardless of the coefficients, there will always be some
value of n where an 2
> bn
, for any values of a and b.
The same argument applies to the non-leading terms. Even if the runtime of Algo
rithm A were n+1000000, it would still be better than Algorithm B for sufficiently
large n.
In general, we expect an algorithm with a smaller leading term to be a better algo
rithm for large problems, but for smaller problems, there may be a crossover point
where another algorithm is better. The location of the crossover point depends on the details of the algorithms, the inputs, and the hardware, so it is usually ignored for
purposes of algorithmic analysis. But that doesnt mean you can forget about it.
If two algorithms have the same leading order term, it is hard to say which is better;
again, the answer depends on the details. So for algorithmic analysis, functions with
the same leading term are considered equivalent, even if they have different coeffi cients.
An order of growth is a set of functions whose growth behavior is considered equiva
lent. For example, 2n, 100n and n+1 belong to the same order of growth, which is
written O(n) in Big-Oh notation and often called linear because every function in
the set grows linearly with n.
All functions with the leading term n2
belong to
O n 2
; they are called
The following table shows some of the orders of growth that appear most commonly
in algorithmic analysis, in increasing order of badness.
Order of growthNameO(1)constant
O log
logarithmic (for any
O n log
O n 2
O n3
O cn
exponential (for any
c) Order of Growth | 247

For the logarithmic terms, the base of the logarithm doesnt matter; changing bases is
the equivalent of multiplying by a constant, which doesnt change the order of growth. Similarly, all exponential functions belong to the same order of growth
regardless of the base of the exponent. Exponential functions grow very quickly, so
exponential algorithms are only useful for small problems.
Exercise 21-1.
Read the Wikipedia page on Big-Oh notation at
Big_O_notation and answer the following questions:1.
What is the order of growth of
n 3
+ n2
? What about
+ n2
? What about
n 3
+ 1000000 n2
What is the order of growth of
+ n n+ 1
? Before you start multiplying,
remember that you only need the leading term.
If f is in O(g), for some unspecified function g, what can we say about af+b?
If f
1 and
2 are in
O(g), what can we say about
1 +
If f
1 is in
O(g) and f
2 is in
O(h), what can we say about
1 +
If f
1 is in
O(g) and f
2 is
O(h) , what can we say about
Programmers who care about performance often find this kind of analysis hard to
swallow. They have a point: sometimes the coefficients and the non-leading terms make a real difference. Sometimes the details of the hardware, the programming lan
guage, and the characteristics of the input make a big difference. And for small prob
lems, asymptotic behavior is irrelevant.
But if you keep those caveats in mind, algorithmic analysis is a useful tool. At least for
large problems, the “better” algorithms is usually better, and sometimes it is much
better. The difference between two algorithms with the same order of growth is usu
ally a constant factor, but the difference between a good algorithm and a bad algo
rithm is unbounded!
Analysis of Basic Python Operations In Python, most arithmetic operations are constant time; multiplication usually takes
longer than addition and subtraction, and division takes even longer, but these run times dont depend on the magnitude of the operands. Very large integers are an
exception; in that case the runtime increases with the number of digits.
Indexing operationsreading or writing elements in a sequence or dictionaryare
also constant time, regardless of the size of the data structure.
248 | Chapter 21: Analysis of Algorithms

A for loop that traverses a sequence or dictionary is usually linear, as long as all of
the operations in the body of the loop are constant time. For example, adding up the elements of a list is linear:
total = 0
for x in t:
total += x
The built-in function sum is also linear because it does the same thing, but it tends to
be faster because it is a more efficient implementation; in the language of algorithmic
analysis, it has a smaller leading coefficient.
As a rule of thumb, if the body of a loop is in
O n a
then the whole loop is in
O n
+ 1
. The exception is if you can show that the loop exits after a constant number
of iterations. If a loop runs k times regardless of n, then the loop is in
O n a
, even for
large k.
Multiplying by k doesnt change the order of growth, but neither does dividing. So if
the body of a loop is in
O n a
and it runs
n/k times, the loop is in
O na
+ 1
, even for
large k.
Most string and tuple operations are linear, except indexing and len, which are con
stant time. The built-in functions min and max are linear. The runtime of a slice oper
ation is proportional to the length of the output, but independent of the size of the
String concatenation is linear; the runtime depends on the sum of the lengths of the operands.
All string methods are linear, but if the lengths of the strings are bounded by a con
stantfor example, operations on single charactersthey are considered constant
time. The string method join is linear; the runtime depends on the total length of the
Most list methods are linear, but there are some exceptions:

Adding an element to the end of a list is constant time on average; when it runs out of room it occasionally gets copied to a bigger location, but the total time for
n operations is O(n), so the average time for each operation is O(1).

Removing an element from the end of a list is constant time.

Sorting is
O nlog n
Analysis of Basic Python Operations | 249

Most dictionary operations and methods are constant time, but there are some excep
The runtime of update is proportional to the size of the dictionary passed as a
parameter, not the dictionary being updated.

keys , values and items are constant time because they return iterators. But if
you loop through the iterators, the loop will be linear.
The performance of dictionaries is one of the minor miracles of computer science.
We will see how they work in “Hashtables” on page 251.
Exercise 21-2.
Read the Wikipedia page on sorting algorithms at
ing_algorithm and answer the following questions:
What is a “comparison sort?” What is the best worst-case order of growth for a
comparison sort? What is the best worst-case order of growth for any sort algo
What is the order of growth of bubble sort, and why does Barack Obama think it is “the wrong way to go?”
What is the order of growth of radix sort? What preconditions do we need to use
What is a stable sort and why might it matter in practice?
What is the worst sorting algorithm (that has a name)?
What sort algorithm does the C library use? What sort algorithm does Python use? Are these algorithms stable? You might have to Google around to find these
Many of the non-comparison sorts are linear, so why does does Python use an
O n log n
comparison sort?
Analysis of Search Algorithms A search is an algorithm that takes a collection and a target item and determines
whether the target is in the collection, often returning the index of the target.
The simplest search algorithm is a “linear search”, which traverses the items of the
collection in order, stopping if it finds the target. In the worst case it has to traverse the entire collection, so the runtime is linear.
250 | Chapter 21: Analysis of Algorithms

The in operator for sequences uses a linear search; so do string methods like find
and count .
If the elements of the sequence are in order, you can use a bisection search, which isO logn
. Bisection search is similar to the algorithm you might use to look a word
up in a dictionary (a paper dictionary, not the data structure). Instead of starting at
the beginning and checking each item in order, you start with the item in the middle
and check whether the word you are looking for comes before or after. If it comes before, then you search the first half of the sequence. Otherwise you search the sec
ond half. Either way, you cut the number of remaining items in half.
If the sequence has 1,000,000 items, it will take about 20 steps to find the word or
conclude that its not there. So thats about 50,000 times faster than a linear search.
Bisection search can be much faster than linear search, but it requires the sequence to be in order, which might require extra work.
There is another data structure called a hashtable that is even fasterit can do a
search in constant timeand it doesnt require the items to be sorted. Python dic
tionaries are implemented using hashtables, which is why most dictionary operations,
including the in operator, are constant time.
To explain how hashtables work and why their performance is so good, I start with a
simple implementation of a map and gradually improve it until its a hashtable.
I use Python to demonstrate these implementations, but in real life you wouldnt write code like this in Python; you would just use a dictionary! So for the rest of this
chapter, you have to imagine that dictionaries dont exist and you want to implement
a data structure that maps from keys to values. The operations you have to implement are:
add(k, v) :
Add a new item that maps from key k to value v. With a Python dictionary, d, this
operation is written d[k] = v.
get(k) :
Look up and return the value that corresponds to key k. With a Python dictio
nary, d, this operation is written d[k] or d.get(k) .
For now, I assume that each key only appears once. The simplest implementation of
this interface uses a list of tuples, where each tuple is a key-value pair:
Hashtables | 251

class LinearMap:
def __init__(self):
self.items = []
def add(self, k, v):
self.items.append((k, v))
def get(self, k):
for key, val in self.items:
if key == k:
return val
raise KeyError
add appends a key-value tuple to the list of items, which takes constant time.
get uses a for loop to search the list: if it finds the target key it returns the corre
sponding value; otherwise it raises a KeyError. So get is linear.
An alternative is to keep the list sorted by key. Then get could use a bisection search,
which is O log n
. But inserting a new item in the middle of a list is linear, so this
might not be the best option. There are other data structures that can implement add
and get in log time, but thats still not as good as constant time, so lets move on.
One way to improve LinearMap is to break the list of key-value pairs into smaller lists.
Heres an implementation called BetterMap, which is a list of 100 LinearMaps. As
well see in a second, the order of growth for get is still linear, but BetterMap is a step
on the path toward hashtables:
class BetterMap:
def __init__(self, n=100):
self.maps = []
for i in range(n):
def find_map(self, k):
index = hash(k) % len(self.maps)
return self.maps[index]
def add(self, k, v):
m = self.find_map(k)
m.add(k, v)
def get(self, k):
m = self.find_map(k)
return m.get(k)
252 | Chapter 21: Analysis of Algorithms

__init__ makes a list of n LinearMap s.
find_map is used by add and get to figure out which map to put the new item in, or
which map to search.
find_map uses the built-in function hash, which takes almost any Python object and
returns an integer. A limitation of this implementation is that it only works with hashable keys. Mutable types like lists and dictionaries are unhashable.
Hashable objects that are considered equivalent return the same hash value, but the
converse is not necessarily true: two objects with different values can return the same
hash value.
find_map uses the modulus operator to wrap the hash values into the range from 0 to
len(self.maps) , so the result is a legal index into the list. Of course, this means that
many different hash values will wrap onto the same index. But if the hash function
spreads things out pretty evenly (which is what hash functions are designed to do), then we expect n/100 items per LinearMap.
Since the runtime of LinearMap.get is proportional to the number of items, we
expect BetterMap to be about 100 times faster than LinearMap. The order of growth
is still linear, but the leading coefficient is smaller. Thats nice, but still not as good as
a hashtable.
Here (finally) is the crucial idea that makes hashtables fast: if you can keep the maxi mum length of the LinearMaps bounded, LinearMap.get is constant time. All you
have to do is keep track of the number of items and when the number of items per
LinearMap exceeds a threshold, resize the hashtable by adding more LinearMaps.
Here is an implementation of a hashtable: class HashMap:
def __init__(self):
self.maps = BetterMap(2)
self.num = 0
def get(self, k):
return self.maps.get(k)
def add(self, k, v):
if self.num == len(self.maps.maps):
self.maps.add(k, v)
self.num += 1
def resize(self):
new_maps = BetterMap(self.num * 2) Hashtables | 253

for m in self.maps.maps:
for k, v in m.items: new_maps.add(k, v)
self.maps = new_maps
Each HashMap contains a BetterMap; __init__ starts with just 2 LinearMaps and ini
tializes num, which keeps track of the number of items.
get just dispatches to BetterMap. The real work happens in add, which checks the
number of items and the size of the BetterMap: if they are equal, the average number
of items per LinearMap is 1, so it calls resize.
resize make a new BetterMap, twice as big as the previous one, and then “rehashes”
the items from the old map to the new.
Rehashing is necessary because changing the number of LinearMaps changes the
denominator of the modulus operator in find_map. That means that some objects
that used to hash into the same LinearMap will get split up (which is what we wanted,
Rehashing is linear, so resize is linear, which might seem bad, since I promised that
add would be constant time. But remember that we dont have to resize every time, so
add is usually constant time and only occasionally linear. The total amount of work to
run add n times is proportional to n, so the average time of each add is constant time!
To see how this works, think about starting with an empty HashTable and adding a sequence of items. We start with two LinearMaps, so the first two adds are fast (no
resizing required). Lets say that they take one unit of work each. The next add
requires a resize, so we have to rehash the first two items (lets call that two more
units of work) and then add the third item (one more unit). Adding the next item costs one unit, so the total so far is six units of work for four items.
The next add costs five units, but the next three are only one unit each, so the total is
14 units for the first eight adds.
The next add costs nine units, but then we can add seven more before the next resize,
so the total is 30 units for the first 16 adds.
After 32 adds, the total cost is 62 units, and I hope you are starting to see a pattern. After n adds, where n is a power of two, the total cost is 2n-2 units, so the average
work per add is a little less than 2 units. When n is a power of two, thats the best case;
for other values of n the average work is a little higher, but thats not important. The
important thing is that it is O(1).
Figure 21-1 shows how this works graphically. Each block represents a unit of work.
The columns show the total work for each add in order from left to right: the first two
adds cost one unit, the third costs three units, etc. 254 | Chapter 21: Analysis of Algorithms

Figure 21-1. e cost of a hashtable add.
The extra work of rehashing appears as a sequence of increasingly tall towers with
increasing space between them. Now if you knock over the towers, spreading the cost of resizing over all adds, you can see graphically that the total cost after n adds is
2 n 2
An important feature of this algorithm is that when we resize the HashTable it grows
geometrically; that is, we multiply the size by a constant. If you increase the size arith
meticallyadding a fixed number each timethe average time per add is linear.
You can download my implementation of HashMap from
code/ , but remember that there is no reason to use it; if you want a map, just
use a Python dictionary.
Glossary analysis of algorithms: A way to compare algorithms in terms of their runtime and/or space requirements.
machine model: A simplified representation of a computer used to describe algorithms.
worst case: The input that makes a given algorithm run slowest (or require the most space).
leading term: In a polynomial, the term with the highest exponent.
crossover point: The problem size where two algorithms require the same runtime or space.
order of growth: A set of functions that all grow in a way considered equivalent for purposes ofanalysis of algorithms. For example, all functions that grow linearly belong to the
same order of growth.
Glossary | 255

Big-Oh notation:Notation for representing an order of growth; for example, O(n) represents the
set of functions that grow linearly.
linear: An algorithm whose runtime is proportional to problem size, at least for large
problem sizes.
quadratic: An algorithm whose runtime is proportional to n2
, where n is a measure of prob
lem size.
search: The problem of locating an element of a collection (like a list or dictionary) ordetermining that it is not present.
hashtable: A data structure that represents a collection of key-value pairs and performssearch in constant time.256 | Chapter 21: Analysis of Algorithms

Aabecedarian, 87, 101
abs function, 62
absolute path, 168, 174
access, 108
accumulator, 119
histogram, 154
list, 112
string, 211
sum, 111
Ackermann function, 72, 137
add method, 200
addition with carrying, 81
algorithm, 81, 82 , 158 , 245
MD5, 176
square root, 82
aliasing, 114, 115 , 120 , 179 , 182 , 206
copying to avoid, 119
all, 226
alphabet, 46
alternative execution, 49
ambiguity, 6
anagram, 121
anagram set, 148, 175
analysis of algorithms, 245, 255
analysis of primitives, 248
and operator, 49
any, 226
append method, 111, 117 , 122 , 211 , 212
arc function, 38
Archimedian spiral, 46
argument, 21, 24 , 26 , 27 , 31 , 116
gather, 142
keyword, 40, 44 , 232 list, 116
optional, 91, 95 , 95 , 114 , 129 , 224
positional, 198, 205 , 232
variable-length tuple, 142
argument scatter, 142
arithmetic operator, 3
assert statement, 192, 193
assignment, 17, 75 , 108
augmented, 111, 119
item, 89, 108 , 140
tuple, 141, 141 , 143 , 148
assignment statement, 11
attribute, 184, 203
class, 208, 218
initializing, 204
instance, 178, 184 , 209 , 218
__dict__, 204
AttributeError, 183, 240
augmented assignment, 111, 119
Austen, Jane, 154
average case, 246
average cost, 254
badness, 247
base case, 53, 57
benchmarking, 161, 163
BetterMap, 252
big, hairy expression, 242
Big-Oh notation, 247, 256
binary search, 122
bingo, 149
birthday, 193
birthday paradox, 121 257

bisect module, 122
bisection search, 122, 251
bisection, debugging by, 81
bitwise operator, 4
body, 24, 31 , 77
bool type, 48
boolean expression, 48, 56
boolean function, 65
boolean operator, 91
borrowing, subtraction with, 81, 191
bounded, 253
bracket operator, 85, 108 , 140
bracket, squiggly, 125
branch, 50, 56
break statement, 78
bubble sort, 245
bug, 7, 9 , 16
worst, 205
built-in function, any, 226, 226
bytes object, 170, 175
calculator, 10, 19
call graph, 132, 136
Car Talk, 105, 105 , 106 , 138 , 149
Card class, 208
card, playing, 207
carrying, addition with, 81, 189 , 190
catch, 174
chained conditional, 50, 57
character, 85
checksum, 172, 176
child class, 213, 218
choice function, 153
circle function, 38
circular definition, 66
class, 4, 177 , 184
Card, 208
child, 213, 218
Deck, 211
Hand, 213
Kangaroo, 205
parent, 213
Point, 178, 199
Rectangle, 180
Time, 187
class attribute, 208, 218
class definition, 177
class diagram, 215, 219class object, 178, 184 , 231
close method, 166, 170 , 172
__cmp__ method, 210
Collatz conjecture, 78
collections, 228, 229 , 231
colon, 24, 236
comment, 15, 18
commutativity, 15, 201
compare function, 62
comparing algorithms, 245
comparison string, 92
tuple, 140, 210
comparison sort, 250
composition, 23, 26 , 32 , 64 , 211
compound statement, 49, 56
concatenation, 15, 18 , 27 , 87 , 89 , 114
list, 110, 117 , 122
condition, 49, 56 , 77 , 238
conditional, 236
chained, 50, 57
nested, 50, 57
conditional execution, 49
conditional expression, 223, 233
conditional statement, 49, 56 , 65 , 224
consistency check, 135, 191
constant time, 254
contributors, xv
conversion, type, 21
copy deep, 183
shallow, 182
slice, 88, 110
to avoid aliasing, 119
copy module, 182
copying objects, 182
count method, 95
counter, 89, 95 , 127 , 134
Counter, 228
counting and looping, 89
Creative Commons, xv
crossover point, 247, 255
crosswords, 99
cumulative sum, 120
data encapsulation, 216, 219
data structure, 147, 148 , 159
database, 169, 175258 | Index

database object, 170
datetime module, 193
dbm module, 169
dead code, 62, 71 , 241
debugger (pdb), 240
debugging, 7, 7 , 9 , 16 , 43 , 55 , 70 , 92 , 104 , 118 ,
134 , 147 , 161 , 173 , 183 , 192 , 204 , 217 , 225 ,
by bisection, 81
emotional response, 7, 243
experimental, 30
rubber duck, 163
superstition, 243
deck, 207
Deck class, 211
deck, playing cards, 211
declaration, 133, 136
decrement, 76, 82
deep copy, 183, 184
deepcopy function, 183
def keyword, 24
default value, 156, 163 , 199
avoiding mutable, 205
defaultdict, 229
definition circular, 66
class, 177
function, 23
recursive, 150
del operator, 113
deletion, element of list, 113
delimiter, 114, 120
designed development, 192
deterministic, 152, 162
development plan, 44
data encapsulation, 216, 219
designed, 190
encapsulation and generalization, 42
incremental, 62, 236
prototype and patch, 188, 190
random walk programming, 162, 243
reduction, 102, 104 , 105
diagram call graph, 136
class, 215, 219
object, 178, 180 , 182 , 185 , 187 , 209
stack, 28, 116
state, 11, 75 , 93 , 108 , 115 , 115 , 130 , 145 , 178 ,
180 , 182 , 187 , 209__dict__ attribute, 204
dict function, 125
dictionary, 125, 125 , 135 , 144 , 240
initialize, 145
invert, 130
lookup, 129
looping with, 128
reverse lookup, 129
subtraction, 156
traversal, 145, 204
dictionary methods, 250
dbm module, 170
dictionary subtraction, 227
diff, 176
Dijkstra, Edsger, 104
dir function, 240
directory, 167, 174
walk, 168
working, 168
dispatch, type-based, 201, 202
divisibility, 48
division floating-point, 47
floor, 47, 56 , 56
divmod, 141, 191
docstring, 43, 44 , 178
dot notation, 22, 32 , 90 , 178 , 197 , 209
Double Day, 194
double letters, 105
Doyle, Arthur Conan, 30
duplicate, 121, 137 , 176 , 227
E element, 107, 119
element deletion, 113
elif keyword, 50
Elkner, Jeff, xii, xv
ellipses, 24
else keyword, 49
email address, 141
embedded object, 180, 184 , 206
copying, 182
emotional debugging, 7, 243
empty list, 107
empty string, 94, 114
encapsulation, 39, 44 , 65 , 82 , 90 , 214
encode, 207, 218
encrypt, 207
end of line character, 174 Index | 259

enumerate function, 144
enumerate object, 144
epsilon, 80
equality and assignment, 75
equivalence, 115, 182
equivalent, 120
error runtime, 16, 53 , 55 , 235
semantic, 17, 235 , 241
shape, 147
syntax, 16, 235
error checking, 69
error message, 9, 16 , 17 , 235
eval function, 83
evaluate, 13
exception, 16, 18 , 235 , 239
AttributeError, 183, 240
IndexError, 86, 93 , 109 , 240
IOError, 169
KeyError, 126, 240
LookupError, 129
NameError, 27, 239
OverflowError, 56
RuntimeError, 54
StopIteration, 225
SyntaxError, 23
TypeError, 86, 88 , 131 , 140 , 142 , 167 , 198 ,
UnboundLocalError, 134
ValueError, 55, 141
exception, catching, 169
execute, 13, 17
exists function, 168
experimental debugging, 30, 162
exponent, 247
exponential growth, 248
expression, 12, 17
big and hairy, 242
boolean, 48, 56
conditional, 223, 233
generator, 225, 226 , 233
extend method, 111
F factorial, 223
factorial function, 66, 69
factory, 233
factory function, 229, 230
False special value, 48Fermats Last Theorem, 57
fibonacci function, 68, 131
file, 165
permission, 169
reading and writing, 166
file object, 99, 105
filename, 167
filter pattern, 112, 120 , 225
find function, 89
flag, 133, 136
float function, 22
float type, 5
floating-point, 4, 8 , 80 , 223
floating-point division, 47
floor division, 47, 56 , 56
flow of execution, 25, 32 , 69 , 70 , 77 , 217 , 239
flower, 45
folder, 167
for loop, 37, 52 , 86 , 109 , 144 , 224
formal language, 5, 9
format operator, 166, 174 , 240
format sequence, 166, 174
format string, 166, 174
frame, 28, 32 , 53 , 68 , 132
Free Documentation License, GNU, xii, xv
frequency, 127
letter, 148
word, 152, 163
fruitful function, 29, 31
frustration, 243
function, 3, 21 , 23 , 31 , 196
abs, 62
ack, 72, 137
arc, 38
choice, 153
circle, 38
compare, 62
deepcopy, 183
dict, 125
dir, 240
enumerate, 144
eval, 83
exists, 168
factorial, 66, 223
fibonacci, 68, 131
find, 89
float, 22
fruitful, 29
getattr, 204260 | Index

getcwd, 167
hasattr, 183, 204
input, 54
int, 21
isinstance, 69, 183 , 201
len, 32, 86 , 126
list, 113
log, 22
math, 22
max, 142, 142
min, 142, 142
open, 99, 100 , 166 , 169 , 169
polygon, 38
popen, 171
programmer defined, 26, 155
randint, 122, 153
random, 152
recursive, 52
reload, 173, 237
repr, 173
reversed, 147
shuffle, 212
sorted, 128, 147
sqrt, 23, 64
str, 22
sum, 142, 226
trigonometric, 22
tuple, 140
type, 183
void, 29
zip, 143
function argument, 26
function call, 21, 31
function composition, 64
function definition, 23, 25 , 31 , 31
function frame, 28, 32 , 53 , 68 , 132
function object, 33
function parameter, 26
function syntax, 197
function type, 24
modifier, 189
pure, 188
function, reasons for, 30
function, tuple as return value, 141
functional programming style, 190, 193
G gamma function, 69
gather, 142, 148 , 232 GCD (greatest common divisor), 73
generalization, 39, 44 , 102 , 191
generator expression, 225, 226 , 233
generator object, 225
geometric resizing, 255
get method, 128
getattr function, 204
getcwd function, 167
global statement, 133, 136
global variable, 133, 136
update, 133
GNU Free Documentation License, xii, xv
greatest common divisor (GCD), 73
grid, 33
guardian pattern, 70, 71 , 92
H Hand class, 213
hanging, 238
HAS-A relationship, 215, 218 , 218
hasattr function, 183, 204
hash function, 131, 136 , 253
hashable, 131, 136 , 145
HashMap, 253
hashtable, 136, 251 , 256
header, 24, 31 , 236
Hello, World, 3
hexadecimal, 178
high-level language, 8
histogram, 127, 127
random choice, 153, 157
word frequencies, 153
Holmes, Sherlock, 30
homophone, 138
hypotenuse, 64
identical, 120
identity, 115, 182
if statement, 49
immutability, 88, 89 , 94 , 116 , 131 , 139 , 146
implementation, 127, 136 , 160 , 203
import statement, 32, 173
in operator, 251
in operator, 91, 102 , 109 , 126
increment, 76, 82 , 189 , 198
incremental development, 71, 236
indentation, 24, 196 , 236
index, 85, 86 , 92 , 94 , 108 , 125 , 240 Index | 261

looping with, 103, 109
negative, 86
slice, 87, 110
starting at zero, 86, 108
IndexError, 86, 93 , 109 , 240
indexing, 248
infinite loop, 77, 82 , 238 , 238
infinite recursion, 53, 57 , 69 , 238 , 239
information hiding, 205
inheritance, 213, 217 , 218 , 231
init method, 199, 204 , 208 , 211 , 213
initialization (before update), 76
initialization, variable, 82
input function, 54
instance, 178, 184
as argument, 179
as return value, 181
instance attribute, 178, 184 , 209 , 218
instantiate, 184
instantiation, 178
int function, 21
int type, 5
integer, 4, 8
interactive mode, 13, 13 , 17 , 29
interface, 40, 43 , 44 , 203 , 218
interlocking words, 122
interpret, 8
interpreter, 2
invariant, 192, 193
invert dictionary, 130
invocation, 90, 95
IOError, 169
is operator, 115, 182
IS-A relationship, 215, 218
isinstance function, 69, 183 , 201
item, 89, 94 , 107 , 125
dictionary, 135
item assignment, 89, 108 , 140
item update, 109
items method, 144
iteration, 77, 82
iterator, 143, 144 , 144 , 147 , 148 , 250
J join, 249
join method, 114, 211
Kangaroo class, 205 key, 125, 135
key-value pair, 125, 135 , 144
keyboard input, 54
KeyError, 126, 240 , 252
keyword, 12, 17 , 236
def, 24
elif, 50
else, 49
keyword argument, 40, 44 , 232
Koch curve, 59
L language
formal, 5
natural, 5
safe, 16
Turing complete, 66
leading coefficient, 247
leading term, 247, 255
leap of faith, 68
len function, 32, 86 , 126
letter frequency, 148
letter rotation, 96, 137
linear, 256
linear growth, 247
linear search, 250
LinearMap, 251
Linux, 30
lipogram, 101
Liskov substitution principle, 218
list, 107, 113 , 119 , 146 , 224
as argument, 116
concatenation, 110, 117 , 122
copy, 110
element, 108
empty, 107
function, 113
index, 109
membership, 109
method, 111
nested, 107, 109
of objects, 211
of tuples, 143
operation, 110
repetition, 110
slice, 110
traversal, 109
list comprehension, 224, 233
list methods, 249262 | Index

literalness, 6
local variable, 27, 31
log function, 22
logarithm, 163
logarithmic growth, 248
logical operator, 48, 49
lookup, 136
lookup, dictionary, 129
LookupError, 129
loop, 37, 44 , 77 , 143
condition, 238
for, 37, 52 , 86 , 109
infinite, 77, 238
nested, 211
traversal, 86
while, 77
loop variable, 224
looping with dictionaries, 128
with indices, 103, 109
with strings, 89
looping and counting, 89
low-level language, 8
ls (Unix command), 171
machine model, 246, 255
maintainable, 203
map pattern, 112, 119
map to, 207
mapping, 135, 158
Markov analysis, 158
mash-up, 159
math function, 22
matplotlib, 163
max function, 142, 142
McCloskey, Robert, 87
md5, 172
MD5 algorithm, 176
md5sum, 176
membership binary search, 122
bisection search, 122
dictionary, 126
list, 109
set, 137
memo, 132, 136
mental model, 242
metaphor, method invocation, 197metathesis, 149
method, 44, 90 , 196 , 204
add, 200
append, 111, 117 , 211 , 212
close, 166, 170 , 172
count, 95
extend, 111
get, 128
init, 199, 208 , 211 , 213
items, 144
join, 114, 211
mro, 217
pop, 113, 212
radd, 202
read, 171
readline, 99, 171
remove, 113
replace, 151
setdefault, 137
sort, 111, 118 , 213
split, 114, 141
string, 95
strip, 100, 151
translate, 151
update, 145
values, 126
void, 111
__cmp__, 210
__str__, 200, 211
method append, 122
method resolution order, 217
method syntax, 197
method, list, 111
Meyers, Chris, xv
min function, 142, 142
Moby Project, 99
model, mental, 242
modifier, 189, 193
module, 22, 32 , 32
bisect, 122
collections, 228, 229 , 231
copy, 182
datetime, 193
dbm, 169
os, 167
pickle, 165, 170
pprint, 135
profile, 161
random, 122, 152 , 212 Index | 263

reload, 173, 237
shelve, 171
string, 151
structshape, 147
time, 122
module object, 22, 172
module, writing, 172
modulus operator, 47, 56
Monty Python and the Holy Grail, 188
MP3, 176
mro method, 217
multiline string, 43, 236
multiplicity (in class diagram), 215, 219
multiset, 228
mutability, 88, 108 , 110 , 116 , 134 , 139 , 146 , 181
mutable object, as default value, 205
namedtuple, 231
NameError, 27, 239
NaN, 223
natural language, 5, 9
negative index, 86
nested conditional, 50, 57
nested list, 107, 109 , 119
newline, 54, 211
Newtons method, 79
None special value, 29, 31 , 62 , 111 , 113
NoneType type, 29
not operator, 49
number, random, 152
Obama, Barack, 245
object, 89, 94 , 114 , 115 , 120
bytes, 170, 175
class, 177, 178 , 184 , 231
copying, 182
Counter, 228
database, 170
defaultdict, 229
embedded, 180, 184 , 206
enumerate, 144
file, 99, 105
function, 33
generator, 225
module, 172
mutable, 181
namedtuple, 231pipe, 175
printing, 196
set, 227
zip, 148
object diagram, 178, 180 , 182 , 185 , 187 , 209
object-oriented design, 203
object-oriented language, 204
object-oriented programming, 177, 195 , 204 ,
odometer, 105
Olin College, xii
open function, 99, 100 , 166 , 169 , 169
operand, 17
operator, 8
and, 49
arithmetic, 3
bitwise, 4
boolean, 91
bracket, 85, 108 , 140
del, 113
format, 166, 174 , 240
in, 91, 102 , 109 , 126
is, 115 , 182
logical, 48, 49
modulus, 47, 56
not, 49
or, 49
overloading, 205
relational, 48, 210
slice, 87, 95 , 110 , 117 , 140
string, 15
update, 111
operator overloading, 200, 210
optional argument, 91, 95 , 95 , 114 , 129 , 224
optional parameter, 155, 199
or operator, 49
order of growth, 246, 255
order of operations, 14, 18 , 242
os module, 167
other (parameter name), 198
OverflowError, 56
overloading, 205
override, 156, 163 , 199 , 210 , 213 , 218
P palindrome, 72, 95 , 103 , 105 , 106
parameter, 26, 27 , 31 , 116
gather, 142
optional, 155, 199 264 | Index

other, 198
self, 197
parent class, 213, 213 , 218
parentheses argument in, 21
empty, 24, 90
parameters in, 26, 27
parent class in, 213
tuples in, 139
parse, 6, 9
pass statement, 49
path, 167, 174
absolute, 168
relative, 168
pattern filter, 112, 120 , 225
guardian, 70, 71 , 92
map, 112, 119
reduce, 112, 119
search, 89, 94 , 101 , 129 , 226
swap, 141
pdb (Python debugger), 240
permission, file, 169
persistence, 165, 174
pi, 23, 83
pickle module, 165, 170
pickling, 170
pie, 45
pipe, 171
pipe object, 175
plain text, 99, 152
planned development, 190
poetry, 6
Point class, 178, 199
point, mathematical, 177
poker, 207, 220
polygon function, 38
polymorphism, 203, 205
pop method, 113, 212
popen function, 171
portability, 8
positional argument, 198, 205 , 232
postcondition, 43, 70 , 218
pprint module, 135
precedence, 242
precondition, 43, 44 , 44 , 70 , 218
prefix, 158
pretty print, 135 print function, 3
print statement, 3, 8 , 200 , 240
problem solving, 1, 8
profile module, 161
program, 1, 8
program testing, 104
programmer-defined function, 26, 155
programmer-defined type, 177, 184 , 187 , 196 ,
200 , 210
Project Gutenberg, 152
prompt, 3, 8 , 54
prose, 6
prototype and patch, 188, 190 , 192
pseudorandom, 152, 162
pure function, 188, 193
Puzzler, 105, 105 , 106 , 138 , 149
Pythagorean theorem, 62
Python 2, 2, 3 , 39 , 48 , 54
Python in a browser, 2
Python, running, 2
PythonAnywhere, 2
Q quadratic, 256
quadratic growth, 247
quotation mark, 3, 5 , 43 , 88 , 236
R radd method, 202
radian, 22
radix sort, 245
rage, 243
raise statement, 129, 136 , 192
Ramanujan, Srinivasa, 83
randint function, 122, 153
random function, 152
random module, 122, 152 , 212
random number, 152
random text, 158
random walk programming, 162, 243
rank, 207
read method, 171
readline method, 99, 171
reassignment, 75, 82 , 108 , 133
Rectangle class, 180
recursion, 51, 52 , 57 , 66 , 68
base case, 53
infinite, 53, 69 , 239
recursive definition, 66, 150 Index | 265

red-black tree, 252
reduce pattern, 112, 119
reducible word, 138, 150
reduction to a previously solved problem, 102,
104 , 105
redundancy, 6
refactoring, 41, 42 , 44 , 216
reference, 116, 116 , 120
aliasing, 115
rehashing, 254
relational operator, 48, 210
relative path, 168, 174
reload function, 173, 237
remove method, 113
repetition, 37
list, 110
replace method, 151
repr function, 173
representation, 177, 180 , 207
return statement, 52, 61 , 243
return value, 21, 31 , 61 , 181
tuple, 141
reverse lookup, 136
reverse lookup, dictionary, 129
reverse word pair, 122
reversed function, 147
rotation, letter, 96, 137
rubber duck debugging, 163
running pace, 10, 19 , 193
running Python, 2
runtime error, 16, 53 , 55 , 235 , 239
RuntimeError, 54, 69
safe language, 16
sanity check, 135
scaffolding, 64, 71 , 135
scatter, 142, 148 , 232
Schmidt, Eric, 245
Scrabble, 149
script, 13, 18
script mode, 13, 13 , 18 , 29
search, 129, 250 , 256
search pattern, 89, 94 , 101 , 226
search, binary, 122
search, bisection, 122
self (parameter name), 197
semantic error, 17, 18 , 235 , 241
semantics, 18, 196sequence, 5, 85 , 85 , 94 , 107 , 113 , 139 , 146
set, 157, 227
anagram, 148, 175
set membership, 137
set subtraction, 227
setdefault, 230
setdefault method, 137
sexagesimal, 190
shallow copy, 182, 184
shape, 148
shape error, 147
shell, 171, 175
shelve module, 171
shuffle function, 212
sine function, 22
singleton, 130, 136 , 139
slice, 94
copy, 88, 110
list, 110
string, 87
tuple, 140
update, 110
slice operator, 87, 95 , 110 , 117 , 140
sort method, 111, 118 , 213
sorted function, 128, 147
sorting, 249, 250
special case, 104, 105 , 190
special value False, 48
None, 29, 31 , 62 , 111 , 113
True, 48
spiral, 46
split method, 114, 141
sqrt, 64
sqrt function, 23
square root, 79
squiggly bracket, 125
stable sort, 250
stack diagram, 28, 28 , 32 , 45 , 53 , 67 , 72 , 116
state diagram, 11, 17 , 75 , 93 , 108 , 115 , 115 , 130 ,
145 , 178 , 180 , 182 , 187 , 209
statement, 13, 17
assert, 192, 193
assignment, 11, 75
break, 78
compound, 49
conditional, 49, 56 , 65 , 224
for, 37, 86 , 109
global, 133, 136266 | Index

if, 49
import, 32, 173
pass, 49
print, 3, 8 , 200 , 240
raise, 129, 136 , 192
return, 52, 61 , 243
try, 169, 184
while, 77
step size, 95
StopIteration, 225
str function, 22
__str__ method, 200, 211
string, 4, 9 , 113 , 146
accumulator, 211
comparison, 92
empty, 114
immutable, 88
method, 90
multiline, 43, 236
operation, 15
slice, 87
triple-quoted, 43
string concatenation, 249
string method, 95
string methods, 249
string module, 151
string representation, 173, 200
string type, 5
strip method, 100, 151
structshape module, 147
structure, 5
subject, 197, 205
subset, 228
subtraction dictionary, 156
with borrowing, 81, 191
suffix, 158
suit, 207
sum, 226
sum function, 142
superstitious debugging, 243
swap pattern, 141
syntax, 5, 9 , 16 , 196 , 236
syntax error, 16, 18 , 235
SyntaxError, 23
temporary variable, 61, 71 , 242
test case, minimal, 241 testing
and absence of bugs, 104
incremental development, 62
is hard, 104
knowing the answer, 63
leap of faith, 68
minimal test case, 241
text plain, 99, 152
random, 158
text file, 174
Time class, 187
time module, 122
token, 5, 9
traceback, 29, 32 , 53 , 55 , 129 , 239
translate method, 151
traversal, 86, 86 , 89 , 92 , 94 , 102 , 102 , 112 , 119 ,
127 , 128 , 143 , 144 , 154
dictionary, 145, 204
list, 109
triangle, 58
trigonometric function, 22
triple-quoted string, 43
True special value, 48
try statement, 169, 184
tuple, 139, 141 , 146 , 148
as key in dictionary, 145, 160
assignment, 141
comparison, 140, 210
in brackets, 145
singleton, 139
slice, 140
tuple assignment, 141, 143 , 148
tuple function, 140
tuple methods, 249
Turing complete language, 66
Turing Thesis, 66
Turing, Alan, 66
turtle typewriter, 46
TurtleWorld, 59
type, 4, 5 , 8
bool, 48
dict, 125
file, 165
float, 5
function, 24
int, 5
list, 107
NoneType, 29 Index | 267

programmer-defined, 177, 184 , 187 , 196 ,
200 , 210
set, 157
str, 5
tuple, 139
type checking, 69
type conversion, 21
type function, 183
type-based dispatch, 201, 202 , 205
TypeError, 86, 88 , 131 , 140 , 142 , 167 , 198 , 240
typewriter, turtle, 46
typographical error, 162
U UnboundLocalError, 134
underscore character, 12
uniqueness, 121
Unix command, ls, 171
update, 76, 80 , 82
database, 170
global variable, 133
histogram, 154
item, 109
slice, 110
update method, 145
update operator, 111
use before def, 25
V value, 4, 8 , 114 , 115 , 135
default, 156tuple, 141
ValueError, 55, 141
values method, 126
variable, 11, 12 , 17
global, 133
local, 27
temporary, 61, 71 , 242
updating, 76
variable-length argument tuple, 142
veneer, 212, 218
void function, 29, 31
void method, 111
vorpal, 66
walk, directory, 168
while loop, 77
whitespace, 55, 100 , 173 , 236
word count, 172
word frequency, 152, 163
word, reducible, 138, 150
working directory, 168
worst bug, 205
worst case, 246, 255
zero, index starting at, 86, 108
zip function, 143
use with dict, 145
zip object, 148
Zipf s law, 163 268 | Index

About the AuthorAllen Downey is a Professor of Computer Science at Olin College of Engineering. He
has taught at Wellesley College, Colby College and U.C. Berkeley. He has a PhD in
Computer Science from U.C. Berkeley and Masters and Bachelors degrees from MIT.
The animal on the cover of
ink Python is the Carolina parrot, also known as the
Carolina parakeet ( Conuropsis carolinensis ). This parrot inhabited the southeastern
United States and was the only continental parrot with a habitat north of Mexico. At
one time, it lived as far north as New York and the Great Lakes, although it was chiefly found from Florida to the Carolinas.
The Carolina parrot was mainly green with a yellow head and some orange coloring
that appeared on the forehead and cheeks at maturity. Its average size ranged from
3133 cm. It had a loud, riotous call and would chatter constantly while feeding. It
inhabited tree hollows near swamps and riverbanks. The Carolina parrot was a very gregarious animal, living in small groups that could grow to several hundred parrots
when feeding.
These feeding areas were, unfortunately, often the crops of farmers, who would shoot
the birds to keep them away from the harvest. The birds social nature caused them to fly to the rescue of any wounded parrot, allowing farmers to shoot down whole flocks
at a time. In addition, their feathers were used to embellish ladies hats, and some par
rots were kept as pets. A combination of these factors led the Carolina parrot to
become rare by the late 1800s, and poultry disease may have contributed to their dwindling numbers. By the 1920s, the species was extinct.
Today, there are more than 700 Carolina parrot specimens preserved in museums
Many of the animals on OReilly covers are endangered; all of them are important to
the world. To learn more about how you can help, go to
The cover image is from Johnsons Natural History. The cover fonts are URW Type
writer and Guardian Sans. The text font is Adobe Minion Pro; the heading font is
Adobe Myriad Condensed; and the code font is Dalton Maags Ubuntu Mono.

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