Angel problem
Angel problem
The angel problem is a question in combinatorial-game-theory proposed by john-horton-conway. The game is commonly referred to as the angels and devils game. The game is played by two players called the angel and the devil. It is played on an infinite chessboard (or equivalently the points of a 2D lattice). The angel has a power k (a natural number 1 or higher), specified before the game starts. The board starts empty with the angel in one square. On each turn, the angel jumps to a different empty square which could be reached by at most k moves of a chess king, i.e. the distance from the starting square is at most k in the infinity norm. The devil, on its turn, may add a block on any single square not containing the angel. The angel may leap over blocked squares, but cannot land on them. The devil wins if the angel is unable to move. The angel wins by surviving indefinitely.
The angel problem is: Can an angel with high enough power win?
There must exist a winning strategy for one of the players. If the devil can force a win then it can do so in a finite number of moves. If the devil cannot force a win then there is always an action that the angel can take to avoid losing and a winning strategy for it is always to pick such a move. More abstractly, the "pay-off set" (i.e., the set of all plays in which the angel wins) is a closed set (in the natural topology on the set of all plays), and it is known that such games are determined. Of course, for any infinite game, if player 2 doesn't have a winning strategy, player 1 can always pick a move that leads to a position where player 2 doesn't have a winning strategy, but in some games, simply playing forever doesn't confer a win to player 1, so undetermined games may exist.
Conway offered a reward for a general solution to this problem ($100 for a winning strategy for an angel of sufficiently high power, and $1000 for a proof that the devil can win irrespective of the angel's power). Progress was made first in higher dimensions. In late 2006, the original problem was solved when independent proofs appeared, showing that an angel can win. Bowditch proved that a 4-angel (that is, an angel with power k = 4) can win and Máthé and Kloster gave proofs that a 2-angel can win.
History The problem was first published in the 1982 book Winning Ways by Berlekamp, Conway, and Guy, under the name "the angel and the square-eater". In two dimensions, some early partial results included:
- If the angel has power 1, the devil has a winning strategy (Conway, 1982). (According to Conway, this result is actually due to Berlekamp.) This can be read at section 1.1 of Kutz's paper.
- If the angel never decreases its y coordinate, then the devil has a winning strategy (Conway, 1982).
- If the angel always increases its distance from the origin, then the devil has a winning strategy (Conway, 1996).
In three dimensions, it was shown that:
- If the angel always increases its y coordinate, and the devil can only play in one plane, then the angel has a winning strategy.
- If the angel always increases its y coordinate, and the devil can only play in two planes, then the angel has a winning strategy.
- The angel has a winning strategy if it has power 13 or more.
- If we have an infinite number of devils each playing at distances <math>d_1 < d_2 < d_3 < \cdots</math> then the angel can still win if it is of high enough power. (By "playing at distance <math>d</math>" 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, Peter Gacs 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 a 3D variant, the angel must increase its y-coordinate with every move, while the devil is limited to blocking squares that lie within three fixed planes. It remains unknown whether the devil has a winning strategy under these conditions.
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 <math>y = -\infty</math> and up to <math>y = \infty</math>. 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 squares 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 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.