and are told the following:
there's a whole number between 10-99
Person A is told the sum of the two digits
Person B is told the multiplication (product?!) of the two digits.
they're both put into a room and the conversation goes as follows:
A: I don't know what the actual number is
B: I don't know what the actual number is
A: I know what the actual number is
B: I also know what the actual number is

and here's the question: what was the original two digit number?

Ok, write the number as ab, we then know that ab can be anything from 10 to 99.

A en B make a list of alll possible sums (1 to 18) and writes the numbers corresponding to that sum behind them.

A says (s)he doesn't know what the number is wich tells them both that the number isn't 10 or 99.

B says (s)he doens't know wich number it is, wich tells them both that the number has no unique product of digits.

After eliminating all such numbers A knows what number it is, because he knows the sum of the digits. In other words, there is one sum with only one number left corresponding to that sum. That number is 20. B sees that as well and knows the number too.

After eliminating all such numbers A knows what number it is, because he knows the sum of the digits. In other words, there is one sum with only one number left corresponding to that sum. That number is 20. B sees that as well and knows the number too.

Unless I have misunderstood the last part of this paragraph: "there is one sum wiht only one number left corresponding to that sum", I believe that it is inaccurate. The idea behind that sentence seems correct, but in reality, both the sums 2 and 17 have only. Two can be made of 20 and 11, 17 can be made of 98 and 89, and unless they hinted to each other what the sum and product were, I cannot see how they would know for certain that 20 was the answer.
I suspect that I have misunderstood the riddle itself in some way, because if I hadn't, I cannot see how you guys could have overlooked what I have just pointed out. If I did misunderstand the riddle, please let me know.

Person B doesn't know the answer immedeately, so that means the number isn't 11.

A realises that and can scratch 11 from his list (as well as 55, 77, and some others). Both 98 and 89 stay on his list. Now if you look at what numbers give the same sum, 20 is the only number with sum 2. All other sums still have more then one possibility of being made (as 17 has both 89 and 98).

Person A know the sum, and knows the answer now, wich means the number has to be 20.

Originally posted by TheMaster37 Person B doesn't know the answer immedeately, so that means the number isn't 11.

A realises that and can scratch 11 from his list (as well as 55, 77, and some others). Both 98 and 89 stay on his list. Now if you look at what numbers give the same sum, 20 is the only number with sum 2. All other sums still have more then one possibility of being made ...[text shortened]... d 98).

Person A know the sum, and knows the answer now, wich means the number has to be 20.