Winning Ways for Your Mathematical Plays, Volume 3

Title page ......Page 1
Date-line ......Page 2
Dedication ......Page 3
Authors ......Page 5
Contents ......Page 7
Preface to Second Edition ......Page 12
Preface ......Page 13
Spade-Work! ......Page 16
1 Whose Game? ......Page 18
Blue-Red Hackenbush ......Page 19
The Tweedledum and Tweedledee Argument ......Page 20
How Can You Have Half a Move? ......Page 21
... And Quarter Moves? ......Page 23
Ski-Jumps for Beginners ......Page 25
Don\'t Just Take the Average! ......Page 27
What Is a Jump Worth? ......Page 28
Do Our Methods Work? ......Page 29
What is a Game? ......Page 31
When Is a Move Good? ......Page 33
Figure 8(d) Is Worth § ......Page 34
References and Further Reading ......Page 35
Which Numbers Are Which? ......Page 36
Simplicity\'s the Answer! ......Page 38
Simplest Forms for Numbers ......Page 39
Cutcake ......Page 41
Maundy Cake ......Page 43
A Few More Applications of the Simplicty Rule ......Page 44
Positive, Negative, Zero, and Fuzzy Positions ......Page 45
Hackenbush Hotchpotch ......Page 46
Sums of Arbitrary Games ......Page 47
The Outcome of a Sum ......Page 48
The Negative of a Game ......Page 50
Cancelling a Game with its Negative ......Page 51
Comparing Hackenbush Positions ......Page 52
The Game of Col ......Page 54
A Star is Born! ......Page 55
Col Contains Such Values ......Page 56
Green Hackenbush, The Game of Nim, and Nimbers ......Page 57
Get Nimble with Nimbers ......Page 59
Childish Hackenbush ......Page 60
Seating Couples ......Page 61
The Sum of Two Finite Games Can Last Forever ......Page 63
A Theorem about Col ......Page 64
Col-lections and Col-lapsings ......Page 65
How Childish Can You Get? ......Page 68
References and Further Reading ......Page 69
Poker-Nim ......Page 70
Northcott\'s Game ......Page 71
Bogus Nim-Heaps and the Mex Rule ......Page 72
The White Knight ......Page 73
Adding Nimbers ......Page 75
Wyt Queens ......Page 76
Reversible Moves in General Games ......Page 77
Deleting Dominated Options ......Page 79
Toads-and-Frogs with Ups and Downs ......Page 80
Game Tracking and Identification ......Page 82
What Are Flowers Worth? ......Page 83
A Gallimaufry of Games ......Page 84
A Closer Look at the Stars ......Page 85
The Values {f | t} and {0| t} ......Page 86
The Upstart Equality ......Page 88
Gift Horses ......Page 89
The Nim-Addition Rule in Several Variations ......Page 90
Wyt Queens and Wythoff\'s Game ......Page 91
Toad Versus Frog ......Page 92
Two Theorems on Simplifying Games ......Page 93
Berlekamp\'s Rule for Hackenbush Strings ......Page 94
References and Further Reading ......Page 95
Kayles ......Page 98
Games With Heaps ......Page 99
Subtraction Games ......Page 100
Ferguson\'s Pairing Property ......Page 103
Other Take-Away Games ......Page 104
Dawson\'s Chess ......Page 105
The Periodicity of Kayles ......Page 107
Other Take-and-Break Games ......Page 108
Dawson\'s Kayles ......Page 109
Guiles ......Page 110
Treblecross ......Page 111
Officers ......Page 112
Grundy\'s Game ......Page 113
Prim and Dim ......Page 114
Double and Quadruple Kayles ......Page 115
Lasker\'s Nim ......Page 116
Standard Form ......Page 117
A Compendium of Octal Games ......Page 118
Sparse Spaces and Common Cosets ......Page 126
Will Grundy\'s Game Be Ultimately Periodic? ......Page 128
Sparse Space Spells Speed ......Page 129
Games Displaying Arithmetic Periodicity ......Page 130
A Non-Arithmetic-Periodicity Theorem ......Page 132
Further References and Reading ......Page 134
Domineering ......Page 136
Switch Games ......Page 138
Cashing Cheques ......Page 139
The Tiniest Games ......Page 142
Modern Management of Cash Flow ......Page 143
The Opening Dissection of Toads-and-Frogs ......Page 144
Seating Boys and Girls ......Page 148
Toads-and-Frogs Completely Dissected ......Page 150
Toads-and-Frogs with Two Spaces ......Page 153
More Domineering Values ......Page 155
References ......Page 160
A Graphic Picture of Farm Life ......Page 162
Don\'t Move In A Number Unless There\'s Nothing Else To Do! ......Page 164
What\'s in it for Me? ......Page 165
The Left and Right Stops ......Page 166
Cooling—and the Thermograph ......Page 168
Cooling Settles the Mean Value ......Page 169
How to Draw Thermographs ......Page 170
When A Player Has Several Options ......Page 171
Foundations for Thermographs ......Page 172
Examples of Thermographs ......Page 173
Who Is To Move From The Final Stop? ......Page 174
The Cheque-Market Exchange ......Page 175
Equitable Games ......Page 176
Excitable Games ......Page 177
The Extended Thermograph ......Page 178
Getting the Right Slant ......Page 179
The Thermostatic Strategy ......Page 180
Thermostrat\'s Not Often Wrong! ......Page 182
Heating ......Page 184
Does The Excitement Show? ......Page 186
Selling Infinitesimal Values To Your Profit-Conscious Friends ......Page 188
Nim, Remoteness and Suspense in Hot Games ......Page 190
Overheating ......Page 191
Cooling the Children\'s Party ......Page 195
But How Do You Cool A Party By One Degree? ......Page 196
Three Snort Lemmas ......Page 197
Why Thermostrat Works ......Page 200
... An On We Go! ......Page 203
Hotstrat, Thermostrat and Sentestrat ......Page 204
References and Further Reading ......Page 205
Green Hackenbush ......Page 206
Green Trees ......Page 208
Fusion ......Page 209
Proving The Fusion Principle ......Page 210
A More Complicated Picture ......Page 212
Impartial Maundy Cake ......Page 213
Hackenbush Hotchpotch ......Page 214
Flower Gardens ......Page 215
The Blue Flower Ploy ......Page 216
Atomic Weights ......Page 217
Atomic Weights of Jungles ......Page 218
Making Tracks in the Jungle ......Page 219
Tracking Down an Animal ......Page 221
Amazing Jungle ......Page 225
Smart Game in the Jungle ......Page 226
Unparted Jungles ......Page 227
Redwood Furniture ......Page 228
Redwood Beds ......Page 232
How Big Is A Redwood Bed? ......Page 233
What\'s The Bottle ......Page 235
Both Ways of Adding Impartial Games ......Page 236
Many-Way Maundy Cake ......Page 237
Solution to Figure 15 ......Page 238
How Hard Was The Bed? ......Page 239
NP-Hardness ......Page 240
References and Further Reading ......Page 242
8 It\'s a Small Small Small Small world ......Page 246
Uppitiness and Uncertainty ......Page 247
Computing Atomic Weights ......Page 248
Eatcake ......Page 250
Splitting The Atom ......Page 251
Turn-and-Eatcake ......Page 252
All You Need To Know About Atomic Weights But Were Afraid To Ask ......Page 253
Childish Hackenbush Hotchpotch ......Page 254
Atomic Weights of Lollipops ......Page 255
Playing Among the Flowers ......Page 257
When is g as uppity as hi ......Page 258
Go Fly A Kite! ......Page 260
All Remote Stars Agree ......Page 261
Large and Small Flowerbeds ......Page 262
Playing Under a Lucky Star ......Page 263
General Multiples of Up ......Page 264
Proof of the Remote Star Rules ......Page 265
Proof That Atomic Weight = Uppitiness ......Page 266
Proper Care of the Eccentric ......Page 268
Galvinized Games ......Page 269
Trading Triangles ......Page 270
Multiples Work! ......Page 272
First for the \"With\" Rule ......Page 273
Now for the \"Without\" Rule: ......Page 274
A Theorem on Incentives ......Page 275
Seating Families of Five ......Page 277
References ......Page 279
Index ......Page 280