Alan and Bert

I told Alan and Bert that I had two different whole numbers in mind, each bigger than 1, but less than 15. I told Alan the product of the two numbers and I told Bert the sum of the two numbers. I explained to both of them what I had done.

Now both these friends are very clever. In fact Bert, who is a bit of a know it all, announced that it was impossible for either of them to work out the two numbers. On hearing that, Alan then worked what the two numbers were!

What was the sum of the two numbers?

Alan and Bert Puzzle Solution

Starting Numbers Sum Product Product can also be made using
6, 5 11 30 10, 3
7, 4 11 28 14, 2
8, 3 11 24 6, 4 and 12, 2
9, 2 11 18 6, 3 and 9, 2
Bert knows that there are four possibilities for the starting numbers - so it is impossible to work out the starting numbers using the sum alone.

As mentioned in the question, he's quite clever - so he looks at the product that would appear if he used each of the four possible combinations. As shown in the table above, the product that appears can also be made from different numbers.

So, he announces that it is impossible for him or Alan to work out what the original numbers were. He seems to be on fairly safe ground.

Alan is a little bit more devious as well as being clever. Armed with this snippet of information he needs to look for a pair of numbers that give a non-unique sum and also give a non-unique product.
11 is the only non-unique sum which always gives a non-unique product. Alan is very clever and very smug.