The Mathematics of Games

This book examines how simple mathematical analysis can throw unexpected light on games of every type games of chance, games of skill, games of chanceandskill, and automatic games. Discusses classic puzzles as peg solitaire and Rubik's cube. Lucid, instructive, and full of surprises, it will fascinate mathematicians and gamesters alike. 1989 edition.

1. Introduction

1

(5)

2. The luck of the deal

Counting made easy

6

(3)

4-3-3-3 and all that

9

(2)

Shuffle the pack and deal again

11

(6)

3. The luck of the die

Counting again made easy

17

(2)

The true law of averages

19

(3)

How random is a toss?

22

(1)

Cubic and other dice

23

(1)

The arithmetic of dice games

24

(4)

Simulation by computer

28

(3)

4. To err is human

Finding a hole in the ground

31

(5)

Finding a hole in the defence

36

(6)

A game of glorious uncertainty

42

(5)

5. If A beats B, and B beats C...

The assessment of a single player in isolation

47

(2)

The estimation of trends

49

(2)

Interactive games

51

(3)

Grades as measures of ability

54

(2)

The self-fulfilling nature of grading systems

56

(3)

The limitations of grading

59

(2)

Cyclic expectations

61

(3)

6. Bluff and double bluff

I've got a picture

64

(2)

An optimal strategy for each player

66

(2)

Scissors, paper, stone

68

(1)

You cut, I'll choose

69

(2)

The nature of bluffing

71

(1)

Analysing a game

72

(5)

7. The analysis of puzzles

Black and white squares

77

(5)

Divisibility by three

82

(3)

Positions with limited potential

85

(3)

Systematic progress within a puzzle

88

(2)

Systematic progress between puzzles

90

(9)

8. Sauce for the gander

A winning strategy at nim

99

(2)

Nim in disguise

101

(5)

All cul-de-sacs lead to nim

106

(2)

Grundy analysis in practice

108

(6)

Some more balancing acts

114

(4)

Playing to lose

118

(2)

9. The measure of a game

Nim with personal counters

120

(2)

Games of fractional measure

122

(3)

General piles

125

(4)

The nature of a numeric game

129

(2)

The measure of a feeble threat

131

(2)

Infinite games

133

(4)

10. When the counting has to stop

The symptoms of a hard game

137

(4)

When you know who, but rot how

141

(4)

The paradox underlying games of pure skill

145

(4)

Round and round in circles

Driving the old woman to bed

149

(4)

Turing games

153

(4)

Turing's paradox

157

(2)

The hole at the heart of mathematics

159

(3)

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The Mathematics of Games

Summary

Table of Contents

Supplemental Materials