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I think the numbers are 3 & 4.
Here goes the explanation:
Lets assume that the numbers are a & b and the product is p and the sum s.
p=a*b
s=a+b
let the person who knows the product be P and the person who knows the sum be S.
P doesn't know the numbers so both a & b cannot be prime. Given that both a & b are not prime there is one more info that P can have abt the product. The lowest possible product is 12. This is the only info that can put restrictions on the number of possible sums.
When S says that he knows that P doesn't know the numbers, it means S also knows that both the numbers are not prime. How does he know that?
The sum of two prime numbers will always be even unless one of the numbers is 2. If S is so sure that both the numbers are not prime, that means the sum must be odd. But then how is S so sure that one of the numbers is not 2. The only reason can be, the sum is such that subtracting 2 from it gives a prime number. So S is sure that both the numbers are not prime (however one can be prime and one can be non-prime) and one of the numbers is not 2.
Now when S tells P that he knows that P doesn't know the numbers, P guesses the numbers. So by this time P had already narrowed the number of possible sums. This is only possible if product given to him is 12.
If the product is 12, possible combinations are 2*6 and 3*4. But from S's statement its clear that he has ruled out 2 as one of the numbers. Also if the numbers are 2 and 6, the sum will be even and S cannot confidently say that both the numbers are not prime. So the only combination remaining would be 3*4. P figures this out. When P figures this out, S also realises that P had narrowed the number of sums down and that means the product must be 12. Hence he also figures that the numbers are 3 and 4.
This is the solution I came out with. Not sure whether correct.
Sam
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