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Prime Perimeter (Posted on 2005-03-11) Difficulty: 3 of 5

You and two other people have numbers written on your foreheads. You are all told that the three numbers are primes and that they form the sides of a triangle with a prime perimeter. You see 5 and 7 on the other two heads and both of the other people agree that they cannot deduce the number on their own foreheads.

What is the number written on your forehead?

See The Solution Submitted by Erik O.    
Rating: 2.8235 (17 votes)

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Charlie's Skepticism | Comment 28 of 89 |
I appreciate your skepticism Charlie, but I think its root lies in contempt for your fellow man's capacity for logic. Perhaps the problem would succeed if it suggested you were sitting at a table with Euclid and Newton.

I think the logic does indeed follow as suggested by pcbouhid. But it does require that we are sitting with bright folks and we all know it. We can certainly deduce that the number on our heads is not a 5 because Person "7" would have been able to deduce their own number by now with the following logic:

Person "7" says to themself: I see two fives. According to the rules I must have a 3 or a 7 on my forehead. If I had a 3 on my head both of my cohorts would see a 5 and a 3 and be wondering if they had 3's or 5' s themselves. It wouldn't take them long to deduce that 3 isn't possible because that would mean one of us would be looking at two 3's. Everyone knows that if they are looking at two 3's their own number must be a five. But no one has been able to draw a conclusion yet, clearly there is no 3 on my head.

Furthermore, we can certainly deduce that our number is not a 7 because person "5" would have been able to deduce their own number by now with this logic:

Person "5" says to themself: I see two sevens. That means my forehead must contain a 3 or a 5. If it is a 3 then that means my tablemates see a 3 and a 7 and would have deduced that they have 7's by now, since it is the only possible prime, triangle, prime perimeter solution. Therefore I must have a 5.

I must have a 11 on my head since enough time has passed for my companions to make accurate deductions otherwise.
  Posted by Eric on 2005-03-12 05:23:59
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