Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim
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Kyle Burke
Abstract
In this paper, we study Transverse Wave, a colorful, impartial combinatorial game played on a two-dimensional grid. We are drawn to this game because of its apparent simplicity, contrasting intractability, and intrinsic connection to two other combinatorial games, one about social influences and another inspired by quantum superpositions. More precisely, we show that Transverse Wave is at the intersection of two other games, the social-influence-derived Friend Circle and superpositionbased Demi-Quantum Nim. Transverse Wave is also connected with Schaefer’s logic game Avoid True from the 1970s. In addition to analyzing the mathematical structures and computational complexity of Transverse Wave, we provide a web-based version of the game. Furthermore, we formulate a basic network-influence game, called Demographic Influence, which simultaneously generalizes Node-Kayles and Demi- Quantum Nim (which in turn contains Nim, Avoid True, and Transverse Wave as particular cases). These connections illuminate a lattice order of games, induced by special-case/generalization relationships, fundamental to both design and comparative analysis of combinatorial games.
Abstract
In this paper, we study Transverse Wave, a colorful, impartial combinatorial game played on a two-dimensional grid. We are drawn to this game because of its apparent simplicity, contrasting intractability, and intrinsic connection to two other combinatorial games, one about social influences and another inspired by quantum superpositions. More precisely, we show that Transverse Wave is at the intersection of two other games, the social-influence-derived Friend Circle and superpositionbased Demi-Quantum Nim. Transverse Wave is also connected with Schaefer’s logic game Avoid True from the 1970s. In addition to analyzing the mathematical structures and computational complexity of Transverse Wave, we provide a web-based version of the game. Furthermore, we formulate a basic network-influence game, called Demographic Influence, which simultaneously generalizes Node-Kayles and Demi- Quantum Nim (which in turn contains Nim, Avoid True, and Transverse Wave as particular cases). These connections illuminate a lattice order of games, induced by special-case/generalization relationships, fundamental to both design and comparative analysis of combinatorial games.
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
