Red Hat, Blue Hat

Date: 11/16/2002 at 19:24:48
From: Ken
Subject: IMP Mathematics

My math teacher tells us that he has gotten an answer to this math 
problem, but he always tells us that an answer is worth nothing unless 
you can prove it. That is our teacher's problem - he can not prove it, 
therefore I don't think that the answer is correct. The question comes 
from the Math Lessons of the Integrated Mathematics Program.

There are 3 children, A, B, and C. The children tell their teacher 
they think the homework they do is just busy work. So the teacher says 
they will get one chance to get an A for the semester without having 
to do any more homework. All they have to do is to answer one of his 
math problems right. 

The problem is that he has 5 hats, 3 red and 2 blue. The children are 
told to stand in line with their eyes blindfolded. The teacher puts 
one hat on each of their heads and then discards the remaining 2 hats 
so they cannot be seen. Then the first child (child A) is told he can 
look at the other two children and, judging by the color of their 
hats, he can guess the color of the hat he wears. He can either guess 
or pass. (If they pass, there will be no effect on their homework, if 
they guess right there will be no more homework, and if they guess 
wrong they have to do all their homework plus grade papers after 
school. So the children will only guess if they are 100% positive they 
are correct.) 

The guess and pass process is carried through with the rest of the 
children. The main problem is that in one particular situation 
student A passes, Student B passes, and then without even opening his 
eyes Student C guess correctly what hat is on his head.

I dont understand how he knows without even looking...

Thank you, 
Ken

Date: 11/18/2002 at 11:44:27
From: Doctor Roy
Subject: Re: IMP Mathematics

Hi,

Thanks for writing to Dr. Math.

Student C must be wearing a red hat.

Let's think about it from the beginning.

Start with student A. Student A doesn't know which hat he has on. That 
must mean at least one (so both B and C or just one of B and C) must 
be wearing a red hat. If both B and C are wearing blue hats, A must 
have a red hat. So, one or both of B and C is wearing a red hat. But A 
could be wearing a red or blue hat in this case, so A doesn't know.

So, let's go to student B. Student B, being very clever in realizing
what student A is thinking, realizes that one or both of himself and C
is wearing a red hat. If student B sees a blue hat on student C, then
student B must be wearing a red hat. But this is impossible, since
student B does not know which hat he has. That means that student B
MUST see a red hat on student C. So, that leaves the possibility that
student B has a blue or a red hat.

That leaves student C. Student C, also being quite clever, realizes
the situation, and realizes that he must have a red hat, since neither
student A nor student B could deduce the colors of their own hats.

That's the solution to the problem. There is no way that student C
could have a blue hat and produce the same results. If student B also
had a blue hat, student A would know he had a red hat. Or if student
B had a red hat, student A would not know his own hat color, but
student B would know that since C had a blue hat, B must have a red 
hat.

I hope this helps.

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

Date: 11/28/2002 at 14:51:53
From: Ken
Subject: Thank you (IMP Mathamatics)

Thank you Doctor Roy, you are a genius. Your explanation was much more 
clear than that of my teacher. Thank you again... if I ever have 
another math question I know exactly whom to ask. 

Thanks
Ken