### (Advanced) Cheryl's Birthday Puzzle

Source: Sent to me by Prateek Chandra Jha (IIT Bombay) Problem: This problem is inspired by the Cheryl's Birthday Puzzle ( FB Post , Guardian Link ). Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information: Both numbers are integers between (including) 1 and 1000 Both numbers may also be identical. Paul is told the product of the two numbers, Sam the sum and Dean the difference. After receiving their number, the following conversation takes place: Paul: I do not know the two numbers. Sam: You did not have to tell me that, I already knew that. Paul: Then I now know the two numbers. Sam: I also know them. Dean: I do not know the two numbers. I can only guess one which may probably be correct but I am not sure. Paul: I know which one you are assuming but it is incorrect. Dean: Ok, I also know the two numbers. What are the two numbers? Disclaimer: Its not a puzzle for 14-15 year olds like Cheryl's