Winning strategies for aperiodic subtraction games

We provide a winning strategy for sums of games of Mark-t, an impartial game played on nonnegative integers where each move consists of subtraction by an integer between 1 and t−1 inclusive, or division by t, rounding down when necessary. Our algorithm computes the Sprague–Grundy values for arbitrary n in quadratic time. This addresses one of the directions of further study proposed by Aviezri Fraenkel. In addition, we characterize the P-positions and N-positions for the game in misère play.