Permutations of counters on a table

Abstract

We consider a game in which a blindfolded player attempts to set $n$ counters lying on the vertices of a rotating regular $n$-gon table simultaneously to $0$. When the counters count$(\mathrm{mod}m)$ we simplify the argument of Bar Yehuda, Etzion, and Moran (1993) showing that the player can win if and only if $n=1$, $m=1$, or $(n,m)=(p^a,p^b)$ for some prime $p$ and $a,b\in \mathbb{N}$. We broadly generalize the result to the setting where the counters can be permuted by any element of a subset of the symmetric group $S\subseteq S_n$, with the original formulation corresponding to $S={\mathbb{Z}}_n$ (rotations of the table).

arXiv comment7 pages