Perfect logicians

Here's a detailed solution.

This doesn't make very much sense. It seems rather obvious that neither party knows the two numbers initially. Why would a proclamation of ignorance transmute into epiphany?

Unless this is to say that the conversation in its self is an irrelevant distraction.

If the numbers had been 2 and 1, both parties would have instantly known. This seems to be the only combination ruled out by the conjecture. This isn't something that is possible. If you look at the link to the online "solutions" it shows you probabilities but not definite numbers.

I have no idea why anyone would think this is remotely solvable.

This lengthy explanation does not lead to a result, it simply shows that the 4 statements made by P and S DO, in fact, give a lot of useful information.

Four Statements:

P: I don't know the two numbers...
S: I know you don't
P: Ah...now I know them...
S: So do I

The fact that the first doesn't know the numbers tells a lot.

There are three cases when it comes to the product that he's given. First is the impossible. He can't be given any of the following as a product: 1,2,3,4,5,7,9,11,13,17,19,23,25... and so on. This is a list of all prime numbers and all perfect squares of prime numbers.

The other two cases are those that he can get that will tell him the two numbers and those that he can get that won't tell him what the two numbers are.

The first would be numbers such as 6,8,10,14,15,16,21, and others. These numbers only have one possibility to be a product of two DISTINCT numbers GREATER than 1. 6 = 3*2, 8 = 4*2, 16 = 8*2 (and 4*4, but that's not possible). By saying that he doesn't know the numbers, he's eliminating this entire group.

The second would be products that do NOT directly tell him what the individual numbers are. Examples would be 18, which could be 6*3 or 9*2. The first guy is telling the second guy that THIS group is where his product falls, which indeed tells the second guy a lot of information.

Now, when the "sum-guy", or sg, says that he knows that the "product-guy", or pg, doesn't know the answer, he's saying a lot also. He's not saying that he, himself doesn't know the answer, since that would be the case for any sum he was given that is higher than 5. However, by stating that he knows that "pg" doesn't know the answer, he's actually telling a lot himself.

wow, this is a long explanation that no one will read.

Anyway, "sg" only has a sum, let's call it x. x can be split up into many combinations. if x = 15, it can be split up into 2,13;3,12;4,11;5,10;6,9;7,8. So "sg" has, at most can narrow it down to (x-3)/2 if it's odd and (x-4)/2 if it's even, choices. By him saying that he KNOWS that the first guy cannot possible know the answer, he's saying that the product of each of the choices does NOT give him enough information. If ANY of the choices that "sg" has, can be multiplied in order to give "pg" enough information, then he can't make the claim.

In the example where "sg" gets 15, one possibility is 2,13. The product of these is 26. If "pg" were to get 26, he would know the answer, since the ONLY N1,N2>1 where N1!=N2 that allows N1*N2 = 26, is 2 and 13. Therefore, "sg" could not make the claim that he knew that "pg" didn't know. Since he DID make the claim, "pg" can safely assume that 15 is not the sum that "sg" got.

These facts can be seen throughout the numbers. Therefore, the solution has to have certain characteristics about it.

1) P: I don't know the two numbers...
(a) The product must have more than one possible group of factors, with each group containing two entries. If this is not true, then P's claim would not be true

2) S: I know you don't
- The sum can NOT contain any "factors" (the two numbers possible to make up the sum) that would not hold true for (a). Since S KNOWS that P does NOT know, then none of the possibilities could give P enough information to solve it on his own.

3) P: Ah...now I know them...
- Since S has just told P that the sum shows that there are no "sum factor" possibilities that could make a product that would give a valid answer, it cuts down his choices. Therefore, there must have only been ONE factor grouping for P that this would be true for. Once S made his claim, the other possibilities for P become impossible and thus, he is left with one solution.

4) S: So do I
- When P tells S that all of his possible factor pairings have become invalid except for one (he says this by claiming to know the answer now), S looks at his own possibilities (of which he does not have many) and all of THOSE are eliminated, due to P's claim, except ONE. This one sum pairing is then his only answer, which of course, matches P's.

Seems the easiest way to do this is a bit of a brute force checking of all sums between 6 and 100 (since this was stated in the question). It looks like the answer link does this through some hidden way (probably software). To make this a good, solid riddle, there should only be one pair of numbers for which these four statements will lead to the solution.

I know there is NO way anyone will read all of this, so I'll end this with a joke:
Q: How do you get an elephant out of the theatre?
A: You can't, it's in their blood!

If you understand/like this joke, then you and I need to become friends, haha.