Assumptions:
1) Zero is not an option
2) Two identical numbers aren’t options:
Mr. Product: I do not know the numbers
The sum of the numbers can’t be a prime number, or 2.
Mr. Sum: I knew you didn’t know the numbers
No permutation of numbers totaling the sum of the two numbers will result in a unique product.
I.e. if Mr. Sum’s number was 56, then Mr. Product’s number could have been 159 (a unique 3 * 53 combination).
Mr. Product: Now I know the numbers
By Mr. Sum’s revealing that at least one possible sum’s permutation’s have no unique results, Mr. Product eliminates all numbers whose sum’s may have unique permutations, leaving only a single “no unique” possibility.
The sums whose every possible combinations of numbers result in a product whose product is shared by another two numbers are: 7, 9, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53
Now, the application of brute-force
Let’s say the numbers are 1 and 6.
The sum is 7, the product is 6.
For Mr. Sum, the numbers can be 1:6, 2:5, 3:4.
For Mr. Product, the numbers can be 1:6, 2:3.
Mr. Sum relizes that Mr. Product can’t know the solution since he may either have 6 (1*6 or 2*3), 10(1*10 or 2*5) or 12 (1*12 or 3*4) as his choices.
Once Mr. Sum reveals that he knew Mr. Product doesn’t know the answer, he eliminates 2:3 as a possibility (since 2:3’s sum is 5, and a sum of five may have resulted from 1:4, and with 1:4 Mr. Product would have known the answer).
Mr. Product has removed 2:3 from the possibilities, leaving only 1:6, so he knows the answer.
If Mr. Product’s number was 10, he couldn’t have eliminated anything, since both 7(1*6) and 11(1*10) are on the above list.
If Mr. Product’s number was 12, he also couldn’t have eliminated anything, since both 7(1*6) and 13(1*12) are on the above list.
Therefore, Mr. Sum relized that Mr. Product eliminated 2*3, leaving 1 and 6.
now I’ll read the resident maven’s solution