left when moving in the forward direction (so if the devil were not blocking off squares, the angel would travel north indefinitely). Note that when going clockwise around a corner, the angel will not move for one step, because the two segments touching the corner have the same square to their right. Máthé's 2-angel proof Máthé introduces the nice devil, which never destroys a square that the angel could have chosen to occupy on an earlier turn. When the angel plays against the nice devil it concedes defeat if the devil manages to confine it to a finite bounded region of the board (otherwise the angel could just hop back and forth between two squares and never lose!). Máthé's proof breaks into two parts: he shows that if the angel wins against the nice devil, then the angel wins against the real devil; he gives an explicit winning strategy for the angel against the nice devil. Roughly speaking, in the second part, the angel wins against the nice devil by pretending that the entire left half-plane is destroyed (in addition to any squares actually destroyed by the nice devil), and treating destroyed squares as the walls of a maze, which it then skirts by means of a "hand-on-the-wall" technique. That is, the angel keeps its left hand on the wall of the maze and runs alongside the wall. One then proves that a nice devil cannot trap an angel that adopts this strategy. The proof of the first part is by contradiction, and hence Máthé's proof does not immediately yield an explicit winning strategy against the real devil. However, Máthé remarks that his proof could in principle be adapted to give such an explicit strategy. Bowditch's 4-angel proof Brian Bowditch defines a variant (game 2) of the original game with the following rule changes: The angel can return to any square it has already been to, even if the devil subsequently tried to block it. A k-devil must visit a square k times before it is blocked. The angel moves either up, down, left or right by one square (a duke move). 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