In this April Fools' Day Riddler we're asked to solve this "impossible" puzzle:
Three very skilled logicians are sitting around a table — Barack, Pete and Susan. Barack 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.”
Barack turns to Susan and asks: “Susan, do you know which numbers I’m thinking of?” Susan looks at her paper and says: “No.”
Barack turns back to Pete and asks: “How about now? Do you know?”
“No, I still don’t,” Pete says.
Barack 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?
Fortunately, it's not actually impossible to solve, and is actually pretty straightforward. We start by examining all of the possible products that Pete could see written on his paper:
We can eliminate any products that appear only once, because otherwise Pete would be able to deduce Barack's two numbers. We then replace all of the products with sums instead, and do the same for Susan - eliminate any sums that only appear once (among the remaining squares that haven't already been eliminated). We repeat this four times each, for the four times Pete and Susan each answer "No" to Barack's question.
Finally, on Pete's fifth turn he says "Yes" so we look for the product that DOES appear only once, as this must be the answer. The following video illustrates the whole process:
As you can see, when Pete finally says "Yes" the only product that appears once is 16, corresponding to the numbers 2 and 8.
For the second part of the puzzle, what would happen if Pete said "No" instead? We would continue as we had been doing and cross off the 2-8 cell instead. Then we'd have this:
Each of the remaining possible sums still appears twice in the table, so Susan would not be able to deduce the answer and would have to answer "No."