Take-away Games

Several authors have considered 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. We prove various results (improving all previously known results) about the sequence of losing positions when f is a linear function.