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Pré-Publication, Document De Travail Année : 2018

Partition games are pure breaking games

Résumé

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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Dates et versions

hal-01723190 , version 1 (05-03-2018)
hal-01723190 , version 2 (23-10-2019)
hal-01723190 , version 3 (06-05-2020)

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Antoine Dailly, Eric Duchene, Urban Larsson, Gabrielle Paris. Partition games are pure breaking games. 2018. ⟨hal-01723190v1⟩

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