Relator games on groups
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Zachary Gates
Abstract
We define two impartial games, the Relator Achievement Game REL and the Relator Avoidance Game RAV. Given a finite group G and generating set S, both games begin with the empty word. Two players form a word in S by alternately appending an element from S ∪ S−1 at each turn. The first player to form a word equivalent in G to a previous word wins the game REL but loses the game RAV. Alternatively, we can think of REL and RAV as make a cycle and avoid a cycle games on the Cayley graph Γ(G, S). We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.
Abstract
We define two impartial games, the Relator Achievement Game REL and the Relator Avoidance Game RAV. Given a finite group G and generating set S, both games begin with the empty word. Two players form a word in S by alternately appending an element from S ∪ S−1 at each turn. The first player to form a word equivalent in G to a previous word wins the game REL but loses the game RAV. Alternatively, we can think of REL and RAV as make a cycle and avoid a cycle games on the Cayley graph Γ(G, S). We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.
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
