You have Plato and Socrates, two transhuman intelligences, over for tea.
You roll two 100-sided dice, producing two numbers x and y, both between 1 and 100.
You tell Plato the product, and Socrates the sum of the numbers, and they have the following dialogue:
Plato: I don’t know the values x and y.
Socrates: I knew you didn’t know. I don’t know them either.
Plato: Now I know x and y.
Socrates: Now I know them too.
How did they work it out? What were x and y?