Purely Periodic Three-move Subtraction Games

Source abstract

A subtraction game is played on a heap of tokens. The players take turns removing s tokens for some s in a fixed set S of positive integers, and the player who cannot move loses. The sequence of Sprague-Grundy values of such a game is eventually periodic. We ask when it is purely periodic, meaning periodic from the very start, for the three-move sets S={a,b,c} with 0 =2, we give an explicit sufficient criterion for pure periodicity. For every non-additive set that satisfies it we determine the least period, all P-positions in closed form, and all nim-values, and the additive case c=a+b is treated separately. We conjecture that the criterion is also necessary, and we prove this for the period a+b whenever c>=2(a+b). The criterion is a finite test on the angle rho = c mod (a+b), built from the P-position pattern of the two-move game {a,b}, and the admissible angles form explicit unions of arcs in Z/(a+b)Z.