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The Angel Game in the Plane


We show that in the game of angel and devil, played on the planar integer lattice, the angel of power 4 can evade the devil. This answers a question of Berlekamp, Conway and Guy. Independent proofs that work for the angel of power 2 have been given by Kloster and by Máthé.

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[1]Berlekamp, E. R., Conway, J. H. and Guy, R. K. (1982) Winning Ways, Vol. 2, Academic Press.
[2]Bollobás, B. (1998) Modern Graph Theory, Springer.
[3]Bollobás, B. and Leader, I. (2006) The angel and devil in three dimensions. J. Combin. Theory Ser. A 113 176184.
[4]Conway, J. H. (1996) The angel problem. In Games of No Chance (Nowakowski, R. J., ed.), Cambridge University Press, pp. 312.
[5]Epstein, R. A. (1967) Theory of Gambling and Statistical Logic, Academic Press.
[6]Gardner, M. (1974) Cram, crosscram and quadraphage: New games having elusive winning strategies. Sci. Amer. 230 106108.
[7]Kloster, O. A solution to the angel problem. To appear in: Theoret Comput. Sci.
[8]Kutz, M. (2005) Conway's angel in three dimensions. Theoret. Comput. Sci. 349 443451.
[9]Kutz, M. and Pór, A. (2006) Angel, devil and king. In Computing and Combinatorics, Vol. 3595 of Lecture Notes in Computer Science, Springer, Berlin, pp. 925934.
[10]Máthé, A. (2007) The angel of power 2 wins. Combin. Probab. Comput. 16 363374.
[11]Silverman, D. L. (1971) Your Move, McGraw-Hill.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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