[FOM] Conway's Angel and Devil problem
joeshipman at aol.com
Tue Jun 19 02:02:17 EDT 2007
Update -- the crumpling is easy even in just 4 dimensions -- the first
of the two moves of the power-2 angel in 2 dimensions is mirrored in
the first two of the 4 dimensions by the power-1 4-d angel, and the
second of the two moves of the power-2 angel in 2 dimensions is made in
the other two of the 4 dimensions by the power-1 4-d angel.
So all that's left is the case of a power-1 angel in deminsion 3 --
here the crumpling also looks possible, because there is enough room at
the core (27 distance <=1 neighbors in 3-d vs 25 distance <=2
neighbors in 2-d, and the growth is cubic rather than quadratic so
there is more crumpling room further out).
From: joeshipman at aol.com
It may have been observed by someone other than me that you can
transform the winning strategy for a 2-d angel of power 2 into a
winning strategy for a 12-d angel of power 1, by crumpling the 2-d
board in 12 dimensions so that each cell in the radius 2 2-d
neighborhood goes to a cell in the radius 1 12-d neighborhood and
2-adjacencies become 1-adjacencies; I'm not sure how much lower one can
make the dimension and still make this work.
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