From: hiraga@Csli.Stanford.EDU (Yuzuru Hiraga)
Subject: Re: Brown/Blue eyes (SPOILER!) -- that is, FINAL SPOILER!
Followup-To: rec.puzzles
Organization: Stanford University CSLI
Date: Mon, 28 Sep 1992 21:21:26 GMT


As of my questions along the blue/brown eye thread:
> What happens if the foreigner says "At least none of you are blue-eyed"?
> What happens if the foreigner lies and says
>	"At least 301 of you are blue-eyed"?

I believe there has been sufficient responses (to rec.puzzles or by mail),
so here goes the SPOILER:




In article <1a2ojtINNk08@gap.caltech.edu> of rec.puzzles,
	rmjarvis@cco.caltech.edu (Robert Michael Jarvis) writes:
>In the first case, the foreigner does not add any new information....
>... nothing untoward will happen in this case.

>The second case is much more disasterous...  ... in two nights,
>that one comment will cause the death of everyone on the island....

Yes, that is the "official correct answer".
The first case is trivial, although would be instructive in marking the
difference with the original comment: "At least one is blue-eyed".
Some comments for the second case:

* Although the existence of brown-eyes played no essential role in the
  original problem, in the above, this is essential (that is,
  for the "disaster" to take place).  Had there been only blue-eyes,
  nothing would happen and all would live happily ever after...
  (Of course the foreigner's comment of 301 becomes more nonsensical,
  because there are only 300 (blue-eyed) inhabitants.)

* Along that line, note that the number 301 is critical.  Had the foreigner
  said any larger number (eg. 302, 400), nothing would have happened.
  In the 301 case, while the blue-eyes can recognize the statement to be
  false, the brown-eyes won't and falsely believe it to be true.
  On the other hand, for a larger number, everyone (both blue and brown)
  would know it to be false, and also know that everyone else knew that.
  So this is only a reverse of the trivial case (replacing
  self-evidently TRUE with self-evidently FALSE).
  Those of you who made the 400 case, sorry, you fell in the trap.

* While the above is the "official answer", other versions are conceivable.
  Some mentioned (by post or mail) that the blue-eyes will go "lynching"
  the foreigner.  My, are you guys aggressive!  :-)
  I was thinking of something more moderate, like some blue-eye speaking
  out: "That's a lie!".  Assuming that everyone can figure out the cause
  of the lynching, the two would be identical.
  But that's not the whole story... there is an aftermath.
  First, all the brown-eyes will be instantaneously relieved (why?).
  As for the blue-eyes, it depends -- on whether the lynching was done
  by only a small number of them, or every blue-eye participated in it
  and everyone knew they would (eg. by some moral code that one should
  not let a lie go through).  The rest is for you to figure out.

  This version, however, is somewhat questionable because the act would
  reveal information on the identity of the inhabitants, violating the
  problem's premise.  But then, one might ask, although it is not permitted
  to tell someone that he's a blue-eye, is it permissible to tell that
  he's a brown-eye?  And doesn't the act of commiting suicide by itself
  betray some information on the identity of the others?
  As for the first one, for sake of safety we should assume that talking
  of eye-color is an absolute taboo.  As for the second, in general
  this is possible and the would-be suicider might fall in a dilemma,
  although this is not the case with our current problem situations.


As a final remark:
In article <83813@bcsaic.boeing.com> of rec.puzzles,
	lbaum@bcsaic.boeing.com (Larry Baum) writes:
>In other words, if I am blue-eyed I must know that there are
>exactly 299 blue-eyed people I have seen.  And I must know
>that all 299 of them know exactly how many they have seen (298
>or 299 depending on my eye color.)
>
>This strains my credulity.

Your doubt is understandable, but then after all, it's only a puzzle!
Pursuing such doubts would eventually lead to something far more nasty --
namely the "frame problem" in AI.
cf. J.M.McCarthy & P.J.Hayes: Some philosophical problems from the
standpoint of artificial intelligence (1968) -- reprinted, e.g. in
Readings in Artificial Intelligence (pp.431-450), Tioga Publishing Co.,
Palo Alto, CA (1981).

So long for now!

-Yuzuru Hiraga (hiraga@csli.stanford.edu)