Winning Ways for Your Mathematical Plays:

This classic on games and how to play them intelligently is being re-issued in a new, four volume edition. This book has laid the foundation to a mathematical approach to playing games. The wise authors wield witty words, which wangle wonderfully winning ways. In Volume 1, the authors do the Spade Work, presenting theories and techniques to dissect games of varied structures and formats in order to develop winning strategies.

Preface to Second Edition

xv

Preface

xvi

Spade-Work!

xix

Whose Game?

1

(18)

Blue-Red Hackenbush

2

(1)

The Tweedledum and Tweedledee Argument

3

(1)

How Can You Have Half a Move?

4

(2)

... And Quarter Moves?

6

(2)

Ski-Jumps for Beginners

8

(2)

Don't Just Take the Average!

10

(1)

What Is a Jump Worth?

11

(1)

Toads-and-Frogs

12

(1)

Do Our Methods Work?

12

(2)

What is a Game?

14

(2)

When Is a Move Good?

16

(1)

Figure 8(d) Is Worth 3/4

17

(1)

References and Further Reading

18

(1)

Finding the Correct Number is Simplicity Itself

19

(34)

Which Numbers Are Which?

19

(2)

Simplicity's the Answer!

21

(1)

Simplest Forms for Numbers

22

(2)

Cutcake

24

(2)

Maundy Cake

26

(1)

A Few More Applications of the Simplicity Rule

27

(1)

Positive, Negative, Zero, and Fuzzy Positions

28

(1)

Hackenbush Hotchpotch

29

(1)

Sums of Arbitrary Games

30

(1)

The Outcome of a Sum

31

(2)

The Negative of a Game

33

(1)

Cancelling a Game with its Negative

34

(1)

Comparing Two Games

35

(1)

Comparing Hackenbush Positions

35

(2)

The Game of Col

37

(1)

A Star is Born!

38

(1)

Col Contains Such Values

39

(1)

Game Trees

40

(1)

Green Hackenbush, The Game of Nim, and Nimbers

40

(2)

Get Nimble with Nimbers

42

(1)

Childish Hackenbush

43

(1)

Seating Couples

44

(2)

Winning Strategies

46

(1)

The Sum of Two Finite Games Can Last Forever

46

(1)

A Theorem about Col

47

(1)

Col-lections and Col-lapsings

48

(3)

Maundy Cake

51

(1)

Another Cutcake Variant

51

(1)

How Childish Can You Get?

51

(1)

References and Further Reading

52

(1)

Some Harder Games and How to Make Them Easier

53

(28)

Poker-Nim

53

(1)

Northcott's Game

54

(1)

Bogus Nim-Heaps and the Mex Rule

55

(1)

The Sprague-Grundy Theory for Impartial Games

56

(1)

The White Knight

56

(2)

Adding Nimbers

58

(1)

Why Queens

59

(1)

Reversible Moves in General Games

60

(2)

Deleting Dominated Options

62

(1)

Toads-and-Frogs with Ups and Downs

63

(2)

Game Tracking and Identification

65

(1)

What Are Flowers Worth?

66

(1)

A Gallimaufry of Games

67

(1)

Who Wins Sums of Ups, Downs, Stars, and Numbers?

68

(1)

A Closer Look at the Stars

68

(1)

The Values (↑ | ↑) and (0| ↑)

69

(2)

The Upstart Equality

71

(1)

Gift Horses

72

(1)

The Nim-Addition Rule in Several Variations

73

(1)

Wyt Queens and Wythoff's Game

74

(1)

Answers to Figures 8, 9, and 11

75

(1)

Toad Versus Frog

75

(1)

Two Theorems on Simplifying Games

76

(1)

Berlekamp's Rule for Hackenbush Strings

77

(1)

References and Further Reading

78

(3)

Taking and Breaking

81

(38)

Kayles

81

(1)

Games With Heaps

82

(1)

P-Positions and N-Positions

83

(1)

Subtraction Games

83

(3)

Ferguson's Pairing Property

86

(1)

Grundy Scales

87

(1)

Other Take-Away Games

87

(1)

Dawson's Chess

88

(2)

The Periodicity of Kayles

90

(1)

Other Take-and-Break Games

91

(1)

Dawson's Kayles

92

(1)

Variations

93

(1)

Guiles

93

(1)

Treblecross

94

(1)

Officers

95

(1)

Grundy's Game

96

(1)

Prim and Dim

97

(1)

Replication of Nim-Values

98

(1)

Double and Quadruple Kayles

98

(1)

Lasker's Nim

99

(1)

Some Remarks on Periodicity

100

(1)

Standard Form

100

(1)

A Compendium of Octal Games

101

(8)

Additional Remarks

109

(1)

Sparse Spaces and Common Cosets

109

(2)

Will Grundy's Game Be Ultimately Periodic?

111

(1)

Sparse Space Spells Speed

112

(1)

Games Displaying Arithmetic Periodicity

113

(2)

A Non-Arithmetic-Periodicity Theorem

115

(2)

Some Hexadecimal Games

117

(1)

Further References and Reading

117

(2)

Numbers, Nimbers and Numberless Wonders

119

(26)

Domineering

119

(2)

Switch Games

121

(1)

Cashing Cheques

122

(3)

Some Simple Hot Games

125

(1)

The Tiniest Games

125

(1)

Modern Management of Cash Flow

126

(1)

Tiny Toads-and-Frogs

127

(1)

The Opening Dissection of Toads-and-Frogs

127

(4)

Seating Boys and Girls

131

(2)

Toads-and-Frogs Completely Dissected

133

(3)

Toads-and-Frogs with Two Spaces

136

(2)

More Domineering Values

138

(5)

References

143

(2)

The Heat of Battle

145

(1)

Snort

145

(1)

A Graphic Picture of Farm Life

145

(2)

Don't Move In A Number Unless There's Nothing Else To Do!

147

(1)

What's in it for Me?

148

(1)

The Left and Right Stops

149

(2)

Cooling---and the Thermograph

151

(1)

Cooling Settles the Mean Value

152

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Winning Ways for Your Mathematical Plays: Volume 1

Summary

Table of Contents

Supplemental Materials