Answer- 3 and 4.

Explanation-
As Sam knows the sum of the numbers (x + y) he would only know the answer immediately.
If the sum was 4 (2 + 2) or 5 (3 + 2),then when Pete (who knows x*y) knew that Sam didn’t know the answer the product must have several numbers that add up to the sum (7 = 3 + 4, 7 = 5 + 2)
When Pete doesn’t know the answer at this point we know the product must have more than one pair of viable factors (12 = 3 * 4, 12 = 6 * 2).

At this point Sam knows the numbers are 3 and 4 because they are the only numbers that meet these criteria.

assume no are a and b
As sam has sum of no a + b = x(which has given )—————(1)
and pate has produnt a*b = y(given)

now (a+b)^2 = a^2 + b^2 + 2 * a *b

find out a^2 + b^2 from above

now find a-b = sqrt(a^2 + b^2 – 2 * a *b)——————–(2)

solve (1) and (2)

n get the a and b … it depends on x(sum) and y(product)

Finally we can say if you know sum (name) and product(name) you know the number(:P)

The numbers are 3 & 4
Explanation: If the sum was 4 (2+2) or  5(2+3) Sam would  know the answer immediately. If the sum was 6 (2 &4 , 3 & 3), Pete would  know the answer immediately because the product is 8 or 9 which has only one combination. So if the sum is 7 (2+5, 3+4), Sam doesn’t know the answer but is is was 2 & 5 Pete would know because the product is  10. So since he doesn’t know he has 12 so he can say he does not know the answer and also know that Sam also does not know. So that how Sam now knows the answer/