mnSP Problem

The Problem:

Two numbers, m & n, are chosen such that:

        2 <= m <= n < = 99.

S is told their sum; P is told their product.

P: "I don't know the numbers."

S: "I knew you didn't know.  I don't know either."

P: "Now I know the numbers."

S: "Now I know them too."

What are the numbers?

Questions:

1. Give an example of m & n such that S cannot deduce their value given the sum.

2. Give an example of m & n such that P cannot deduce their value given the product.

3. Give an example of m & n such that S knows that P does not know their value.

4. Give an example of m & n such that P can deduce their value after learning that S knows that P does not know their value.

5. At what point might it be useful to write a program to obtain the answer to any of the above questions?  What would the program do?

6. What are the values of m & n?