Pavel and Sara roll two, fair six-sided dice (with faces labeled from 1 to 6) but do not look at the result. A third-party observer whispers the product of the face-up numbers to Pavel and the sum of the face-up numbers to Sara.
Pavel and Sara are perfectly rational and truth-telling, and they both know this.
Pavel says, “With the information I have, I am unable to deduce the sum of the two numbers rolled.”
Sara responds, “Interesting! With the information I have, I am unable to deduce the product of the two numbers rolled.”
Pavel responds, “Wow! I still cannot deduce the sum. But I’m sure you know the product by now!”
What is the product?