1. Introduction

Several works, including [Berlekamp et al. 1982; Epp and Ferguson 1980; Schwenk 1970], have studied take-away games where the players alternately remove a positive number of counters from a single pile, the player removing the last counter being the winner. On his initial move, the first player may remove any number of counters, so long as he leaves at least one. On each subsequent move, a player may remove at most f(n) counters, where n is the number of counters removed by his opponent on the preceding move. Thus, any mapping f from the positive integers to themselves defines such a take-away game. Epp and Ferguson [1980] considered the case where f is nondecreasing and f (1) ≥ 1. For any such f, let H1 = 1, H2 , • • • be the sizes of the initial pile from which the first player has no winning strategy; we call these the losing positions.