From: johnson@sleipnir.mnesouth.corp.mot.com ("Johnson")
Subject: Re: Problematic smurfs!
Reply-To: johnson@sleipnir.mnesouth.corp.mot.com
Organization: Motorola Wireless Enterprises, Schaumburg, IL
Date: Tue, 1 Sep 1992 19:33:20 GMT

In article <65049@cup.portal.com>, ts@cup.portal.com (Tim W Smith) writes:
=)I've heard the problem stated in terms of a village of perfect logicians.
=)Each has either a red dot or a blue dot on the back of his neck.
=)
=)Each does not know the color of his own dot.  Furthermore, anyone who
=)finds out the color of his own dot will take an overdose of sleeping
=)pills before going to bed that night and die.  The body will be
=)discovered by any living logicians the next morning, and all will
=)know of the death.
=)
...logician problem deleted for bandwidth...
=)
=)In general, when someone who sees N-1 blue dots sees that the N-1 people
=)don't kill themselves on day N-1, that person deduces their own blue
=)dot and dies on day N.  When the people with red dots see that N visible
=)blue dotted people die on day N, they deduce their own red dots, and die
=)on day N+1.
=)
=)Now for the interesting problem.  Suppose there were at least two blue
=)dots.  Then everyone already knows that there is at least one blue dotted
=)person in the village.  The Stranger seems to have given them no information,
=)so why was his statement fatal?
=)
Take the case of two blue dots.  Each of the blue-dotted people know that
there exists another, but they also know that the other does not know it.
After the stranger tells them that there exists a blue dot, the other would:
   a) die immediately, knowing that his must be the only one, or
   b) not die immediately, since there exists another one.
In case b, the second then knows that he must have the other dot, and then
dies.

However, these are more or less degenerate cases in your puzzle.  If there
are three or more blue dots, the visitor has in fact added no information,
since all of the wearers are aware that some blue dots exist.  No blue dots
does not create an inconsistency.  One fails to, since that person has no
way of knowing that there exists a blue dot.  Two also fails, because each
person is an observer who can vacillate between the one-dot case and the
two-dot case until the other blue-dot determines his own uniqueness or
lack thereof.  At three, however, the system breaks down.  In this case,
the puzzle would begin at the time this peculiar suicide pact was begun,
rather than requiring the intervention of the stranger; probably everybody
would be dead by the time he passes through.  Whichever color was in the
minority would be wiped out on the Nth night, with the remainder on the
following night.  I suspect the puzzle was rephrased from a different
version, and this information was somehow muddled.  At any rate, the real
puzzle here is why a village of perfect logicians would behave in such a
bizzare manner.
-- 
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|  "Johnson"                              | "                               " |
|  johnson@sleipnir.mnesouth.corp.mot.com |               -- Marcel Marceau   |
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|  Disclaimer:  It's just my Tourette's Syndrome acting up again.             |
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