[FOM] Conway's Angel and Devil problem

Robbie Lindauer rlindauer at gmail.com
Thu Jun 21 18:32:24 EDT 2007

Can this question be posed in such a way as to allow transfinite  
values for the power of the angel?

Alternatively, are there any "mathematically interesting" questions  
along those lines already developed?


Robbie Lindauer

On Jun 18, 2007, at 10:18 PM, joeshipman at aol.com wrote:

> 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
> in
> Wikipedia.
> http://en.wikipedia.org/wiki/Angel_problem
> The problem remained unsolved until recently, when four (!)  
> independent
> and almost simultaneous solutions appeared, showing that the angel  
> wins.
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