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From: jaw@aurora.UUCP (James A. Woods)
Newsgroups: sci.math
Subject: Re: angels and devils
Message-ID: <514@aurora.UUCP>
Date: Tue, 21-Oct-86 18:43:01 EDT
Article-I.D.: aurora.514
Posted: Tue Oct 21 18:43:01 1986
Date-Received: Wed, 22-Oct-86 06:31:02 EDT
References: <2056@princeton.UUCP>
Organization: NASA Ames Research Center, Mt. View, Ca.
Lines: 25

# "Resist the devil, and he will flee" -- James 4:7
# "Philosophy will clip an angel's wings. " -- John Keats
# "The devil is an angel too." -- Miguel de Unamuno (1864-1936)

     The one-dimensional case is easy; the devil picks a sufficiently large
distance from the angel at the initial point, methodically destroys 
100 planets in a line, watches the angel hop wherever she pleases, repeats
the same on the other side, then entraps her with a random squeeze between
the barriers.

     For the n-dimensional case, we defer to the intriguing exercise in
Berlekamp, Conway, and Guy's "Winning Ways, part 2 (Games in Particular)".
Conway, et. al., teases us with a claim by T. Koerner that the angel
can escape on the hypercube, but how this might apply to 2D is nebulous.
This kind of result has the flavor of the Poincare Conjecture, where higher
dimensional results trickle in first (Smale, n>=5, Freedman, n=4, ?, n=3).
[Mr. T -- do you believe this has been fully resolved?]

     The book by Berlekamp (world's Dots-and-Boxes champ, among other things),
Conway (poser, and part solver of the computation-universality of Life),
and Guy (Alpine climbing Sprague-Grundy theorist), is highly recommended
for its refreshing style, and merits discussion in the likes of Scientific
American's Mathematical Recreations.
 
     -- James A. Woods (ames!jaw)