This was a difficult puzzle. Many people sent me the answers, though none of them were completely correct. One grade 11 student sent me the correct answer but the reasoning was not completely correct. So unfortunately no names on the honor board this time. Sirisha, who contributed the puzzle, also sent the answer which was very nicely laid out and hence I will take the liberty of posting her solution.
The correct answer is 2 and 2. The best way to do this is to create a matrix of the possibilities of two numbers, both for sum, as well as the product.
In step 1, we should be able to eliminate all the possibilities where A would get to know at least one of the numbers, for example, if the sum was ‘0’, then A would know that the two numbers are ‘0’ and ‘0’. Similarly, if the sum is ‘1’, then A would know that the two numbers are ‘0’ and ‘1’. So, by removing these possibilities, one gets the following matrix for ‘A’:
Similarly for B, if the product is such that it can be achieved by a unique combination (e.g. 64 can only be achieved by 8 and 8) or if one of the numbers can be figured out, e.g. if the product is ‘0’, then one knows that at least one of the numbers has to be ‘0’. If we eliminate all such possibilities, one gets the following set of possibilities for B. Note that a lot of combinations get eliminated here and also note that the products left appear at least twice e.g. 36 can be achieved by 9 and 4 and also by 6 and 6.
Now that A says that he knows the answer, it can only be because if we look at the corresponding sums of two numbers left after elimination of so many possibilities, the sum of the two numbers has to be unique. IN the matrix below, note that only those possibilities are represented that correspond to the product combinations left after B said that he cannot guess the answer. Within these sums, only the sum 4, 12 and 13 appear once. For example, if the sum of the two numbers was 11, A would still not have been able to give the answer as it could still be 9 and 2 or 8 and 3. Which means the sum has to be either 4, 12 or 13.
Interestingly, for B, the sum 12 in the above matrix is formed by 6 and 6, which leads to a product of 36, and 13 if formed by 9 and 4, also yielding 36. Since B is able to give the answer in the end as well, it means that the only possibility could have been 2 and 2.
Hope you enjoyed the puzzle!