Let S be a set of positive integers, and let D be a set of integers larger than 1. The game

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 $d\not\equiv 1(\mathrm{mod}t)$, but left the case $d\equiv 1(\mathrm{mod}t)$ open.

We solve this problem by calculating the Sprague–Grundy function of

for $d\equiv 1(\mathrm{mod}t)$, for all $t,d\ge 2$. We also calculate the Sprague–Grundy function of

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

, whose Sprague–Grundy function seems to behave erratically and does not show any clear pattern. We prove that each value $0,1,2$ occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.

中文翻译：

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一些 i-Mark 游戏

让S是一组正整数，让D是一组大于 1 的整数。 游戏

是 Sopena (2016) 推出的一种公正的组合游戏，使用单堆令牌进行游戏。在每一回合中，玩家可以减去$秒\in 秒$ 从桩，或将桩的大小除以 $d\in D$, 如果桩的大小可以被d整除。Sopena 部分分析了游戏$秒=[1,吨-1]$ 和 $D=\{d\}$ 为了 $d\not\equiv 1(\mathrm{模组}吨)$，但离开了这个案子 $d\equiv 1(\mathrm{模组}吨)$ 打开。

我们通过计算 Sprague-Grundy 函数来解决这个问题

为了 $d\equiv 1(\mathrm{模组}吨)$， 对所有人 $吨,d\ge 2$. 我们还计算了 Sprague-Grundy 函数

对于所有k，并表明它表现出相似的行为。最后，按照 Sopena 的建议看游戏$|D|>1$，我们为游戏推导出了一些部分结果

，其 Sprague-Grundy 函数似乎表现不规则，并且没有显示任何清晰的模式。我们证明每个值$0,1,2$ 在其 SG 序列中无限频繁地出现，连续出现之间具有最大间隙长度。