Can you guess?

I actually stumbled upon the correct answer (2 + 3 = 5) by guessing correctly on my first try, but I'll still provide the reasoning.

"Ben doesn't know which two numbers Ash had picked" means that the sum, s, can be written in more than one way, like 5 = 2 + 3 = 1 + 4. "[Ben is] sure that Charles [doesn't] know either" means that for every possible combination of two values, their product can be written in more than one way, so Charles cannot be sure exactly what the two numbers are. For example, if 5 = 1 + 4, then 4 = 1 * 4 = 2 * 2; if 5 = 2 + 3, then 6 = 2 * 3 = 1 * 6. This also implies that the numbers are small, because their product can be written in more than one way as the product of two numbers between 1 and 9.

"Charles knows now" means that Ben's statement had eliminated all but one options. Charles sees the product 6, and has the two options of 2 * 3 or 1 * 6, which means Ben has either 5 or 7. If Ben had 7 and 7 = 2 + 5, Ben would not have been certain that Charles could not be sure of the numbers, because 10 = 2 * 5 cannot be written in any other way. Therefore, Ben must have 5, which Charles confirms by stating that 6 > 5.

Ben, knowing that he has the options 1 + 4 or 2 + 3, knows the numbers now because (1 * 4) < 5 but (2 * 3) > 5, so it must be 5 = 2 + 3.

Lets say the two numbers are $x,y$ , with $S = x + y$ and $P = xy$ .

"Ben doesn't know" $\implies S \neq 2, 3, 17, 18$ .

"Ben knows Charles doesn't know" $\implies S$ cannot be:

• 1) $1$ more than a prime less than $10$ .

• 2) broken into $2$ numbers, whose product can be written as a unique product subject to $x,y < 10$ . hence if it cannot be written as $5+y$ , as $5y$ can only be written as $5 \times y$ , the same with $7+y)$ . Hence $S < 6$ .

Therefore, $S = 5$ .

$P = 4$ or $6$ . Charles, then learns $S = 5$ and deduces the numbers.

$P>S \implies P=6$ and $(x,y) = (2,3)$ .

x+y= B = Ben's number, xy= C = Charles' number

The only time a+b > ab is when either x or y is equal to 1. This means that Ben's number can only be bigger than Charles' when he has the sum of 1 and another number. When Charles said that his number was greater than Ben's, Ben knew what the answer was. This tells us that for Ben there was a possibility that either x or y was equal to 1. If there was a possibility of that scenario, B could not have been greater than 10. We also know that Ben is sure he knows what numbers Ash picked after excluding the scenario in which x or y was equal to 1. This means that B can be made up of x+1 and only one other way. We know B can't be greater than 10 and also has to be made up of x+1 and only one other way. The amount of unique sums (x and y not being switched around) there are in a number is equal to half of that number, or if it's an odd number, that odd number minus 1 and then half of that. And so, the only numbers where this is equal to two are 4 and 5. We know it's not 4 because 2x2 is the only other option from 1x3, and 2x2=2+2, and so is not greater than Charles' number.

In this solution it's actually not all that important that B cannot be greater than 10, but I put it in anyway in the scenario process by elimination is preferred, because you can actually also solve this by eliminating all other numbers but 5 when you get to that point.