Strategy for a class of games with dynamic ties

Abstract

Let 0 < 2 b ≤ a < c be integers. Two players play alternately with a pile of stones. Each player at his turn selects one move from the following two: (i) Remove k stones from the pile subject to 1 ≤ k ≤ a or c + 1 ≤ k ≤ c + a . (ii) If the number m of stones in the pile satisfies m ≡ 2 b (mod 2 a ), add a stones to the pile. The player making the last move wins. If there is no last move, the game is a (dynamic) tie. The Generalized Sprague—Grundy function G is determined, thus giving the strategy of play for the game and its disjunctive compound. An algorithm requiring O( a 2 ) steps for computing G is given. It turns out that G = G (a, b, c) is of a rather complicated form. The main interest of the paper is in presenting a complete strategy for a class of games with dynamic ties.