Philosophers’ Phutball, invented in the towers of Cambridge by Conway and company, and named after a much-beloved Monty Python sketch of a similar name, is a two-player game played on a m×n board. The official Phutball pitch, as described by Berlekamp–Conway–Guy [BCG03], is 19×15 (that is, there are 19 rows and 15 columns); however, Phutball is usually played on a 19×19 Go board.

The rules of the game are fairly simple. One of the grid points is occupied by the ball, usually a black Go stone. Alfred, the first player, wins if the ball crosses the topmost row or ends up in the topmost row at the end of a turn. Betty, going second, wins if the ball crosses the bottommost row or ends up in the bottommost row at the end of a turn. On their turn, the players may either place a chap, usually a white Go stone, or move the ball. The ball is moved by jumping over a line of chaps in one of the eight possible directions, and those chaps are immediately removed; and multiple jumps are allowed, although not required. However, in all our diagrams, we will represent the ball by a gray stone and the chaps by black stones (Phutball played with reversed colors is called floodlit Phutball).

Despite the simplicity of the rules, the game is fairly complicated. Phutball on an n×n board is PSPACE-hard [Der10] and to even check if one has a win in one is NP-complete [DDE02]. Several variants of Phutball have been analyzed in great detail, such as “directional Phutball” [Loo08] when the players are constrained to jump in certain directions, or “one-dimensional Phutball” [GN02] played on an m ×1 board. Even the one-dimensional game is surprisingly complicated: It has only been analyzed fully in [GN02] when the players are forbidden to place “off-parity” chaps.