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 , 2 c } 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 , 2 c } , c ≥ 4 . The result easily extends to the case of C = { 1 , 2 c 1 , 2 c 2 , 2 c 3 , … } , where c 1 , c 2 , … ≥ 2 .