Some i-Mark games

Abstract

Let \(S\) be a set of positive integers, and let \(D\) be a set of integers larger than \(1\). The game \(i\)-Mark\((S,D)\) is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract \(s \in S\) from the pile, or divide the size of the pile by \(d \in D\), if the pile size is divisible by \(d\). Sopena partially analyzed the games with \(S=[1, t-1]\) and \(D=\{d\}\) for \(d \not\equiv 1 \pmod t\), but left the case \(d \equiv 1 \pmod t\) open. We solve this problem by calculating the Sprague-Grundy function of \(i\)-Mark\(([1,t-1],\{d\})\) for \(d \equiv 1 \pmod t\), for all \(t,d \geq 2\). We also calculate the Sprague-Grundy function of \(i\)-Mark\((\{2\},\{2k + 1\})\) for all \(k\), and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with \(|D|>1\), we derive some partial results for the game \(i\)-Mark\((\{1\}, \{2, 3\})\), whose Sprague-Grundy function seems to behave erratically and does not show any clean pattern. We prove that each value \(0,1,2\) occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.