Partition games are pure breaking games

Abstract : Taking-and-breaking games are combinatorial games played on heaps of tokens, where both players are allowed to remove tokens from a heap and/or split a heap into smaller heaps. Subtraction games, octal and hexadecimal games are well-known families of such games. We here consider the set of pure breaking games, that correspond to the family of taking-and-breaking games where splitting heaps only is allowed. The rules of such games are simply given by a list L of positive integers corresponding to the number of sub-heaps that a heap must be split into. Following the case of octal and hexadecimal games, we provide a computational testing condition to prove that the Grundy sequence of a given pure breaking game is arithmetic periodic. In addition, the behavior of the Grundy sequence is explicitly given for several particular values of L (e.g. when 1 is not in L or when L contains only odd values). However, despite the simplicity of its ruleset, the behavior of the Grundy function of the game having L = {1, 2} is open.
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Contributeur : Antoine Dailly <>
Soumis le : lundi 5 mars 2018 - 13:38:05
Dernière modification le : jeudi 19 avril 2018 - 14:38:06
Document(s) archivé(s) le : mercredi 6 juin 2018 - 15:11:20


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  • HAL Id : hal-01723190, version 1
  • ARXIV : 1803.02621


Antoine Dailly, Eric Duchene, Urban Larsson, Gabrielle Paris. Partition games are pure breaking games. 2018. 〈hal-01723190〉



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