A convergence technique for the game i-MARK

The game of $i\mathrm{-MARK}$ 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 $i\mathrm{-MARK}(S,D)$ with $\left|D\right|=1$. Friman and Nivasch also obtained some partial results for the case $i\mathrm{-MARK}\left(\right\{1\},\{2,3\left\}\right)$. In this paper we present a convergence technique that gives polynomial-time algorithms for the Sprague–Grundy sequence of many instances of $i\mathrm{-MARK}$ with $\left|D\right|>1$. In particular, we prove our technique works for all games $i\mathrm{-MARK}(\left\{1\right\},\{d_1,d_2\})$.