Properties of K-Welter's game

Abstract Place n coins on distinct squares of a semi-infinite strip. The squares are numbered with the nonnegative integers 0,1,2,3,… from the left end of the strip. Each of the two players alternately moves a coin to a lower unoccupied square, at most k squares from its present position, where k is a fixed positive integer. The player first unable to move (because the coins are jammed in the lowest possible numbered squares 0,1,2,..., n −1) loses and his opponent wins. The main interest of this paper is in investigating interesting properties of the game, presenting a solution for the case of three coins(> n =3) for any k (by giving a strategy with a polynomial algorithm) and investigating properties of the Sprague-Grundy function for this case, including its polynomial computation for the subcase k =2 m −2.