From: cjdcosta@eos.ncsu.edu (CHRISTOPHER J DCOSTA)
Subject: Re: Problematic smurfs!
Originator: cjdcosta@zepplin.ece.ncsu.edu
Keywords: smurfs
Reply-To: cjdcosta@eos.ncsu.edu (CHRISTOPHER J DCOSTA)
Organization: North Carolina State University, Project Eos
Date: Mon, 31 Aug 1992 03:33:48 GMT


spacefox@acs.bu.edu (Godfrey Degamo) writes:
>-----------------------------------------------------------------------------
>     A group of 1000 smurfs wear either a red or blue hat on their head.
>They can only wear one color. The color hat on their head is known to all, but
>the wearer.  The wearer of the hat has no means whatsoever for obtaining the
>color of his hat.  It is never the case that all 1000 smurfs will wear
>the same color hat.
>     One day the mayor, not apart of the 1000 smurfs, decides to call a
>town meeting.  All 1000 smurfs are gathered into the town hall.  The mayor
>is not wearing a hat.  During the meeting, the mayor asks that all the
>smurfs wearing red hats to stand up.
>     There is a bit of commotion in the crowd, but no one stands up.
>     The mayor requests the same demand a bit more sternly.
>     Again, commotion, but no one stands up.
>     The mayor, irate, demands the red hatted smurfs to stand up.
>     Then, a certain amount of smurfs rise.
>
>find:
>     The exact amount of smurfs that rise.  
>         (Hint: This number will be greater than one.)
>     The color of the hats of the smurfs that rise.
>


OK, we know that at least ONE smurf MUST wear a red hat. Also we must assume
that all the smerfs are capable of reasoning, are honest, and would carry out
their mayor's request if they were able to do so. Now:

<1> Since no one stood up after the mayor's first request, there must be MORE 
    THAN ONE smurf wearing a red hat. 

    If there was only one red-hatted smurf, then he would have looked around
    when the mayor made the first request, spotted everyone else with blue 
    hats, and deduced that he HAD to be the only red hatted smurf, since all
    couldn't have blue hats. But he didn't stand up, so there must have been
    more than one red hat.

<2> Since no one stood up after the mayor's second request, there must be MORE
    THAN TWO smurfs with red hats.

    If there were exactly two, then one of the red hatted smurf (Say smurf 
    'A') would have been watching the other red hatted smurf (say smurf 'B') 
    during the first request. 'A' would then conclude that: since 'B' did not 
    stand up after the first request, 'B' must have observed SOMEONE ELSE with 
    a red hat and was therefore unsure of his (B's) hat color by the reasoning 
    in <1>. Since 'A' can see everybody else's hat except his own, and sees 
    all blue hats except for the one on 'B's head, 'A' would have been able 
    to conclude that his own hat was red. And 'A' would have stood up at the 
    second request. Assuming 'B' is capable of the same reasoning as 'A','B' 
    would have stood up as well at the second request. But neither did. So 
    there must have been MORE THAN TWO red hats.

<3> Now let's assume that there were three red hats, worn by 'A', 'B', and 'C'
    If all three have been reasoning along our lines, then:

    <a> They would each have initially seen two blue hats in the audience. 
    They each did not initially know the color of their own hat. But they 
    could at least conclude that there were EITHER TWO, OR THREE red hats 
    in all. 

    <b> After the first and second requests, they would have reached the same
    conclusion we reached in <2>, i.e. that there were MORE THAN TWO red hats.

Now, we know that 'some' smurfs stood up at the third request.

Look at conclusions <a> and <b>. The only possible answer is THREE RED HATS.
Assuming all the smerfs were capable of reasoning this out, then all three 
would have stood up at the third request. And all of them would of course
be wearing red hats. The 1000 figure in irrelevant, save that it's more than
three.
        

>more importantly,
>     Explain how you deduced your answer.
>----------------------------------------------------------------------------


That line sounds a lot like something you'd see in an undergraduate quiz. Or 
maybe I'm doing Godfrey's homework here.. Oh well. Maybe he deserves this help
for his resourcefulness. Of course, if I'm wrong, he'll face the music. :-)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
|   Christopher J. D'Costa           cjdcosta@eos.ncsu.edu |
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