• # Students with Hats

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• ### Puzzle Prime

October 12, 2020 at 5:39 pm

Professor Vivek decided to test three of his students, Frank, Gary and Henry. The teacher took three hats, wrote on each hat a positive integer, 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?
— Yes, my number is 5.

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

Puzzle from the archives.

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Initially, the problem was quite different, but it got modified after Vivek pointed out that its solution was not unique. We present the original formulation below, along with Vivek’s remarks.

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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:

â€” Frank, do you know the number on your hat?
â€” No, I donâ€™t.
â€” Gary, do you know the number on your hat?
â€” No, I donâ€™t.
â€” Henry, do you know the number on your hat?
â€” No, I donâ€™t.

Then the teacher started another round of questioning:

â€” Frank, do you know the number on your hat?
â€” No, I donâ€™t.
â€” Gary, do you know the number on your hat?
â€” No, I donâ€™t.
â€” Henry, do you know the number on your hat?
â€” Yes, it is 144.

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

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