# Crushing logic

I meet Samuel and Pablo, two logical mathematicians, my friends, to whom I propose the following challenge:

I have chosen two numbers greater than 1 and less than 100. To my friend Samuel I say to the ear the result of the sum of both numbers and to Paul I say the result of multiplying both numbers. Samuel knows that Pablo knows the product and Pablo in turn knows that Samuel knows the sum of the two numbers but neither knows the numbers that I have chosen. After this Samuel and Pablo meet and the following conversation takes place between them:

Pablo: I don't know what the numbers are.
Samuel: I knew you couldn't know.
Pablo: Ah, so then I know what numbers they are.
Samuel: Well then me too.

What are the numbers that I have chosen?

Extracted from www.zurditorium.com

#### Solution

The solution is 4 and 13. I have given Pablo the product of both numbers, which is 52, from which he deduces that the numbers are 2 and 26 or 4 and 13 so he still cannot know the solution.

I have given Samuel the sum of both numbers which is 17. Since 17 cannot be put as a sum of 2 prime numbers, he knows that Paul cannot deduce the numbers. Once he tells Pablo that he already knew that he would not know, Paul dismisses case 2 and 26 since the sum cannot be 28 because if it were, Samuel could not have ruled out that the solution was 23 and 5 and so I couldn't have said so much that Paul wasn't going to know what numbers they are.

With this Paul already knows what numbers they are. Samuel would also know since he can rule out any other pair of numbers that add up to 17 since in these cases, Paul would not have found the solution. The other possible cases are

• 2 and 15. The product would be 30 so Pablo could not rule out 2 × 15 or 5 × 6.
• 3 and 14. The product would be 42 so Pablo could not rule out 3 × 14 or 2 × 21.
• 5 and 12. The product would be 60 so Pablo could not rule out 5 × 12 or 3 × 20.
• 6 and 11. The product would be 66 so Pablo could not rule out 6 × 11 or 2 × 33.
• 7 and 10. The product would be 70 so Pablo could not rule out 7 × 10 or 2 × 35.
• 8 and 9. The product would be 72 so Pablo could not rule out 8 × 9 or 3 × 24.

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