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

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Combinatorial Games

0.1 Basic Terminology

Problems

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

Problems

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

Problems

3 Motivational Interlude: Sums of Games

3.1 Sums

3.2 Comparisons

3.3 Equality and Identity

3.4 Case Study: Domineering Rectangles

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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