One-Dimensional Phutball

Summary

ABSTRACT. We consider the game of one-dimensional phutball. We solve the case of a restricted version called Oddish Phutball by presenting an explicit strategy in terms of a potential function.

1. Introduction

J. H. Conway's Philosophers Football, otherwise known as Phutball [1], is played by two players, Left and Right, who move alternately. The game is usually played on a 19 by 19 board, starts with a ball on a square, one side is designated the Left goal line and the opposite side the Right goal line. A move consists of either placing a stone on an unoccupied square or jumping the ball over a (horizontal, vertical or diagonal) line of stones one end of which is adjacent to the square containing the ball. The stones are removed immediately after being jumped. Jumps can be chained and a player does not have to jump when one is available. A player wins by getting the ball on or over the opponent's goal line. Demaine, Demaine and Eppstein [2] have shown that deciding whether or not a player has a winning jump is NP complete.

In this paper we consider the 1-dimensional version of the game. For example:

is a position on a finite linear strip of squares. Initially, there is a black stone, •, which represents the ball. A player on his turn can either place a white stone, o, on an empty square, or jump the ball over a string of contiguous stones one end of which is adjacent to the ball; the ball ends on the next empty square. The stones are removed immediately upon jumping. A jump can be continued if there is another group adjacent to the ball's new position. Left wins by jumping the ball onto or over the rightmost square —Right's goalline; Right wins by jumping on over the leftmost square — Left's goalline

It would seem clear that

• • Your position can only improve by having a stone placed between the ball and the opponent's goalline; and

• • Your position can only improve if an empty square between the ball and the opponent's goalline is deleted.