Circular Nim games CN(7, 4)
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Matthieu Dufour
Abstract
Circular Nim is a two-player impartial combinatorial game consisting of n stacks of tokens placed in a circle. A move consists of choosing k consecutive stacks and taking at least one token from one or more of the stacks. The last player able to make a move wins. The question of interest is: Who can win from a given position if both players play optimally? This question is answered by determining the set of P-positions from which the next player is bound to lose, no matter what moves the player makes. We will completely characterize the set of P-positions for n = 7 and k = 4, adding to the known results for other games in this family. The interesting feature of the set of P-positions of this game is that it splits into different subsets, unlike the structures for the previously solved games in this family.
Abstract
Circular Nim is a two-player impartial combinatorial game consisting of n stacks of tokens placed in a circle. A move consists of choosing k consecutive stacks and taking at least one token from one or more of the stacks. The last player able to make a move wins. The question of interest is: Who can win from a given position if both players play optimally? This question is answered by determining the set of P-positions from which the next player is bound to lose, no matter what moves the player makes. We will completely characterize the set of P-positions for n = 7 and k = 4, adding to the known results for other games in this family. The interesting feature of the set of P-positions of this game is that it splits into different subsets, unlike the structures for the previously solved games in this family.
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
