$\mathcal{C}=\{1,2c\}$的\textsc{cut}的算术周期性
The Arithmetic-Periodicity of \textsc{cut} for $\mathcal{C}=\{1,2c\}$
摘要：
\textsc{cut}是一类在有限数量的有限堆令牌上进行的划分游戏。\textsc{cut}的每个版本都由割集$\mathcal{C}\subseteq\mathbb{N}$指定。合法移动包括选择一个桩并将其划分为$d+1$非空桩，其中$d\in\mathcal{C}$。没有代币从游戏中移除。结果表明，任何具有$c\geq 2$的$\mathcal{C}=\{1,2c\}$的nim集都是算术周期的，这回答了\cite{par}的一个公开问题。关键的一步是证明$\mathcal{C}=\{1,6\}$的\textsc{cut}的nim-集和$\mathcal{C}=\{1,2c\}, c\geq 4$的\textsc{cut}的nim-集之间存在对应关系。该结果很容易推广到$\mathcal{C} = \{1, 2c_1, 2c_2, 2c_3, ...\}$的情况，其中$c_1,c_2, ... \geq 2$。
英文摘要：
\textsc{cut} is a class of partition games played on a finite number of finite piles of tokens. Each version of \textsc{cut} is specified by a cut-set $\mathcal{C}\subseteq\mathbb{N}$. A legal move consists of selecting one of the piles and partitioning it into $d+1$ nonempty piles, where $d\in\mathcal{C}$. No tokens are removed from the game. It turns out that the nim-set for any $\mathcal{C}=\{1,2c\}$ with $c\geq 2$ is arithmetic-periodic, which answers an open question of \cite{par}. The key step is to show that there is a correspondence between the nim-sets of \textsc{cut} for $\mathcal{C}=\{1,6\}$ and the nim-sets of \textsc{cut} for $\mathcal{C}=\{1,2c\}, c\geq 4$. The result easily extends to the case of $\mathcal{C} = \{1, 2c_1, 2c_2, 2c_3, ...\}$, where $c_1,c_2, ... \geq 2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
