The Max–Welter game

On a semi-infinite strip of squares rightward numbered 0,1,2,…0,1,2,… with at most one coin in each square, in Welter’s game, two players alternately move a coin to an empty square on its left. Jumping over other coins is legal. The player who first cannot move loses. We examine a variant of Welter’s game, that we call Max–Welter, in which players are allowed to move only the coin furthest to the right. We solve the winning strategy and describe the positions of Sprague–Grundy value 1. We propose two theorems classifying some special cases where calculating the Sprague–Grundy value of a position of size kk becomes easier by considering another position of size k−1k−1. We establish two results on the periodicity of the Sprague–Grundy values. We then show that the Max–Welter game is classified in a proper subclass of tame games that Gurvich calls strongly miserable.