An old puzzle from the Car Talk radio show on NPR:
You and I each draw a number from a magic hat. We don’t know each other’s numbers, but the hat ensures that they’re adjacent positive integers.
An interlocutor now begins to ask each of us in turn whether he knows the other’s number. Eventually, one of us will answer yes. Why?
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Suppose I’m asked first and I answer no. This tells you that I’m not holding the number 1, because if I were I’d know for certain that you held 2.
The interlocutor turns to you and asks whether you know my number. You say no. This tells me that you don’t hold 2, by the same principle — I know that you just inferred that I’m not holding 1, so if you held 2 you’d know that I must have 3.
And so on. Together we’ll work our way up the number line until one of us says yes.
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