A convergence technique for the game i-Mark

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.