i-MARK: A New Subtraction Division Game

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 moveeither to a heap of $n-s$ tokens for some $s\in S$, or to a heap of $n/d$ tokensfor 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 Buhlerand studied by Aviezri Fraenkel and Alan Guo, for which it is allowed to move from a heap of $n$ tokensto 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.Comment: A few typos have been corrected, including the statement of Theorem

Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence

Nous d\'ecrivons quelques r\'esultats r\'ecents sur la suite de Thue-Morse, ainsi que des questions ou conjectures, dont l'une, due \`a Shevelev, est r\'esolue dans cet article. We describe some recent results on the Thue-Morse sequence. We also list open questions and conjectures, one of which is due to Shevelev and proved in this paper.Comment: Proof of Shevelev's conjecture fixe