Editing Angel problem (section)

==== "Guardian" proof ====

The proof, which shows that in a three-dimensional version of the game a high powered angel has a winning strategy, makes use of "guardians".  For each cube of any size, there is a guardian that watches over that cube.  The guardians decide at each move whether the cube they are watching over is unsafe, safe, or almost safe.  The definitions of "safe" and "almost safe" need to be chosen to ensure this works. This decision is based purely on the density of blocked points in that cube and the size of that cube.

If the angel is given no orders, then it just moves up.  If some cubes that the angel is occupying cease to be safe, then the guardian of the biggest of these cubes is instructed to arrange for the angel to leave through one of the borders of that cube. If a guardian is instructed to escort the angel out of its cube to a particular face, the guardian does so by plotting a path of subcubes that are all safe.  The guardians in these cubes are then instructed to escort the angel through their respective subcubes. The angel's path in a given subcube is not determined until the angel arrives at that cube.  Even then, the path is only determined roughly.  This ensures the devil cannot just choose a place on the path sufficiently far along it and block it.

The strategy can be proven to work because the time it takes the devil to convert a safe cube in the angel's path to an unsafe cube is longer than the time it takes the angel to get to that cube.

This proof was published by [[Imre Leader]] and [[Béla Bollobás]] in 2006.<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>  A substantially similar proof was published by [[Martin Kutz]] in 2005.<ref name="kutz2004" /><ref name="kutz2005">Martin Kutz, Conway's Angel in three dimensions, ''Theoret. Comp. Sci.'' 349(3):443–451, 2005.</ref>