Red and Blue Hats

Date: 03/20/2001 at 20:37:09
From: Belvona Johnson
Subject: Logic problem

Here is the question (from page 205 in the Prentice Hall Geometry 
book):

Alan, Ben, and Cal are seated as shown with their eyes closed. 
Three hats are placed on their heads from a box that contains three 
red and two blue hats. 

They open their eyes and look forward. 

Alan says, "I cannot deduce what color hat I'm wearing." 
Hearing that, Ben says, "I cannot deduce what color hat I'm wearing, 
either."
Cal then says, "I know what color I'm wearing!"

How does Cal know what color his hat is?

Here is what I know. If Alan does not know what color hat he is 
wearing, then Ben and Cal can't both be wearing blue hats, so they 
could either both be wearing red hats, or one blue and one red. 

If Ben can't decide either, then I think Cal's hat is most likely red, 
but I am not sure. Can you give me any hints, or clues to help me 
solve the guestion?  

Thank you,
Belvona Johnson

Date: 03/21/2001 at 15:09:40
From: Doctor Rob
Subject: Re: Logic problem

Thanks for writing to Ask Dr. Math, Belvona.

You made a good start by realizing that Ben and Cal cannot both have
blue hats.  The possibilities after Alan's statement are:

   Alan   Ben   Cal
    B      B     R
    B      R     B
    B      R     R
    R      B     R
    R      R     B
    R      R     R

Now if Ben can see that Cal's hat is blue, he can deduce that his hat 
is red, because Ben and Cal cannot both have blue hats. Since Cal 
cannot deduce this either, the second and fifth possibilities above 
are out.  That leaves this:

   Alan   Ben   Cal
    B      B     R
    B      R     R
    R      B     R
    R      R     R

You can see that in every remaining case, Cal is wearing a red
hat.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/