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Apologies if this is not appropriate for the site; I thought people might enjoy it.

In the spirit of Cheryl's rational gifts, here is an epistemic logic puzzle that I used on the final exam in my logic class this semester. I'll post a solution later, if necessary, but I expect we'll get some quality answers very soon.

Suppose that Alice and Bob are each given a different fraction, of the form $\frac{1}{n}$, where $n$ is a positive integer, and it is commonly known to them that they each know only their own number and that it is different from the other one. The following conversation ensues.

JDH    I privately gave you each a different rational number of the form $\frac{1}{n}$, where $n$ is a positive integer. Who has the larger number?

Alice    I don't know.

Bob    I don't know either.

Alice    I still don't know.

Bob    Suddenly, now I know who has the larger number.

Alice    In that case, I know both numbers.

What numbers were they given?

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If Alice had 1/1, she'd know the winner at her first turn.

If Bob had 1/1 or 1/2, he'd know the winner at his first turn.

If Alice had 1/2 or 1/3, she'd know the winner at her second turn.

Now Bob knows who wins, so he must have 1/3 or 1/4. If Alice's number is smaller than 1/4, she can't tell which Bob has. So Alice has 1/4, and Bob 1/3.

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  • Alice I don't know. So Alice does not have $\frac11$

  • Bob I don't know either. So Bob does not have $\frac11$ or $\frac12$

  • Alice I still don't know. So Alice does not have $\frac12$ or $\frac13$

  • Bob Suddenly, now I know who has the larger number. So Bob has $\frac13$ or $\frac14$

  • Alice In that case, I know both numbers. If Alice knows which Bob has then she has the other, and (from earlier) she does not have $\frac13$

So Alice has $\frac14$ and Bob has $\frac13$

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If Alice had $1/2$ she'd know. So she doesn't.

Bob now knows that. If he had $1/2$ or $1/3$ he'd know. So he doesn't.

If Alice had $1/3$ she'd know. So she doesn't.

Bob now knows. So he must have $1/4$ and have the smaller number.

Alice knows that and knows her number, so she knows both.

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