The game of i-Mark is an impartial combinatorial game introduced by Sopena (2016). The game is parametrized by two sets of positive integers $S$, $D$, where $\min D\ge 2$. From position $n\ge 0$ one can move to any position $n-s$, $s\in S$, as long as $n-s\ge 0$, as well as to any position $n/d$, $d\in D$, as long as $n>0$ and $d$ divides $n$. The game ends when no more moves are possible, and the last player to move is the winner. Sopena, and subsequently Friman and Nivasch (2021), characterized the Sprague-Grundy sequences of many cases of i-Mark$(S,D)$ with $|D|=1$. Friman and Nivasch also obtained some partial results for the case i-Mark$(\{1\},\{2,3\})$. In this paper we present a convergence technique that gives polynomial-time algorithms for the Sprague-Grundy sequence of many instances of i-Mark with $|D|>1$. In particular, we prove our technique works for all games i-Mark$(\{1\},\{d_1,d_2\})$. Keywords: Combinatorial game, impartial game, Sprague-Grundy function, convergence, dynamic programming.

翻译：i-Mark游戏是由Sopena（2016年）引入的一种无偏组合博弈。该博弈由两个正整数集合$S$和$D$参数化，其中$\min D\ge 2$。从位置$n\ge 0$出发，玩家可以移动到任意满足$n-s\ge 0$的位置$n-s$（其中$s\in S$），也可以移动到任意满足$n>0$且$d$整除$n$的位置$n/d$（其中$d\in D$）。当无法再进行移动时游戏结束，最后移动的玩家获胜。Sopena以及随后的Friman与Nivasch（2021年）刻画了$|D|=1$时多种i-Mark$(S,D)$实例的Sprague-Grundy序列。Friman与Nivasch还对i-Mark$(\{1\},\{2,3\})$情形获得了部分结果。本文提出一种收敛技术，可为$|D|>1$的多种i-Mark实例的Sprague-Grundy序列提供多项式时间算法。特别地，我们证明该技术适用于所有i-Mark$(\{1\},\{d_1,d_2\})$博弈。关键词：组合博弈，无偏博弈，Sprague-Grundy函数，收敛，动态规划。