Alan and Bob are trying to figure out two numbers. They know that both numbers are integers between 1 and 100 (but not 1 or 100). Alan knows the product of the numbers, and Bob knows the sum. Their conversation goes as follows:

Alan: I can't tell what the two numbers are.
Bob: I knew you couldn't.
Alan: Ok, now I know the numbers.
Bob: Now I know them, too.

What are the two numbers?

1 The numbers are not both prime, or Alan, knowing their product, would know the numbers.

2. Bob knew that they could not both be prime. Since the sum of any two primes is an even number, unless one of the primes is 2, then the sum Bob knows must be odd, but it must also not be two more than a prime number.

3. After hearing Bob's statement, Alan knows the number. This means that there are three prime factors of the number p(1), p(2), and 2. The original numbers must be either [2*p(1) and p(2)] or [p(10 and 2*p(2)] Since Alan can now eliminate one of those pairs [say 2*p(1) and p(2)], that means that [2*p(1) + p(2) is 2 more than some third prime, p(3): [2*p(1) + p(2) = p(3) +2]

4) Since Bob knows the original numbers after Alan's last statement, Every way of breaking the sum that bob knows into the form [2*p(1) + p(2)], except one, must allow that [p(1) +2*p(2)] also follow rule 2 above, which would therefore not have allowed Alan to make his deduction.

Posted by TomM on 2002-10-15 03:50:31