Ordinal sums, clockwise hackenbush, and domino shave
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Alda Carvalho
Abstract
We present two rulesets, domino shave and clockwise hackenbush. The first is somehow natural and has, as special cases, stirling shave and Hetyei’s Bernoulli game. Clockwise hackenbush seems artificial, yet it is equivalent to domino shave. From the pictorial form of the game and a knowledge of hackenbush, the decomposition into ordinal sums is immediate. The values of clockwise bluered hackenbush are numbers, and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is {x | } or { | x}, and x is a number. That formula generalizes van Roode’s signed binary number method for blue-red hackenbush.
Abstract
We present two rulesets, domino shave and clockwise hackenbush. The first is somehow natural and has, as special cases, stirling shave and Hetyei’s Bernoulli game. Clockwise hackenbush seems artificial, yet it is equivalent to domino shave. From the pictorial form of the game and a knowledge of hackenbush, the decomposition into ordinal sums is immediate. The values of clockwise bluered hackenbush are numbers, and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is {x | } or { | x}, and x is a number. That formula generalizes van Roode’s signed binary number method for blue-red hackenbush.
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
