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 Prime Perimeter (Posted on 2005-03-11)

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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 full solution | Comment 7 of 90 |

Since the numbers are the sides of a triangle, my number is between 3 [ > (7-5)] and 11 [ < (7+5)]. So my number (a prime) could be only 3, 5, 7 and 11.

My number couldn't be 3, because the perimeter (3 + 5 + 7) = 15, isn't a prime.

If my number is a 5, the one that have a 7 will reason : "I'm seeing two 5's, and so my number is a 3 or a 7 (remember the prime perimeter). If my number is a 3, then the other two could deduce that both have a 5, since if one of them had a 3, the other would know that he had to have a 5. Since no one announced his number, my number is not a 3, so my number is 7". Since the one that have a 7 didn't say nothing, my number is not a 5.

If my number is a 7, the one that have a 5 will reason : "I'm seeing two 7's, so my number is a 3 or a 5 (remember the prime perimeter). If my number is 3, the other two would know that they both had a 7, and since no one announced nothing, my number is a 5". Since the one that have a 5 didn't say nothing, my number is not a 7, too.

So, my number is 11.

 Posted by pcbouhid on 2005-03-11 19:27:14

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