Lessons in Play : An Introduction to Combinatorial Game Theory, Second Edition book cover
2nd Edition

Lessons in Play
An Introduction to Combinatorial Game Theory, Second Edition

ISBN 9781482243031
Published April 22, 2019 by A K Peters/CRC Press
344 Pages 148 B/W Illustrations


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Book Description

This second edition of Lessons in Play reorganizes the presentation of the popular original text in combinatorial game theory to make it even more widely accessible. Starting with a focus on the essential concepts and applications, it then moves on to more technical material. Still written in a textbook style with supporting evidence and proofs, the authors add many more exercises and examples and implement a two-step approach for some aspects of the material involving an initial introduction, examples, and basic results to be followed later by more detail and abstract results.


  • Employs a widely accessible style to the explanation of combinatorial game theory
  • Contains multiple case studies
  • Expands further directions and applications of the field
  • Includes a complete rewrite of CGSuite material

Table of Contents

Combinatorial Games

0.1 Basic Terminology


1 Basic Techniques

1.1 Greedy

1.2 Symmetry

1.3 Parity

1.4 Give Them Enough Rope!

1.5 Strategy Stealing

1.6 Change the Game!

1.7 Case Study: Long Chains in Dots & Boxes


2 Outcome Classes

2.1 Outcome Functions

2.2 Game Positions and Options

2.3 Impartial Games: Minding Your Ps and Ns

2.4 Case Study: Roll The Lawn

2.5 Case Study: Timber

2.6 Case Study: Partizan Endnim


3 Motivational Interlude: Sums of Games

3.1 Sums

3.2 Comparisons

3.3 Equality and Identity

3.4 Case Study: Domineering Rectangles


4 The Algebra of Games

4.1 The Fundamental Definitions

4.2 Games Form a Group with a Partial Order

4.3 Canonical Form

4.4 Case Study: Cricket Pitch

4.5 Incentives


5 Values of Games

5.1 Numbers

5.2 Case Study: Shove

5.3 Stops

5.4 A Few All-Smalls: Up, Down, and Stars

5.5 Switches

5.6 Case Study: Elephants & Rhinos

5.7 Tiny and Miny

5.8 Toppling Dominoes

5.9 Proofs of Equivalence of Games and Numbers


6 Structure

6.1 Games Born by Day 2

6.2 Extremal Games Born By Day n

6.3 More About Numbers

6.4 The Distributive Lattice of Games Born by Day n

6.5 Group Structure


7 Impartial Games

7.1 A Star-Studded Game

7.2 The Analysis of Nim

7.3 Adding Stars

7.4 A More Succinct Notation

7.5 Taking-and-Breaking Games

7.6 Subtraction Games

7.7 Keypad Games


8 Hot Games

8.1 Comparing Games and Numbers

8.2 Coping with Confusion

8.3 Cooling Things Down

8.4 Strategies for Playing Hot Games

8.5 Norton Products


9 All-Small Games

9.1 Cast of Characters

9.2 Motivation: The Scale of Ups

9.3 Equivalence Under □

9.4 Atomic Weight

9.5 All-Small Shove

9.6 More Toppling Dominoes

9.7 Clobber


10 Trimming Game Trees

10.1 Introduction

10.2 Reduced Canonical Form

10.3 Hereditary-Transitive Games

10.4 Ordinal Sum

10.5 Stirling-Shave

10.6 Even More Toppling Dominoes


Further Directions

1 Transfinite Games

2 Algorithms and Complexity

3 Loopy Games

4 Kos: Repeated Local Positions

5 Top-Down Thermography

6 Enriched Environments

7 Idempotents

8 Mis`ere Play

9 Scoring Games

A Top-Down Induction

A.1 Top-Down Induction

A.2 Examples

A.3 Why is Top-Down Induction Better?

A.4 Strengthening the Induction Hypothesis

A.5 Inductive Reasoning


B CGSuite

B.1 Installing CGSuite

B.2 Worksheet Basics

B.3 Programming in CGSuite’s Language

C Solutions to Exercises

D Rulesets

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Michael Albert - University of Otago

Richard Nowakowski - Dalhousie University

David Wolfe - Dalhousie University


"The wisdom and joy outshining from this 2nd edition, beat even the original. The helpful preludes for student and instructor, prefacing each chapter, have  elevated subtly in additional reader-friendliness; new subsections and a new case study were added.  An interesting new Chapter 10 trades complex yet complete computation of a game’s strategy, with a simplified slightly approximate winning strategy. The last chapter, which awards  the reader with a flavor of cutting edge research, was updated with a section on scoring games. The book is a must for novice and expert alike."

Aviezri Fraenkel, Weizmann Institute of Science, Israel

"In this second edition of Lessons in Play, the authors have corrected errors, updated the bibliography, and added a new chapter on trimming game trees. Like the first edition, this new edition is beautifully typeset and illustrated."

Brian Borchers, Editor, MAA Reviews

In this second edition of Lessons in Play: An Introduction to Combinatorial Game Theory, authors Albert , Nowakowski, and White provide a reorganized text presenting a variety of two-player finite games, discussed in theory as well as application. The theoretical material is presented in a clear and concise theorem/proof format and includes problems and exercises to aid readers’ understanding. Solutions are provided at the end of the book. Multiple examples from actual games are provided throughout, including Boxcars, Clobber, Cutthroat, Dots and Boxes, Hackenbush, and Toppling Dominoes. Throughout the text, the authors also provide in-depth case studies on specific games. A unique feature of this book is that each chapter begins by presenting a series of “prep problems” with notes to the instructor so students can preview the material prior to reading the chapter. Overall, this book is an excellent beginning read for anyone interested in learning about combinatorial games, assuming at least some background in abstract algebra.

S. L. Sullivan, Catawba College

Praise for the previous edition

This is an excellent introductory book to beginning game theory, written in an easily understandable manner yet advanced enough not to be considered trivial.
Books Online, July 2007

The first book to present combinatorial game theory in the form of a textbook suitable for students at the advanced undergraduate level … The authors state and prove theorems in a rigorous fashion [and] the presentation is enlivened with many concrete examples … an outstanding textbook … It will also be of interest to more advanced readers who want an introduction to combinatorial game theory.
Brian Borchers, June 2007

The theory is accessible to any student who has a smattering of general algebra and discrete math. Generally, a third year college student, but any good high school student should be able to follow the development with a little help.
Sir Read a Lot, May 2007

Lessons in Play is an enticing introduction to the wonderful world of combinatorial games. Using a rich collection of cleverly captivating examples and problems, the authors lead the reader through the basic concepts and on to several innovative extensions. I highly recommend this book.
Elwyn R. Berlekamp

A neat machine, converting novices into enthusiastic experts in modern combinatorial game theory.
Aviezri Fraenkel

Combinatorial games are intriguing, challenging, and often counter-intuitive, and are rapidly being recognized as an important mathematical discipline. Now that we have the attractive and friendly text Lessons in Play in hand, we can look forward to the appearance of many popular upper-division undergraduate courses, which encourage instructors to learn alongside their students.
Richard K. Guy

… If you have Winning Ways, you must have this book.
Andy Liu



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