# [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

Thanks,

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
> 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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```