On the Sprague-Grundyfunction of Exact k-Nim

Moore’s generalization of the game of Nim is played as follows. Given two integer parameters n, k such that 1 ≤ k ≤ n, and n piles of tokens. Two players take turns. By one move a player reduces at least one and at most k piles. The player who makes the last move wins. The P-positions of this game were characterized by Moore in 1910 and an explicit formula for its Sprague-Grundy function was given by Jenkyns and Mayberry in 1980, for the case n = k+1 only. We modify Moore’s game and introduce Exact k-Nim in which each move reduces exactly k piles. We give a simple polynomial algorithm computing the Sprague-Grundy function of Exact k-Nim in case 2k > n and an explicit formula for it in case 2k = n. The last case shows a surprising similarity to Jenkyns and Mayberry’s solution even though the meaning of some of the expressions are quite different.