The arithmetic-periodicity of CUT for C={1,2c}

Abstract

CUT is a class of partition games played on a finite number of finite piles of tokens. Each version of CUT is specified by a cut-set C⊆N. A legal move consists of selecting one of the piles and partitioning it into d+1 nonempty piles, where d∈C. No tokens are removed from the game. It turns out that the nim-set for any C={1,2c} with c≥2 is arithmetic-periodic, which answers an open question of Dailly et al. (2020). The key step is to show that there is a correspondence between the nim-sets of CUT for C={1,6} and the nim-sets of CUT for C={1,2c},c≥4. The result easily extends to the case of C={1,2c1,2c2,2c3,…}, where c1,c2,…≥2.