Grundy numbers of impartial three-dimensional chocolate-bar games
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Ryohei Miyadera
Abstract
Chocolate-bar games are variants of the Chomp game. Let Z≥0 be a set of nonnegative numbers, and let x, y, z ∈ Z≥0. A three-dimensional chocolate bar is comprised of a set of 1 × 1 × 1 cubes with a “bitter” or “poison” cube at the bottom of the column at position (0, 0). For u, w ∈ Z≥0 such that u ≤ x and w ≤ z, the height of the column at position (u, w) is min(F(u, w), y) + 1, where F is an increasing function. We denote such a chocolate bar as CB(F, x, y, z). Two players take turns to cut the bar along a plane horizontally or vertically along the grooves, and eat the broken pieces. The player who manages to leave the opponent with a single bitter cube is the winner. In a prior work, we characterized the function f for a two-dimensional chocolatebar game such that the Sprague-Grundy value of CB(f , y, z) is y ⊕ z. In this study, we characterize the function F such that the Sprague-Grundy value of CB(F, x, y, z) is x ⊕ y ⊕ z.
Abstract
Chocolate-bar games are variants of the Chomp game. Let Z≥0 be a set of nonnegative numbers, and let x, y, z ∈ Z≥0. A three-dimensional chocolate bar is comprised of a set of 1 × 1 × 1 cubes with a “bitter” or “poison” cube at the bottom of the column at position (0, 0). For u, w ∈ Z≥0 such that u ≤ x and w ≤ z, the height of the column at position (u, w) is min(F(u, w), y) + 1, where F is an increasing function. We denote such a chocolate bar as CB(F, x, y, z). Two players take turns to cut the bar along a plane horizontally or vertically along the grooves, and eat the broken pieces. The player who manages to leave the opponent with a single bitter cube is the winner. In a prior work, we characterized the function f for a two-dimensional chocolatebar game such that the Sprague-Grundy value of CB(f , y, z) is y ⊕ z. In this study, we characterize the function F such that the Sprague-Grundy value of CB(F, x, y, z) is x ⊕ y ⊕ z.
Chapters in this book
- 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
Chapters in this book
- 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
