Can You Solve The Impossible Puzzle?


#1

I got this one from 538 this morning and really liked solving it:

Three very skilled logicians are sitting around a table — Adam, Pete and Susan. Adam says: “I’m thinking of two natural numbers between 1 and 9, inclusive. I’ve written the product of these two numbers on this paper that I’m giving to you, Pete. I’ve written the sum of the two numbers on this paper that I’m giving to you, Susan. Now Pete, looking at your paper, do you know which numbers I’m thinking of?”

Pete looks at his paper and says: “No, I don’t.”

Adam turns to Susan and asks: “Susan, do you know which numbers I’m thinking of?” Susan looks at her paper and says: “No.”

Adam turns back to Pete and asks: “How about now? Do you know?”

“No, I still don’t,” Pete says.

Adam keeps going back and forth, and when he asks Pete for the fifth time, Pete says: “Yes, now I know!”

First, what are the two numbers? Second, if Pete had said no the fifth time, would Susan have said yes or no at her fifth turn?

If you think you have the answer, you can also submit it here with your explanation if you care to.