The Angel wins

The angel-devil game is played on an infinite two-dimensional “chessboard” Z. The squares of the board are all white at the beginning. The players called angel and devil take turns in their steps. When it is the devil’s turn, he can turn a square black. The angel always stays on a white square, and when it is her turn she can fly at a distance of at most J steps (each of which can be horizontal or vertical) steps to a new white square. Here J is a constant. The devil wins if the angel does not find any more white squares to land on. The result of the paper is that if J is sufficiently large then the angel has a strategy such that the devil will never capture her. This deceptively easy-sounding result has been a conjecture, surprisingly, for about thirty years. Several other independent solutions have appeared also in this summer: see the Wikipedia. Some of them prove the result for an angel that can make up to two steps (including diagonal ones). The solution opens the possibility to solve a number of related problems and to introduce new, adversarial concepts of connectivity.