论文标题
一些I-Mark游戏
Some i-Mark games
论文作者
论文摘要
让$ s $是一组正整数，让$ d $是一组大于$ 1 $的整数。游戏$ i $ -mark $（s，d）$是Sopena（2016）引入的公正组合游戏，该游戏带有一堆令牌。在每个回合中，如果桩尺寸可除以$ d $，则玩家可以从堆中减去S $中的$ s \ in s $，或将桩的大小除以$ d \。 Sopena用$ s = [1，t-1] $和$ d = \ {d \} $部分分析了游戏，用于$ d \ not \ equiv 1 \ pmod t $，但留下了case $ d \ equiv 1 \ pmod t $打开。 我们通过计算$ i $ -mark $（[1，t-1]，\ {d \}）$的sprague-grundy函数来解决此问题。我们还计算所有$ k $的$ i $ -mark $（\ {2 \}，\ {2k + 1 \}）$的sprague-grundy函数，并表明它表现出相似的行为。最后，按照Sopena提出了$ | d |> 1 $的看法，我们为游戏$ i $ -mark $（\ {1 \}，\ {2，3 \}）$获得了一些部分结果，其sprague-grundy函数似乎不错，并且没有表现出任何干净的模式。我们证明，每个值$ 0,1,2 $在其SG序列中经常发生，连续出现之间的间隙长度最大。
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.
