论文标题
\ textsc {cut}的算术 - periodicity for $ \ mathcal {c} = \ {1,2c \} $
The Arithmetic-Periodicity of \textsc{cut} for $\mathcal{C}=\{1,2c\}$
论文作者
论文摘要
\ textsc {cut}是在有限数量的有限令牌上玩的一类分区游戏。 \ textsc {cut}的每个版本均由cut-set $ \ mathcal {c} \ subseteq \ mathbb {n} $指定。法律举措包括选择其中一块，然后将其划分为$ d+1 $ nonepty桩，其中$ d \ in \ Mathcal {C} $。没有令牌从游戏中删除。事实证明，任何$ \ MATHCAL {C} = \ {1,2C \} $带有$ C \ geq 2 $的NIM-SET是arithmetic-periodic，它回答了\ cite {par}的开放问题。关键步骤是证明\ textsc {cut}的Nim-sets对于$ \ Mathcal {c} = \ {1,6 \} $与\ textSc {cut}的NIM-sets之间的nim-sets之间存在对应关系。结果很容易扩展到$ \ MATHCAL {C} = \ {1，2C_1，2C_2，2C_3，... \} $，其中$ C_1，C_2，... \ GEQ 2 $。
\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 $c_1,c_2, ... \geq 2$.
