一些i-Mark游戏
Some i-Mark games
摘要：
令$S$是正整数的集合，并且令$D$是大于$1$的整数的集合。$i$-Mark $(S,D)$是一款由Sopena（2016）推出的公平组合游戏，使用一堆代币进行游戏。在每一轮中，玩家可以从牌堆中减去$s \in S$，或者如果牌堆的大小可以被$d$整除，则可以将牌堆的大小除以$d \in D$。Sopena部分分析了使用$S=[1, t-1]$和$D=\{d\}$的$d \not\equiv 1 \pmod t$游戏，但留下了$d \equiv 1 \pmod t$的案例。我们通过计算$i$的Sprague-Grundy函数-Mark $([1,t-1],\{d\})$ for $d \equiv 1 \pmod t$，for all $t,d \geq 2$来解决这个问题。我们还计算了所有$k$的$i$-Mark $(\{2\},\{2k + 1\})$的Sprague-Grundy函数，并表明它表现出类似的行为。最后，根据索佩纳的建议，看看游戏与$|D|>1$，我们得出了一些部分结果的游戏$i$-马克$(\{1\}, \{2, 3\})$，其Sprague-Grundy函数似乎表现不稳定，并没有显示任何干净的模式。我们证明，每个值$0,1,2$发生无限经常在其SG序列，连续出现之间的最大间隙长度。
英文摘要：
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.
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