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I am sitting with two other people who are wearing hats with the prime numbers 5 and 11. My hat has a prime number also, but I don't know what it is, nor do each of them know their numbers. It is also known that our three numbers represent side lengths forming a triangle having a perimeter length which is also a prime number. They are each asked if they know their number, and each says no. I answer no also. They are asked again, and say no, and now it is my turn. What is my number?

Note: I have figured out that the only possible numbers I can have are 7 and 13, but I can't seem to narrow it down to which is my number.

J. W. Tanner
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Ken Bannister
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    The trick in these problems is to think it through carefully from the others' points of view. Say you have a $7$. Then the $5-$person sees $(11,7)$. How do they reason? And so on. – lulu Aug 11 '23 at 19:41
  • Thanks, lulu, I will try doing that. I can see where if "5" does see (11,7) on the first round, then she will have to say "yes, I know my number" because her hat would have to be 5 since 5+11+7 =23, satisfying the requirement the triangle's perimeter length is a prime number. If "5" sees (11,13), once again "yes," because her number has to be 5 (7 doesn't work). I don't know, I seem to be going in circles. – Ken Bannister Aug 12 '23 at 14:37
  • Anyway, I ended up with 7, but not totally certain. Could be 13. – Ken Bannister Aug 15 '23 at 16:22

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