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" (that is, 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 does not have a winning strategy, then player 1 can...