Sum and Product Redux

Logic Level 4

Dan has written up 2 distinct positive integers, both of them are no greater than 10. Dan told Sam and Pam the sum and product of these 2 integers, respectively. Both Sam and Pam knew about the information in this paragraph.

The following conversation occurs.

Sam: I know that Pam doesn't know the 2 integers.
Pam: I know that Sam doesn't know the 2 integers.
Sam: I know that Pam doesn't know the 2 integers.
Pam: I know that Sam doesn't know the 2 integers.
.
.
.

This goes on and on for as long as possible until one of them shouted "Now I know the 2 integers!" and the other later shouted, "Me too!"

Assuming that Sam and Pam are perfect logician who never lied, which of the following is not a possible sum of the 2 integers?


Inspiration.

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