[FOM] Conway's Angel and Devil problem

joeshipman@aol.com joeshipman at aol.com
Tue Jun 19 01:18:02 EDT 2007

The best of these solutions show that the angel of power 2 (and 
therefore all higher powers) wins in dimension 2 (and therefore all 
higher dimensions).

The devil always wins in dimension 1 agains any power of angel, and 
wins in dimension 2 against an angel of power 1 (chess king).

So the only unsolved question is "what is the lowest dimension in which 
an angel of power 1 wins, or is there none?"

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.

The first case of this is, and the one that seems to correspond to an 
interesting game, is "does the devil win in dimension 3 against an 
angel of power 1?" There we can start by asking for a lower bound on 
the size of the board on which the devil can force a win.

-- JS

-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
Subject: [FOM] Conway's Angel and Devil problem

Conway's angel/devil problem was first published, I believe, in Winning
Ways some 25 years ago.  A nice description of the problem may be found 


The problem remained unsolved until recently, when four (!) independent
and almost simultaneous solutions appeared, showing that the angel wins.

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