2nd Edition

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

    346 Pages 148 B/W Illustrations
    by A K Peters/CRC Press

    346 Pages 148 B/W Illustrations
    by A K Peters/CRC Press

    344 Pages 148 B/W Illustrations
    by A K Peters/CRC Press

    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

    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


    Michael Albert - University of Otago

    Richard Nowakowski - Dalhousie University

    David Wolfe - Dalhousie University