I disagree with answer 2 and 9. Because:

first person know the product is 18. He don't know the answer now. He just could guess the set of the answer:

1x18 imposible, coz 1 not allowed

2x9 could be the answer

3x6 could be the answer

By know the sum is less than 14, he still don't know the answer coz

2+9 < 14

3+6 < 14

So he don't know the answer and couldn't say wether he know the answer

Ok assume the numbers were 3 and 6.

State 1:

Sum kid knows the sum is 9. Possible combinations:

2:7 Product 14

3:6 product 18

4:5 product 20

State 2:

The 2:7 gives a product of 14. A product of 14 can only derive from 2 and 7. Product kid would need no help to figure out the numbers.

4:5 gives a product of 20 which has the combinations of State 3

3:6 which has a product of 18 and the combinations of State 4

State 3:

2+10=12 In this case the product boy assumes that the sumn boy knows the sum is 12. What could be going in the sum boys mind right now?

2+10=12 Product 20 Could be the answer

3+9=12 Nope product is 27 giving away the answer

4+8=12 Product 36 Could be the answer

5+7=12 Nope product is 35 giving away the answer

6+6=12 Product 36 Could be the answer

State 4:

3+6 = 9 Could be the answer

2+9 = 11 Could be the answer

Both have a sum less than 14

This gets too confusing to continue, but all you need to understand is that none of the boys can eliminate enough combinations to have only one combination left to speak the answer.

So the boys have to talk to each other to get some more info.

The sum boys says the sum is less than 14. After this statement, both boys know the numbers. At this point the product boy knows the numbers because he states so. The sum boy eliminates the combination he had that would have left the product boy in doubt, thus he figures out the numbes too.

BUT in the above example, the product boy doesn't figure out the numbers. Check state 3 and state 4. Both of them have combination that sum up to less that 14. So the product boy cant eliminate any combination. The sum at this situation doesn't help him. But in our example, the product boy had figured out the numbers after he know the sum was less than 14. So 3 and 6 are not the combination since the leave the boys still struggling to figure out the solution.