I found this number puzzle elsewhere online, and since my solution was very computational I figured someone here might enjoy it. I've actually modified it a bit. I would link to the original except it contains my answer.


Person A picks two whole numbers between 1 and 100, excluding both 1 and 100. The numbers are not necessarily distinct. Person A tells Person S the sum of the numbers and Person P their product. S and P have the following conversation, where everything they say is true:

P: "I know only the product and not the sum."
S: "I know that you know only the product and not the sum."
P: "Now I know the numbers."

Easier version: if P's product was 182, what were the numbers?
Harder version: After S and P's conversation ends, A remarks, "Amazing! If you repeated this exact conversation with any other pair of numbers, the sum would be different!" What were the numbers?