Playing Bynum’s game cautiously
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L. R. Haff
Abstract
Several sequences of infinitesimals are introduced for the purpose of analyzing a restricted form of Bynum’s game or “Eatcake”. Two of these have terms with uptimal values (à la Conway and Ryba, the 1970s). All others (eight) are specified by “uptimal+ forms,” i. e., standard uptimals plus a fractional uptimal. The game itself is played on an n × m grid of unit squares, and here we describe all followers (submatrices) of the 12 × 12 grid. Positional values of larger grids become intractable. However, an examination of n × n squares, 2 ≤ n ≤ 21, reveals that all but three of them are equal to ∗, the exceptions being the 10×10, 14×14, and 18×18 cases. Nonetheless, the exceptional cases have “star-like” characteristics: they are of the form ±(G), confused with both zero and up, and less than double-up.
Abstract
Several sequences of infinitesimals are introduced for the purpose of analyzing a restricted form of Bynum’s game or “Eatcake”. Two of these have terms with uptimal values (à la Conway and Ryba, the 1970s). All others (eight) are specified by “uptimal+ forms,” i. e., standard uptimals plus a fractional uptimal. The game itself is played on an n × m grid of unit squares, and here we describe all followers (submatrices) of the 12 × 12 grid. Positional values of larger grids become intractable. However, an examination of n × n squares, 2 ≤ n ≤ 21, reveals that all but three of them are equal to ∗, the exceptions being the 10×10, 14×14, and 18×18 cases. Nonetheless, the exceptional cases have “star-like” characteristics: they are of the form ±(G), confused with both zero and up, and less than double-up.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents XIII
- The game of flipping coins 1
- The game of blocking pebbles 17
- Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim 39
- A note on numbers 67
- Ordinal sums, clockwise hackenbush, and domino shave 77
- Advances in finding ideal play on poset games 99
- Strings-and-Coins and Nimstring are PSPACE-complete 109
- Partizan subtraction games 121
- Circular Nim games CN(7, 4) 139
- Misère domineering on 2 × n boards 157
- Relator games on groups 171
- Playing Bynum’s game cautiously 201
- Genetically modified games 229
- Game values of arithmetic functions 245
- A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups 281
- Recursive comparison tests for dicot and dead-ending games under misère play 309
- Impartial games with entailing moves 323
- Extended Sprague–Grundy theory for locally finite games, and applications to random game-trees 343
- Grundy numbers of impartial three-dimensional chocolate-bar games 367
- On the structure of misère impartial games 389
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents XIII
- The game of flipping coins 1
- The game of blocking pebbles 17
- Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim 39
- A note on numbers 67
- Ordinal sums, clockwise hackenbush, and domino shave 77
- Advances in finding ideal play on poset games 99
- Strings-and-Coins and Nimstring are PSPACE-complete 109
- Partizan subtraction games 121
- Circular Nim games CN(7, 4) 139
- Misère domineering on 2 × n boards 157
- Relator games on groups 171
- Playing Bynum’s game cautiously 201
- Genetically modified games 229
- Game values of arithmetic functions 245
- A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups 281
- Recursive comparison tests for dicot and dead-ending games under misère play 309
- Impartial games with entailing moves 323
- Extended Sprague–Grundy theory for locally finite games, and applications to random game-trees 343
- Grundy numbers of impartial three-dimensional chocolate-bar games 367
- On the structure of misère impartial games 389
