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From: dhecker@sjuvax.UUCP
Newsgroups: sci.math
Subject: Re: angels and devils
Message-ID: <244@sjuvax.UUCP>
Date: Wed, 5-Nov-86 11:49:22 EST
Article-I.D.: sjuvax.244
Posted: Wed Nov  5 11:49:22 1986
Date-Received: Sat, 8-Nov-86 03:51:52 EST
References: <1056@navajo.STANFORD.EDU>
Reply-To: dhecker@sjuvax.UUCP (Dr. David Hecker)
Organization: St. Joseph's Univ., Phila. PA
Lines: 38

In article <1056@navajo.STANFORD.EDU> avg@navajo.UUCP (Allen Van Gelder) writes:
>
>Now a theorem:
>If the devil can always force the angel to move to a planet in 2 or more
>moves that she was already on or could have reached in 1 move,
>then the angel can be trapped.
>Proof:
>If the angel has an escaping strategy, then assume that she uses an
>optimal escaping strategy.
>Suppose the angel travels from P1 to P2 in 2 or more moves, and could
>have arrived there in 1 move; then the strategy is not optimal because
>the devil has gotten at least 1 "free" move.  Similarly, if P1=P2,
>then the devil has gotten at least 2 free moves.  Thus, if the devil
>can always force this to occur, then the angel has no escaping strategy.
>QED
>That isn't supposed to be rigorous; it's just supposed to convince you.

I'm not convinced.  Part of the angel's strategy may be to backtrack
after it sees what the devil is doing.  You are assuming that the
angel's strategy is completely independent of what the devil does.


>To formalize this a little more, say the angel can "see" a planet if
>she can move to it in one turn.  Keep track of planets the angel has seen
>and when they were FIRST seen.  At each turn the angel must move to
>a planet she sees but has never seen before.

This note adds a perspective to the problem I hadn't thought of before.
How far can the angel see ????   I would assume, as I said above, that
the angel's strategy will be dependent upon the devil's moves.  But, if
the angel can't see the devil's moves, this may restrict the possible
strategies for the angel, since it doesn't know what the devil is
going on !!!

                              David Hecker
                              St. Joseph's University
                              Phila., Pa.