Dynamic One and Two Pile Nim Games Using Generalized Bases

Several authors have written papers dealing with a class of combinatorial games consisting of one-pile and two-pile counter pickup games for which the maximum number of counters that can be removed on each move changes during the play of the game. See Holshouser [8],[7], and Flanigan [4]. The maximum number of counters that can be removed can depend upon a number of factors including the number of counters removed on the last move, the sizes of the piles, and the move number in the game. In this paper we consider a two-pile game in which the maximum number of counters that can be removed depends on both the size of the last move and difference between the two pile sizes of the previous position. Consider the following game where f : N0 × N → N0 is a function. Two players alternate removing positive numbers of counters from two piles of counters. On each move the moving player chooses a pile and removes counters from the chosen pile. Initially, the player moving first can remove from one pile at most k counters. On each subsequent move, a player can remove from one pile a maximum of f(x− y, t) counters, where x and y, x ≥ y, are the pile sizes in