Phutball Endgames are Hard

TL;DR

This paper proves that determining winning moves in the game Phutball is NP-complete, highlighting its computational complexity, unlike similar problems in checkers which are polynomial-time solvable.

Contribution

The paper establishes the NP-completeness of deciding immediate winning moves in Phutball, a significant complexity result for this combinatorial game.

Findings

Determining immediate winning moves in Phutball is NP-complete.

Unlike checkers, Phutball's move decision problem is computationally hard.

The result contrasts Phutball's complexity with simpler similar games.

Abstract

We show that, in John Conway's board game Phutball (or Philosopher's Football), it is NP-complete to determine whether the current player has a move that immediately wins the game. In contrast, the similar problems of determining whether there is an immediately winning move in checkers, or a move that kings a man, are both solvable in polynomial time.