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.
Features
- 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
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
A.3 Why is Top-Down Induction Better?
A.4 Strengthening the Induction Hypothesis
A.5 Inductive Reasoning
Problems
B CGSuite
B.1 Installing CGSuite
B.2 Worksheet Basics
B.3 Programming in CGSuite’s Language
C Solutions to Exercises
D Rulesets
