The Problem of the Cubs Caps The problem is this: A teacher comes to class with

Question

The Problem of the Cubs Caps
The problem is this: A teacher comes to class with a box and shows the contents of the
box to the students. It contains three Cubs caps, two Cardinals caps, and nothing else: the
Cubs caps are true blue (for goodness), the Cardinals ones fire red (for evil). There are
only three students in this class and the teacher tells them that she is going to blindfold
each one of them and then place one of the five caps on each of their heads. The
remaining two caps will then be placed back in the box so no one can see them when the
blindfolds are removed. The teacher then tells the students that if anyone can tell what
color cap she has on her head, then the teacher will give her 10 extra credits points.
However, the students are not allowed to guess the color of their hats, they must be able
to prove they have that cap on.

The teacher removes the blindfold from the first student who is now able to see the color
of the caps on the other two students’ heads, but he cannot see his own cap. The first
student looks carefully at the other two caps, thinks for a while, and says he does not
know the color of his hat. (The student does not say aloud the color of the hats he sees on
the other two students’ heads.) The teacher then removes the blindfold from the second
student who looks at the other two students’ caps, thinks for a while, and says he does not
know the color of his cap either. (As before, this student does not say aloud the color of
the caps he sees on the other two students’ heads.) Just as the teacher is about to remove
the blindfold from the third student, the third student says that she knows exactly the
color of the cap on her head, and she doesn’t even need to see the caps of the other two
students.
The information given regarding this problem is to be considered accurate. No
information is being held back, no tricks are being played, and no word-games are used.
All the information necessary to solve the problem is contained in the description of the
problem.

There are three possibilities to consider. Which of the three is correct?

1. She cannot possibly know what color cap she has on her head.
2. She has a Cubs cap and can prove it.
3. She has a Cardinals cap and can prove it.

For the correct answer and a precise explanation (250-500 words

Explanation / Answer

The third student has a blue cubs cap on his head and he can prove it with the following reasoning:

The first student could not tell the answer, which means that he didn't see two red hats. (Because if he had seen two red hats on the 2nd and 3rd student, then he could have told that he has a blue hat because there are only 2 red hats.)

The second student realises this fact that the only reason for the first student to be not able to answer is that he didn't see two red hats. Now if the second student sees a red hat on the third student, he would automatically understand that he himself has a blue hat ( because he knows the fact that the first student didn't see 2 red hats).

Thus, the conclusion is that for both the first and second students to be in doubt, the first student should not see two red hats, and the second student should definitely not see a red hat on the third student.

So, the third student realises that he has a blue cap on his head.

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