What Color Hat am I Wearing?

Date: 4/17/96 at 18:22:24
From: KHU
Subject: A Hat of a Different Color

The wise teacher offered the three noisiest students a deal.  He 
showed them that he had two red hats and three blue hats.

The deal worked like this:

The three students would close their eyes, and while their eyes 
were closed, the teacher would put a hat on each of their heads 
(and hide the other two hats).

Then, one at a time, the students would open their eyes, look at 
the other two students' heads, and try to guess which color hat 
was on their own head.

Any students that guessed correctly would have no homework to do 
the rest of the semester.  But any students that guessed wrong 
would not only have to do their own homework, but they would have 
to help grade everyone else's work also.

The students drew numbers to see who would guess first.  Then they 
closed their eyes and the wise teacher put a hat on each one's 
head.  Arturo, who was to go first, opened his eyes, looked at the 
others' heads, and said he didn't really want to play.  He 
couldn't tell for sure and he didn't want to guess in case he was 
wrong.  Next, Belicia opened her eyes and looked at the others' 
heads.  She also thought about the fact that Arturo had said he 
couldn't tell.  Then she said she didn't want to risk it either.  
She couldn't tell for sure.

Carletta was third.  She just stood there with her eyes still 
closed tightly and a big grin on her face.  "I know what color hat 
I have on," she said. 

And she gave the right answer.

Your problem is to figure out what color hat Carletta had on and 
how she knew for sure.  Remember: Carletta didn't even look!

Date: 4/18/96 at 20:18:7
From: Doctor Patrick and Doctor Ian
Subject: Re: A Hat of a Different Color

Hi,

Suppose we enumerate all the possible arrangements:

   Arturo    Belicia    Carletta
   -------   -------    --------
      b         b          b          No red hats
      r         b          b          One red hat
      b         r          b           "
      b         b          r           "
      r         r          b          Two red hats
      r         b          r           "
      b         r          r
      
The only way for Arturo to guess what color his own hat would be
is if he can see two red hats.  That would let him know that he
must be wearing a blue hat.  Since he doesn't make that guess, 
we can eliminate that possibility:

   Arturo    Belicia    Carletta
   -------   -------    --------
      b         b          b          
      r         b          b          
      b         r          b           
      b         b          r           
      r         r          b          
      r         b          r          
      
Similarly, if Belicia sees two red hats, she knows she's wearing 
a blue hat.  But she doesn't make that guess, so we can eliminate
that possibility:

   Arturo    Belicia    Carletta
   -------   -------    --------
      b         b          b          
      r         b          b          
      b         r          b          
      b         b          r          
      r         r          b          
      
Now, note that as soon as Arturo refuses to guess, Belicia and Carletta 
both know something: at least one of them is wearing a blue hat!  
(Possibly both of them are.)

So imagine you're Belicia for a moment.  You know that there is at
least one blue hat between you and Carletta.  So if you see 
Carletta in a red hat, you know your hat is blue, and you can
win the game.  But you don't draw that conclusion, which means
that Carletta doesn't have a red hat.  So we can eliminate 
that possibility.  

   Arturo    Belicia    Carletta
   -------   -------    --------
      b         b          b          
      r         b          b          
      b         r          b          
      r         r          b          

Now if you're Carletta, and you've been following along, you
know that you must have a blue hat!  And you don't even have
to look to know that.  

This seems like a lot of work for Carletta, to keep a table
like this in her head.  Fortunately, she doesn't really have
to do that.  Here's how she can reason her way to the answer
without constructing a table:

  1) If Arturo had seen two red hats, he would have
     guessed that his own hat was blue.  So he saw
     at least one blue hat. 
     
  2) If Belicia had seen Carletta wearing a red hat,
     she would have guessed that her own hat was blue
     (since at least one of them had to have the blue
     hat that Arturo had seen). 
     
  3) So Carletta already knows that she isn't wearing 
     a red hat.  Which means she must have a blue one. 

Doctors Patrick and Ian -The Math Forum