angel can still win if it is of high enough power. (By "playing at distance " we mean the devil is not allowed to play within this distance of the origin). Finally, in 2006 (not long after the publication of Peter Winkler's book Mathematical Puzzles, which helped publicize the angel problem) there emerged four independent and almost simultaneous proofs that the angel has a winning strategy in two dimensions. Brian Bowditch's proof works for the 4-angel, while Oddvar Kloster's proof and András Máthé's proof work for the 2-angel. Additionally, Péter Gács has a claimed proof which requires a much stronger angel; the details are fairly complex and have not been reviewed by a journal for accuracy. The proofs by Bowditch and Máthé have been published in Combinatorics, Probability and Computing. The proof by Kloster has been published in Theoretical Computer Science. Further unsolved questions In 3D, given that the angel always increases its y-coordinate, and that the devil is limited to three planes, it is unknown whether the devil has a winning strategy. Proof sketches Kloster's 2-angel proof Oddvar Kloster discovered a constructive algorithm to solve the problem with a 2-angel. This algorithm is quite simple and also optimal, since, as noted above, the devil has a winning strategy against a 1-angel. We start out by drawing a vertical line immediately to the left of the angel's starting position, down to and up to . This line represents the path the angel will take, which will be updated after each of the devil's moves, and partitions the board's squares into a "left set" and a "right set." Once a square becomes part of the left set, it will remain so for the remainder of the game, and the angel will not make any future moves to any of these squares. Every time the devil blocks off a new square, we search over all possible modifications to the path such that we move one or more squares in the right set which the devil has blocked off into the left set. We will only do this if the path increases in length by no more than twice the number of blocked square moved into the left set. Of such qualifying paths, we choose one that moves the greatest number of blocked off squares into the left set. The angel then makes two steps along this path, keeping the path to its