From: radams@BNR.CA (Rodney Adams)
Subject: Re: Problematic smurfs!
Reply-To: "Rodney Adams" <RADAMS@BNR.CA>
Organization: Bell-Northern Research, Ltd.
Distribution: usa
Date: Fri, 04 Sep 1992 17:38:28 GMT

In article <79009@ut-emx.uucp>, alexpace@ccwf.cc.utexas.edu (Alex L. Pace) writes:
|> 
|> In article <15117@claris.com> mike_steiner@qm.claris.com (Mike Steiner) writes:
|> >In article <1992Sep01.152304.11249@bnr.ca>, radams@BNR.CA (Rodney Adams) writes:
|> >
|> >>  It is with three individuals each wearing one of five hats - three
|> >>  brown, two black.  They are standing in a line and can only see
|> >>  those in front of them.  The last in line can see the color of
|> >>  the hats of the two in front of him.  The second in line can see
|> >>  only the one in front of him, and the first in line can't see
|> >>  anyone's hat.  They can't talk or leave until they are certain what
|> >>  color hat they have on.  After a long time the person first in
|> >>  line leaves.  What color hat does he have on?
|> >
|> >The third guy doesn't see two black hats; otherwise, he would leave.  If the
|> >second guy saw a black hat, he'd know that he had a brown hat (because the
|> >third guy would have left), and he'd leave.  So, the first guy knows that he
|> >has a brown hat, and leaves.
|>
By the way, congratulations, Mike.  I honestly had you in mind when I brought
this problem over from sci.math.
 
|> HOW CAN THE SECOND GUY SEE A BLACK HAT ON THE HEAD OF THE 1ST WHEN THE
|> FIRST IS ACTUALLY WEARING A BROWN ONE?
|> 

The second guy doesn't see a black hat on the first guy.  Mike uses RAA,
reductio ad absurdum, to show this:  IF the third guy saw two black hats,
then a conflict would exist with the original problem.  We thus conclude that
premise is false--the first guy did not see two black hats.  Likewise, IF the
second quy saw a black hat, a conflict (i.e., logical impossibility) would
also exist.  Therefore, we can conclude that the second guy does not see a
black hat.  Using this reasoning, the first guy can now determine that he is
wearing a brown hat (if the second guy does not see a black hat, then he must
see a brown one).
 
|> What if the thrid guy sees two brown hats and does not leave because his
|> may be either brown or black.  The second guy also sees a brown hat and
|> knows that third guy didnt leave.  As far as he knows, there are 2 reasons
|> that the 3rd guy didnt leave: 
|>       A - Both the 1st and 2nd hats are brown (as is the case)
|>       B - The first guy has a brown hat and the 2nd has a black hat,
|>           this would also lead to uncertainty on the part of the 3rd.
|> So, the 2nd guy doesnt leave.  The first guy doesnt leave either.
|> 
You're right about the first and second guys, although the first guy
then has enough information to leave, as shown above.  HOWEVER, I think 
I see the reason you might be confused--your numbering differs from the 
original problem.  The first person in a line is the one facing away
from the people behind him.  Switch your numbering for the first and
third guys and see if you can figure it out.

And they make jokes about us Aggies.... ;) 

----------------------------------------------------------------------

Rodney G. Adams
The opinions expressed above are not necessarily those of my employer.
As if you didn't already know that....
"We are all captives of the pictures in our heads - our belief that 
the world we experience is the world that really exists."
                                                     - Walter Lippmann