The Arithmetic-Periodicity of \textsc{cut} for $\mathcal{C}=\{1,2c\}$

Abstract

\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 $c_1,c_2, ... \geq 2$.