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Games of No Chance
MSRI Publications
Volume 29, 1996

The Angel Problem
http://www.msri.org/publications/books/Book29/files/conway.pdf
JOHN H. CONWAY

Abstract. Can the Devil, who removes one square per move from an infinite chessboard, strand the Angel, who can jump up to 1000 squares per move? It seems unlikely, but the answer is unknown. Andreas Blass and I have proved that the Devil can strand an Angel who's handicapped in one of several ways. I end with a challenge for the solution the general problem.

1. Introduction
The Angel and the Devil play their game on an infinite chessboard, with one square for each ordered pair of integers (x; y). On his turn, the Devil may eat any square of the board whatsoever; this square is then no longer available to the Angel. The Angel is a "chess piece" that can move to any uneaten square (X; Y ) that is at most 1000 king's moves away from its present position (x; y)-in other words, for which |X x| and |Y y| are at most 1000. Angels have wings, so that it does not matter if any intervening squares have already been eaten. The Devil wins if he can strand the Angel, that is, surround him by a moat
of eaten squares of width at least 1000. The Angel wins just if he can continue to move forever.

What we have described is more precisely called an Angel of power 1000. The Angel Problem is this:

Determine whether an Angel of some power can defeat the Devil.

Berlekamp showed that the Devil can beat an Angel of power one (a chess King) on any board of size at least 32 x33. However, it seems that it is impossible for the Devil to beat a Knight, and that would imply that an Angel of power two (which is considerably stronger than a Knight) will win. But can you prove it?