Dynamic Single-Pile Nim Using Multiple Bases

In the game G0 two players alternate removing positive numbers of counters from a single pile and the winner is the player who removes the last counter. On the first move of the game, the player moving first can remove a maximum of k counters, k being specified in advance. On each subsequent move, a player can remove a maximum of f(n, t) counters where t was the number of counters removed by his opponent on the preceding move and n is the preceding pile size, where f : N × N → N is an arbitrary function satisfying the condition (1): ∃t ∈ N such that for all n, x ∈ N , f(n, x) = f(n + t, x). This note extends our paper [5] that appeared in E-JC. We first solve the game for functions f : N × N → N that also satisfy the condition (2): ∀n, x ∈ N , f(n, x + 1) − f(n, x) ≥ −1. Then we state the solution when f : N ×N → N is restricted only by condition (1) and point out that the more general proof is almost the same as the simpler proof. The solutions when t ≥ 2 use multiple bases.