Imagine a chessboard stretching endlessly in all directions. In this unusual game however, there are only two actors: the angel, who is our hero of the play, and the devil. The angel moves in a manner similar to a chess king, except that the distance he may travel is determined by some fixed power k. For example, an angel with power 3 can on each turn travel 3 squares from its starting location. The devil can move to any square he likes, and in turn burns that same square, making it permanently inaccessible for the angel. Our question is: is it eventually possible for the devil to trap the angel, leaving it with no squares to fly to, or will the angel always be able to escape?

This problem was first posed by the English mathematician John Horton Conway in 1982. Quite surprisingly perhaps, we can frame this puzzle as a game theory problem. The devil has a winning strategy if it can always force a win within a finite number of moves, delivering the perfect reply to every move of the angel. This is the case when the angel has a power of 1, and the proof is relatively elementary. The angel has a winning strategy if the devil doesn’t. Intuitively, you might say that the devil always wins, as it can start from a distance far away from the angel, and start building its wall. Conway offered a prize of $1000 to the person who could prove this. This prize, however, was never awarded, as it soon became apparent that the situation was far more subtle than anyone could have imagined.

Before diving into the main question, it makes sense to consider some subcases, to get a feel for the strategies and obstacles of both players. We call an angel who is always required to step forward a Fool, i.e., a Fool can only move from (x,y) to (X,Y) for which Y>y. In this case, the devil always has a winning strategy, irrespective of the angel’s power. If the devil just starts at a long enough distance, then he will be able to build a wall in time, if he keeps the angel’s horizontal movement in mind. This distance grows exponentially large for larger powers of k. What would your guess be for an angel with a power of 2? The answer is around 8,589,934,592 squares away! Conway also considered some other variations of the “Fool” in his paper, for example the Fool who always increases his distance from the origin. But for each subcase he considered, he could find a way for the devil to keep the angel enclosed in his cage. It seemed that all hope for fans of the Divine was lost.

That was until 2006, when suddenly, almost simultaneously, several independent proofs were published. One of them proved that an angel with power 4 could always win, and two of them even proved it for an angel with only power 2! One of them, a proof by Andras Máthé, is a bit easier to understand than the other one. Broadly speaking, in his proof he first defines the Nice devil as a devil who never burns a square that the angel could have gone to on an earlier turn. This implies that the angel will never run out of squares to go to if it just keeps hopping between the same two squares, so the winning condition of the devil is changed to just confining the angel to a finite bounded region of the board. Máthé then showed that if there exists a winning strategy against the Nice devil, then there must also exist a winning strategy against the normal devil. He was able to prove this winning strategy by letting the angel be a “wall-hugger”. That is, every time the angel would face an obstacle, he would try to go around it, effectively surfing around the boundary of the blocked area. The devil simply wouldn’t be able to keep up with the angel’s speed. Finally, the problem was solved.

All’s well that ends well, right? Not exactly. What we’ve just discussed is only the standard version of the angel problem, but as you can imagine, there are several variants of this problem. A simple modification would be to “buff” the devil, giving it M squares he can burn on each turn. Or you could change the chessboard to be three-dimensional. All of these problems fall under the family of so-called pursuit-evasion problems, a fascinating area within mathematics and computer science. So many questions to answer, but one thing is certain: the chase between angel and devil will never end.