Once every decade or so I come on and say, “I am sorry I made a mistake.” However, this is not one of those decades, although it may be called an oversight or even careless; no, I think careless is going overboard…
I am sorry for the delay in confirming the answer; I saw Lizzie had posted some days ago and thought there had been no response since then. Had I known that you two had been going at it like Possums in springtime I could have expedited matters " My bad!
Mark:
NUMBERS
How about 4 and 13
First, create a list of all your possible pairs and write them out with their sums and products as well. I believe there are (97*97 + 97)/2 = 4753 such pairs, right? (I am ignoring the "sum < 100" hint.)
2,2,4,4
2,3,5,6
2,4,6,8
2,5,7,10
2,6,8,12
...
3,3,6,9
3,4,7,12
...
4,4,8,16
4,5,9,20
...
etc.
Step 1) We know it is not a pair of primes, or Mr. P would know them immediately from the product.
So eliminate all those from the possibilities, but keep them in a new list of "Prime Pairs" for future reference.
Step 2) We know it is not a pair whose sum could also come from a pair of primes, or S would not be sure that P doesn't know. So check the list of Prime Pairs and get a listing of all the sums. Now eliminate any pairs in the Possibilities that have a sum in that list. So here for example, you lose 3,4 because it has the same sum as the prime pair 2,5.
Step 3) Now, we know that with this information and knowing the product, Mr. P knows the answer, so therefore the product must only have one pair of possible multiplicants whereby their sum cannot be expressed as the sum of a pair of primes. So find the list of products that were eliminated in Step 2. Now find all the remaining Possibilities that share one of those same products and only have one pair left that makes that product. These are the only products Mr. P could have.
Step 4) If there is more than one possibility left, your last step is to go through all the Possibilities eliminated in Step 3, and check their sums. Now find all the remaining Possibilities that share one of those same sums and only have one pair left that makes that sum. These are the only sums Mrs. S could have.
Hopefully at this point you are left with only 1 possible pair, and that is your solution. (4,13)
iwnhg
thpls
tomap