## The Angel Problem_{Oddvar Kloster} |
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## ContentsThe problem |
## VariationsAs well as solving the main Angel Problem, the new solutions also yield some results for other variations of the game: - The King of power 2 can win. Even though the Angel can jump over an eaten square, he does not need to; it suffices that he is able to squeeze diagonally between two eaten squares. This is shown in my paper and also claimed in Máthé's.
- A Rook of power 9 can win, if it is allowed to jump over eaten squares.
- In 3D, the Angel of power 1 can win.
## RooksThe Rook is a chess piece that can only move to squares in the same row or column as its current position. We can define several flavours of rooks.
So far, nothing new. But let us define the
It seems likely that one can also adapt Máthé's proof for the Angel, to prove that there are winning strategies for Jumping Rooks of powers lower than 9. ## The Angel in 3DWhile the Angel Problem in 2D was long unsolved, the analogous game where the Angel is allowed to move in 3D, has been known for some time to be a win for the Angel. This has been proven independently by Bollobás and Leader ("The Angel and Devil in three dimensions", Journal of Combinatorial Theory, Series A 113 (2006) 176 – 184), and by Martin Kutz. The previously best known bound on the Angel's required power seems to be that of Kutz, 13. Of course, now that we know that the Angel of power 2 can win in 2D, it is obvious that only power 2 is required in 3D as well. But we can do better! My 2D strategy can be adapted to yield a winning strategy for the Angel of power 1 in 3D. As presented in the paper, the 2D strategy works in a game where
the Angel has power 2 and the Devil eats one whole square in each
turn. But let us look at the game on a twice as fine time scale. Then
the Angel has power 1, and the Devil eats only half a square in each
turn. Of course, the Devil must eat both halves of a square to make
it inaccessible to the Angel, but we do not require him to eat both
in consecutive turns. To win this game, the Angel can use the exact
same strategy as in the original game, and the proof of this fact is
completely analogous to that given in the paper (note that
j) now can take half-integer
values).Then to 3D. Let us restrict the Angel of power 1 in 3D to stay
within the planes |