Strategy for a class of games with dynamic ties

Let 0 < 2b ≤ 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 ≡ 2b (mod 2a), 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(a²) 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.