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Article Dans Une Revue Theoretical Computer Science Année : 2016

i-MARK: A New Subtraction Division Game

Résumé

Given two finite sets of integers $S\subseteq\NNN\setminus\{0\}$ and $D\subseteq\NNN\setminus\{0,1\}$, the impartial combinatorial game $\IMARK(S,D)$ is played on a heap of tokens. From a heap of $n$ tokens, each player can move either to a heap of $n-s$ tokens for some $s\in S$, or to a heap of $n/d$ tokens for some $d\in D$ if $d$ divides $n$. Such games can be considered as an integral variant of \MARK-type games, introduced by Elwyn Berlekamp and Joe Buhler and studied by Aviezri Fraenkel and Alan Guo, for which it is allowed to move from a heap of $n$ tokens to a heap of $\lfloor n/d\rfloor$ tokens for any $d\in D$. Under normal convention, it is observed that the Sprague-Grundy sequence of the game $\IMARK(S,D)$ is aperiodic for any sets $S$ and $D$. However, we prove that, in many cases, this sequence is almost periodic and that the set of winning positions is periodic. Moreover, in all these cases, the Sprague-Grundy value of a heap of $n$ tokens can be computed in time $O(\log n)$. We also prove that, under mis\`ere convention, the outcome sequence of these games is purely periodic.
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Dates et versions

hal-01198929 , version 1 (14-09-2015)
hal-01198929 , version 2 (07-11-2015)

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Citer

Eric Sopena. i-MARK: A New Subtraction Division Game. Theoretical Computer Science, 2016, 627, pp.90-101. ⟨hal-01198929v2⟩

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