Angel Problem - Sketch of Bowditch's Proof (4-angel)

Sketch of Bowditch's Proof (4-angel)

Brian Bowditch defines a variant (game 2) of the original game with the following rule changes:

1. The angel can return to any square it has already been to, even if the devil subsequently tried to block it.
2. A k-devil must visit a square k times before it is blocked.
3. The angel moves either up, down, left or right by one square (a duke move).
4. To win, the angel must trace out a circuitous path (defined below).

A circuitous path is a path where is a semi-infinite arc (a non self-intersecting path with a starting point but no ending point) and are pairwise disjoint loops with the following property:

• where is the length of the ith loop.

(Note that to be well defined must begin and end at the end point of and must end at the starting point of )

Bowditch considers a variant (game 1) of the game with the changes 2 and 3 with a 5-devil. He then shows that a winning strategy in this game will yield a winning strategy in our original game for a 4-angel. He then goes on to show that an angel playing a 5-devil (game 2) can achieve a win using a fairly simple algorithm.

Bowditch claims that a 4-angel can win the original version of the game by imagining a phantom angel playing a 5-devil in the game 2.

The angel follows the path the phantom would take but avoiding the loops. Hence as the path is a semi-infinite arc the angel does not return to any square it has previously been to and so the path is a winning path even in the original game.