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?

(In reply to re: full solution as understood by Sachin)

"Scenario 1. A=5, B =5 and C =7

In this case C will see 2 fives which means the only possible values for C are 3 or 7. If C's value is 3, B could have guessed his numbers as he is seeing 3 and 5, and only possible third number is 5. But B couldn't, which leaves C's number as 7. Since even C could not guess his number as 7. So scenario 1 is not possible."

But if B were seeing 3 and 5 he could assume his own was either 3 or 5, not just 5.

"Scenario 2. A = 7, B =5 and C =7

In this case B will see 2 sevens which means the only possible values for B are 3 or 5. If B's value is 3, C could have guessed his number as he is seeing 3 and 7, and only possible third number is 7. But C couldn't, which leaves B's number as 5. Since even B could not guess his number as 5. So scenario 2 is also not possible."

If C were seeing 3 and 7, then 3 would be another possibility for C to consider possible besides the 7.

So is still think it would take at least two rounds of thinking, that included the third person (me, you, depending on who's reading--Yes I mistakenly said narrator before--I meant the person doing this thinking and coming to a conclusion).

Posted by Charlie on 2005-03-12 03:01:36