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Larry tells Marry and Jerry that he is thinking of two consecutive integers from 1 to 10. He tells Marry one of the numbers and then tells Jerry the other number. Then occurs a conversation between Marry and Jerry:

Marry: I don't know your number.

Jerry: I don't know your number either.

Marry: Ah, I now know your number.

Assuming both of them use correct logic, what is the sum of all possible numbers Marry could have?

What I have tried:

Marry's #s: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Jerry's #s: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Since Marry doesn't know Jerry's number, Marry's number could not have been 1 or 10.

Jerry's number then could not have been either 2 or 9 because then since he already knows Marry's # is not 1, then Marry's number would have been three. Same logic for 9.

So I am left with the possibilities as follows:

Marry's #s: 2, 3, 4, 5, 6, 7, 8, 9

Jerry's #s: 1, 3, 4, 5, 6, 7, 8, 10

I'm stuck here!

Help would be appreciated!

Also, it would also be nice if you would help me on this question(Transferring bases of numbers.) too!

Thanks!

Max0815

Max0815
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2 Answers2

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Note that the same reasoning applies to Jerry that you applied to Marry: Jerry's number cannot be 1 or 10, or otherwise Jerry would know Marry's number.

So: Marry could indeed have 2 (or 9): Marry would initially indeed not know Jerry's number (since it would be 1 or 3 ... (or 8 or 10), but then since Jerry says he does not know Marry's (which makes sense if Jerry's is 3, for then for all Jerry knows Marry's is 2 or 4 and indeed in either case Mary would not know) Marry knows Jerry's cannot be 1 (or 10), and thus is 3 (or 8).

Marry could also have 3 (8): Marry knows Jerry has 2 or 4 (7 or 9), but if Jerry has 2, then Jerry would know Marry must have 3 (for if Marry had 1 she would know Jerry has 2). So, Jerry saying he does not, she knows Jerry has 4 (7).

Finally, Marry cannot have 4 (7): Jerry would have 3 or 5, but either way Jerry would not know, so Marry learns nothing from Jerry saying he does not know. Mary having 5 (6) also leaves too many options open for her to know Jerry's number on her second turn.

So, Mary's number is 2,3,8, or 9. Sum is 22

Bram28
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Marry also knows that the numbers are consecutive. Therefore, if her number is $n$, she knows that Jerry's number is either $n+1$ or $n-1$. And after the fact that Jerry still doesn't know her number narrows down the field still further. Jerry's number can't be 1 or 10 because then he'd know Marry's number without help. Jerry's number also can't be 2 or 9 because then the fact that Marry doesn't know Jerry's number would tell Jerry that Marry's number has to be 3 or 8, respectively.

If Jerry's inability to pinpoint Marry's number allows Marry to pinpoint Jerry's number, then her ability to eliminate 2 or 9, or the knowledge that Jerry's number isn't 1 or 10, must be new information that allows her to pinpoint her own number. Marry's number therefore has to be 2, 3, 8, or 9 and Jerry's number has to be 3, 4, 7, or 8.

Robert Shore
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