Editing Angel problem (section)

== History ==
The problem was first published in the 1982 book ''Winning Ways'' by Berlekamp, Conway, and Guy,<ref>{{citation
 | last1 = Berlekamp | first1 = Elwyn R. | author1-link = Elwyn Berlekamp
 | last2 = Conway | first2 = John H. | author2-link = John Horton Conway
 | last3 = Guy | first3 = Richard K. | author3-link = Richard K. Guy
 | contribution = Chapter 19: The King and the Consumer
 | pages = 607–634
 | publisher = Academic Press
 | title = Winning Ways for your Mathematical Plays, Volume 2: Games in Particular
 | year = 1982}}.</ref> 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 [[Elwyn Berlekamp|Berlekamp]].) This can be read at section 1.1 of Kutz's paper.<ref name="kutz2004">Martin Kutz, The Angel Problem, Positional Games, and Digraph Roots, [http://dx.doi.org/10.17169/refubium-14116 PhD Thesis] FU Berlin, 2004</ref>
* 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.<ref>B. Bollobás and I. Leader, ''The angel and the devil in three dimensions.'' Journal of Combinatorial Theory, Series A. vol. 113 (2006), no. 1, pp. 176&ndash;184</ref>
* 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 <math>d_1 < d_2 < d_3 < \cdots</math> then the angel can still win if it is of high enough power. (By "playing at distance <math>d</math>" 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|Brian Bowditch's]] [http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf proof] works for the 4-angel, while Oddvar Kloster's [https://web.archive.org/web/20160107125925/http://home.broadpark.no/~oddvark/angel/Angel.pdf proof] and András Máthé's [http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf proof]  work for the 2-angel. Additionally, [[Peter Gacs]] has a [http://www.cs.bu.edu/~gacs/papers/angel.pdf claimed proof] {{Webarchive|url=https://web.archive.org/web/20160304030433/http://www.cs.bu.edu/~gacs/papers/angel.pdf |date=2016-03-04 }} which requires a much stronger angel; the details are fairly complex and have not been reviewed by a journal for accuracy. The proofs by Bowditch and Máthé have been published in ''[[Combinatorics, Probability and Computing]]''. The proof by Kloster has been published in ''[[Theoretical Computer Science (journal)|Theoretical Computer Science]]''.