Angel Problem - History

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).
  • 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 then the angel can still win if it is of high enough power. (By "playing at distance " 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. Péter Gács's proof works only for a much larger constant. The proofs by Bowditch and Máthé have been published in Combinatorics, Probability and Computing (edited by Béla Bollobás and Imre Leader). The proof by Kloster has been published in Theoretical Computer Science.

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