# Students with Hats

One teacher decided to test three of his students, Frank, Gary and Henry. The teacher took three hats, wrote on each hat an integer number greater than 0, and put the hats on the heads of the students. Each student could see the numbers written on the hats of the other two students but not the number written on his own hat.

The teacher said that one of the numbers is sum of the other two and started asking the students:

— Frank, do you know the number on your hat?

— No, I don’t.

— Gary, do you know the number on your hat?

— No, I don’t.

— Henry, do you know the number on your hat?

— No, I don’t.

Then the teacher started another round of questioning:

— Frank, do you know the number on your hat?

— No, I don’t.

— Gary, do you know the number on your hat?

— No, I don’t.

— Henry, do you know the number on your hat?

— Yes, it is 144.

What were the numbers which the teacher wrote on the hats?

**SOLUTION**

The numbers are 36, 108, 144.

After the first round of questioning, all students knew that all three numbers were different. After Frank and Gary got questioned for the second time, Henry could conclude that his number was not two times larger or smaller than any of the other two numbers. This was all the information he had, and since he answered correctly, then he must have noticed that one of the following is correct:

– the difference of Frank and Gary’s numbers is twice the smaller one

– the difference of Frank and Gary’s numbers is half the bigger one

– the sum of Frank and Gary’s number is twice one of their numbers

The third option is impossible since it will imply that Frank and Gary had the same numbers. The second option is impossible because it will imply that the numbers Henry had the same number as Frank or Gary. Therefore, the three numbers were X, 3X, 4X, where 4X = 144. This implies that X = 36 and 3X = 108.

The students knew BEFORE the first round of questioning that the numbers were different, because the teacher had told them this (of course the teacher might have been a seven-legged octopus, which would have prevented any solution being reached at all). The solution as presented displays runaway intuition at work – the sort of pattern finding exercise that might indicate the likelihood of a correct answer in a multiple choice IQ test if one was unable to find it in any other way, but it does not suffice here. Henry merely quessed. True he guessed correctly, but the number of options was severely limited and his success means nothing. Henry had no information whatever to show that his number was not 72, i.e., that the two numbers he could see (36 and 108) were not part of a sum (36) and a total (108), the latter being reached by adding his possible 72 to 108. Henry possessed enough information to know that but two options existed as to his number: either 72 or 144. That he guessed correctly is a chance result by very definition of the word.

Hi Element. The teacher did not tell the students that their numbers are different. They concluded this themselves, because if there were 2 hats with the same number, than the third student would have figured out that his number is equal to their sum. The rest of the reasoning also did not evolve any guessing.