Answer

Mr. Sum said, "I knew you didn't know what the numbers were." The only way for Mr. Sum to know this is if all the possible pairs of numbers implied by the sum had at least one non-prime number.

Therefore, we are looking for sums that cannot be broken down into the sum of two prime numbers. The following sums are possible:

11, 17, 23, 27, 29, 35, 37, 47, 51 ...

For example, if the sum was 18, Mr. Sum would reason, "The two numbers might be 11 and 7, the product of which is 77. If  Mr. Product knew the product was 77, he would know that the only two possible numbers were 11 and 7; therefore, I cannot say with certainty that I know that he doesn't know what the numbers are."

Let's assume that the sum is 17 (because I already tried using a sum of 11 and it didn't work!). Then, from Mr. Sum's point of view, the following pairs are possible:

Note that all of the pairs contain at least one non-prime number, which means that the products imply more than one possible pair of numbers. Therefore, Mr. Sum says, "Mr. Product won't know what the numbers are."

This means that Mr. Product now realizes that the sum could not be divided into two primes. After looking at the possible pairs of numbers implied by the product, he eliminates those whose sum can also be expressed as a sum of two primes, i.e. he eliminates those that aren't on the above list. After he has finished, he has reached one possible solution, and says, "Now I know what the numbers are."

Mr. Sum looks at his numbers, and realizes that Mr. Product has gone through a process of elimination and reached only one possible solution. Therefore, he will check all of his pairs of numbers, and see which ones yield only one possible solution after a similar process of elimination. The possible products are:

The products yield the following possible solutions:

It is clear that 52 is the only product that yields only one possible solution; therefore, it must be the product. Therefore, the numbers are 13 and 4.

Solution by David Heetderks

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