of the ith loop. (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. 3D version of the problem "Guardian" proof The proof, which shows that in a three-dimensional version of the game a high powered angel has a winning strategy, makes use of "guardians". For each cube of any size, there is a guardian that watches over that cube. The guardians decide at each move whether the cube they are watching over is unsafe, safe, or almost safe. The definitions of "safe" and "almost safe" need to be chosen to ensure this works. This decision is based purely on the density of blocked points in that cube and the size of that cube. If the angel is given no orders, then it just moves up. If some cubes that the angel is occupying cease to be safe, then the guardian of the biggest of these cubes is instructed to arrange for the angel to leave through one of the borders of that cube. If a guardian is instructed to escort the angel out of its cube to a particular face, the guardian does so by plotting a path of subcubes that are all safe. The guardians in these cubes are then instructed to escort the angel through their respective subcubes. The angel's path in a given subcube is not determined until the angel arrives at that cube. Even then, the path is only determined roughly. This ensures the devil cannot just choose a place on the path sufficiently far along it and