1. Introduction

John Conway's game Phutball [1-3,13], also known as Philosopher's Football, starts with a single black stone (the ball) placed at the center intersection of a rectangular grid such as a Go board. Two players sit on opposite sides of the board and take turns. On each turn, a player may either place a single white stone (a man) on any vacant intersection, or perform a sequence of jumps. To jump, the ball must be adjacent to one or more men. It is moved in a straight line (orthogonal or diagonal) to the first vacant intersection beyond the men, and the men so jumped are immediately removed (Figure 1). If a jump is performed, the same player may continue jumping as long as the ball continues to be adjacent to at least one man, or may end the turn at any point. Jumps are not obligatory: one can choose to place a man instead of jumping. The game is over when a jump sequence ends on or over the edge of the board closest to the opponent (the opponent's goal line) at which point the player who performed the jumps wins. It is legal for a jump sequence to step onto but not over one's own goal line. One of the interesting properties of Phutball is that any move could be played by either player, the only partiality in the game being the rule for determining the winner.

It is theoretically possible for a Phutball game to return to a previous position, so it may be necessary to add a loop-avoidance rule such as the one in Chess (three repetitions allow a player to claim a draw) or Go (certain repeated positions are disallowed). However, repetitions do not seem to come up much in actual practice.