Nine Lottery Balls

Ali and Zoe reach into a bag that they know contains nine lottery balls numbered 1 to 9. They each take one ball out to keep and they look at it secretly. Then, they make the following statements, in order:

Ali: I don’t know whose number is bigger.
Zoe: I don’t know whose number is bigger either.
Ali: I still don’t know whose number is bigger.
Zoe: Now I know that my number is bigger!

Assuming Ali and Zoe are perfect logicians, what is Zoe’s smallest possible number?

Answer

Zoe’s smallest possible number is 6.

Based on the first statement of Ali, it indicates that he has neither 1 nor 9. If he had either 1 or 9 then he would know that Zoe must have a bigger or smaller number.

Now Zoe, based on Ali’s first statement, knows that Ali doesn’t have 1 or 9. Zoe’s first statement indicates that she does not have 2 or 8 (neither 1 nor 9). If she had 1, 2, 8 or 9, then she could have concluded that Ali has a bigger or smaller number.

Now Ali knows that Zoe doesn’t have 1, 2, 8 or 9. Ali’s second statement indicates that he does not have 3 or 7 and also not 1, 2, 8 or 9.

Zone can conclude that Ali doesn’t have 1, 2, 3, 7, 8 or 9. In short, Ali must have either 4, 5 or 6.

Now when Zoe says that she has a bigger number then it must be either 6, 7, 8 or 9 and Ali having 4 or 5. Zoe can’t say confidently that she has a bigger number if she had a 4 or 5, as it could be smaller than what Ali could have.

So Zoe’s smallest possible number is a 6.

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