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W(hat) (Ha)t.... Ha!

Three students are standing in a line single-file.

The teacher has five hats, three of the hats are green, and the other two hats are yellow.

The teacher places a hat (at random) on each of the student's heads.

Then, the students are asked to use logic to figure out the color of the hat on their head, while keeping their eyes forward.(That is, the front student cannot see anyone else's hat, the second student can see only the first student's hat, the third student can see the hats on both of the students in front of him).

No student can see the colors of the two hats that the teacher is still holding.

After about thirty seconds the student first in line answers with a color and is correct.

What is the color of the hat that she is wearing and how did she know?


Source: Thanks to Carly J. (WWU), who got the problem from her "big sis... taking a logic class at the UW."

Hint: Role play using objects (e.g. checkers, playing cards, bananas and apples) to represent the two yellow hats and the three green hats. Try combinations and pretend you are each student...what would you know....?


Solution Commentary: Carly J.'s solution: Student one must be wearing a green hat. Since student three could not answer, there can be only two color combinations of hats in front of him--one green and one yellow, or two green.

If there were two yellow hats, student three would know he must have a green hat on his head.

Aware of student three's silence, student two knows that student one and himself are wearing one of these combinations (two green hats or one hat of each color).

Since student two also is silent, student one can deduce that her own hat is green. If student two was looking at a yellow hat in front of him, he would be able to answer instead of being silent.