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January 2015 - Alice and Bob


There are two different positive integers a and b. Both are between 1 and 10 inclusive. Alice got told the sum and Bob got told the difference. Alice and Bob are perfect logicians and are both truthful. The following conversation occurred.

Alice: I don't know the numbers.
Bob: Neither do I.
Alice: I knew that.
Bob: Even if you told me one of the numbers, I'm not sure I could figure out the other one.
Alice: I knew that.
Bob: Now I know that 10 is not one of the numbers.
Alice: Now I know that 3 is not one of the numbers.
Bob: Now I know the numbers.
Alice: So do I.

What is the product of the numbers?

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