Some i-Mark games
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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 ∈ S from the pile, or divide the size of the pile by d ∈ D, if the pile size is divisible by d. Sopena partially analyzed the games with S=[1, t-1] and D={d} for d ≢1 t, but left the case d ≡ 1 t open. We solve this problem by calculating the Sprague-Grundy function of i-Mark([1,t-1],{d}) for d ≡ 1 t, for all t,d ≥ 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.
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