Winning Ways for your mathematical plays

Elwyn R. Berlekamp, John H. Conway, Richard K. Guy1568811306, 9781568811307, 156881142X, 1568811438, 1568811446

Nearly a quarter of a century ago, three mathematicians created the first and definitive book on mathematical games. Now, this classic is being republished in a completely revised edition, broken down into four volumes to accommodate new developments and to reorganize the material in the most accessible way. In Volume 1, the authors present theories and techniques to dissect games of varied structures and formats in order to develop winning strategies. The irreverent yet highly effective style of the book, as reflected in some of the reviews, makes reading a profitable pleasure. The inclusion of many examples helps the reader to put the mathematical analysis to immediate use.

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Table of contents :
Title page ……Page 1
Date-line ……Page 2
Dedication ……Page 3
Authors ……Page 5
Contents ……Page 7
Preface to Second Edition ……Page 13
Preface ……Page 14
Spade-Work! ……Page 17
1 Whose Game? ……Page 19
Blue-Red Hackenbush ……Page 20
The Tweedledum and Tweedledee Argument ……Page 21
How Can You Have Half a Move? ……Page 22
… And Quarter Moves? ……Page 24
Ski-Jumps for Beginners ……Page 26
Don’t Just Take the Average! ……Page 28
What Is a Jump Worth? ……Page 29
Do Our Methods Work? ……Page 30
What is a Game? ……Page 32
When Is a Move Good? ……Page 34
Figure 8(d) Is Worth frac{3}{4} ……Page 35
References and Further Reading ……Page 36
Which Numbers Are Which? ……Page 37
Simplicity’s the Answer! ……Page 39
Simplest Forms for Numbers ……Page 40
Cutcake ……Page 42
Maundy Cake ……Page 44
A Few More Applications of the Simplicty Rule ……Page 45
Positive, Negative, Zero, and Fuzzy Positions ……Page 46
Hackenbush Hotchpotch ……Page 47
Sums of Arbitrary Games ……Page 48
The Outcome of a Sum ……Page 49
The Negative of a Game ……Page 51
Cancelling a Game with its Negative ……Page 52
Comparing Hackenbush Positions ……Page 53
The Game of Col ……Page 55
A Star is Born! ……Page 56
Col Contains Such Values ……Page 57
Green Hackenbush, The Game of Nim, and Nimbers ……Page 58
Get Nimble with Nimbers ……Page 60
Childish Hackenbush ……Page 61
Seating Couples ……Page 62
The Sum of Two Finite Games Can Last Forever ……Page 64
A Theorem about Col ……Page 65
Col-lections and Col-lapsings ……Page 66
How Childish Can You Get? ……Page 69
References and Further Reading ……Page 70
Poker-Nim ……Page 71
Northcott’s Game ……Page 72
Bogus Nim-Heaps and the Mex Rule ……Page 73
The White Knight ……Page 74
Adding Nimbers ……Page 76
Wyt Queens ……Page 77
Reversible Moves in General Games ……Page 78
Deleting Dominated Options ……Page 80
Toads-and-Frogs with Ups and Downs ……Page 81
Game Tracking and Identification ……Page 83
What Are Flowers Worth? ……Page 84
A Gallimaufry of Games ……Page 85
A Closer Look at the Stars ……Page 86
The Values {$uparrow$|$uparrow$} and {0|$uparrow$} ……Page 87
The Upstart Equality ……Page 89
Gift Horses ……Page 90
The Nim-Addition Rule in Several Variations ……Page 91
Wyt Queens and Wythoff’s Game ……Page 92
Toad Versus Frog ……Page 93
Two Theorems on Simplifying Games ……Page 94
Berlekamp’s Rule for Hackenbush Strings ……Page 95
References and Further Reading ……Page 96
Kayles ……Page 99
Games With Heaps ……Page 100
Subtraction Games ……Page 101
Ferguson’s Pairing Property ……Page 104
Other Take-Away Games ……Page 105
Dawson’s Chess ……Page 106
The Periodicity of Kayles ……Page 108
Other Take-and-Break Games ……Page 109
Dawson’s Kayles ……Page 110
Guiles ……Page 111
Treblecross ……Page 112
Officers ……Page 113
Grundy’s Game ……Page 114
Prim and Dim ……Page 115
Double and Quadruple Kayles ……Page 116
Lasker’s Nim ……Page 117
Standard Form ……Page 118
A Compendium of Octal Games ……Page 119
Sparse Spaces and Common Cosets ……Page 127
Will Grundy’s Game Be Ultimately Periodic? ……Page 129
Sparse Space Spells Speed ……Page 130
Games Displaying Arithmetic Periodicity ……Page 131
A Non-Arithmetic-Periodicity Theorem ……Page 133
Further References and Reading ……Page 135
Domineering ……Page 137
Switch Games ……Page 139
Cashing Cheques ……Page 140
The Tiniest Games ……Page 143
Modern Management of Cash Flow ……Page 144
The Opening Dissection of Toads-and-Frogs ……Page 145
Seating Boys and Girls ……Page 149
Toads-and-Frogs Completely Dissected ……Page 151
Toads-and-Frogs with Two Spaces ……Page 154
More Domineering Values ……Page 156
References ……Page 161
A Graphic Picture of Farm Life ……Page 163
Don’t Move In A Number Unless There’s Nothing Else To Do! ……Page 165
What’s in it for Me? ……Page 166
The Left and Right Stops ……Page 167
Cooling—and the Thermograph ……Page 169
Cooling Settles the Mean Value ……Page 170
How to Draw Thermographs ……Page 171
When A Player Has Several Options ……Page 172
Foundations for Thermographs ……Page 173
Examples of Thermographs ……Page 174
Who Is To Move From The Final Stop? ……Page 175
The Cheque-Market Exchange ……Page 176
Equitable Games ……Page 177
Excitable Games ……Page 178
The Extended Thermograph ……Page 179
Getting the Right Slant ……Page 180
The Thermostatic Strategy ……Page 181
Thermostrat’s Not Often Wrong! ……Page 183
Heating ……Page 185
Does The Excitement Show? ……Page 187
Selling Infinitesimal Values To Your Profit-Conscious Friends ……Page 189
Nim, Remoteness and Suspense in Hot Games ……Page 191
Overheating ……Page 192
Cooling the Children’s Party ……Page 196
But How Do You Cool A Party By One Degree? ……Page 197
Three Snort Lemmas ……Page 198
Why Thermostrat Works ……Page 201
… An On We Go! ……Page 204
Hotstrat, Thermostrat and Sentestrat ……Page 205
References and Further Reading ……Page 206
Green Hackenbush ……Page 207
Green Trees ……Page 209
Fusion ……Page 210
Proving The Fusion Principle ……Page 211
A More Complicated Picture ……Page 213
Impartial Maundy Cake ……Page 214
Hackenbush Hotchpotch ……Page 215
Flower Gardens ……Page 216
The Blue Flower Ploy ……Page 217
Atomic Weights ……Page 218
Atomic Weights of Jungles ……Page 219
Making Tracks in the Jungle ……Page 220
Tracking Down an Animal ……Page 222
Amazing Jungle ……Page 226
Smart Game in the Jungle ……Page 227
Unparted Jungles ……Page 228
Redwood Furniture ……Page 229
Redwood Beds ……Page 233
How Big Is A Redwood Bed? ……Page 234
What’s The Bottle ……Page 236
Both Ways of Adding Impartial Games ……Page 237
Many-Way Maundy Cake ……Page 238
Solution to Figure 15 ……Page 239
How Hard Was The Bed? ……Page 240
NP-Hardness ……Page 241
References and Further Reading ……Page 243
8 It’s a Small Small Small Small world ……Page 247
Uppitiness and Uncertainty ……Page 248
Computing Atomic Weights ……Page 249
Eatcake ……Page 251
Splitting The Atom ……Page 252
Turn-and-Eatcake ……Page 253
All You Need To Know About Atomic Weights But Were Afraid To Ask ……Page 254
Childish Hackenbush Hotchpotch ……Page 255
Atomic Weights of Lollipops ……Page 256
Playing Among the Flowers ……Page 258
When is $g$ as uppity as $h$? ……Page 259
Go Fly A Kite! ……Page 261
All Remote Stars Agree ……Page 262
Large and Small Flowerbeds ……Page 263
Playing Under a Lucky Star ……Page 264
General Multiples of Up ……Page 265
Proof of the Remote Star Rules ……Page 266
Proof That Atomic Weight = Uppitiness ……Page 267
Proper Care of the Eccentric ……Page 269
Galvinized Games ……Page 270
Trading Triangles ……Page 271
Multiples Work! ……Page 273
First for the “With” Rule ……Page 274
Now for the “Without” Rule: ……Page 275
A Theorem on Incentives ……Page 276
Seating Families of Five ……Page 278
References ……Page 280
Index ……Page 281