On the Sprague–Grundyfunction of Exact[formula omitted]-Nim

Abstract

Moore’s generalization of the game of Nim is played as follows. Let n and k be two integers such that 1≤k<n. Given n piles of tokens, two players move alternately, removing tokens from at least one and at most k of the piles. The player who makes the last move wins. The game was solved by Moore in 1910; an explicit formula for its Sprague–Grundy function was given by Jenkyns and Mayberry in 1980, but only for the case n=k+1. We introduce another generalization of Nim, called Exactk- Nim, in which exactly k piles are reduced by each move. We give an explicit formula for the Sprague–Grundy function of Exactk- Nim for the case 2k≥n. If n=2k, our formula is surprisingly similar to Jenkyns and Mayberry’s one.