I played a little logical game with my two highly intelligent godsons, Proddy and Addy.

I had in mind three different digits chosen from 1 to 8 and I whispered the product of the three to Proddy and the sum to Addy. I explained all this to them and our conversation went as follows:

Me: “Proddy, can you now work out what my three numbers are?”

Proddy: “No.”

Me: “Now do you think that Addy will be able to work out what my numbers are?”

Proddy: “No, he will not be able to work them out.”

Addy: “Now I know what the numbers are.”

What are they?

Credit due to Aunt Susan Denham.

I think the numbers are either (1,3,4) or (2,3,5).

The only way this makes sense to me is if Proddy (who is no prodigy) is answering the 2nd question without considering that Addy has heard the answer to the 1st question.

If Addy knows that the total is 8, then the numbers could be (1,2,5) or (1,3,4), so Proddy answers no to the 2nd question. But as soon as Proddy says no to question 1, Addy knows that the numbers cannot be (1,2,5), because in this case Proddy can work out the numbers, as only one set of numbers have a product of 10.

Similarly, if Proddy knows that the total is 10, then the numbers could be (1,2,7) or (1,3,6) or (1,4,5) or (2,3,5). But as soon as Proddy says no to question 1, Addy knows that the numbers must be (2,3,5), because in the other three cases case Proddy can work out the numbers, as only one set of numbers have a product of 14 or 18 or 20.

If my interpretation is correct, then the problem would have been better if the order of the questions to Proddy were reversed, since then we would not need to assume that Proddy is not a prodigy.