Ali and Zoe reach into a bag that they know contains nine lottery balls numbered 1-9. 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 perfectly logical, what is Zoe's smallest possible number?
18
8
8
To find Zoe's smallest possible number, let's analyze the statements made by Ali and Zoe.
Statement 1: Ali says, "I don't know whose number is bigger."
From this statement, we can infer that Ali's number is not the smallest since if his number were the smallest, he would know that Zoe's number is bigger. This means that Zoe's number must be smaller than Ali's number.
Statement 2: Zoe says, "I don't know whose number is bigger either."
This statement suggests that Zoe has eliminated the possibility of having the smallest number because if she had the smallest number, she would know her number is smaller than Ali's. Therefore, Zoe's number is now limited to 2 through 9.
Statement 3: Ali says, "I still don't know whose number is bigger."
Since Ali did not gain any new information from Zoe's statement, his number cannot be the smallest either. This narrows down Zoe's number to 3 through 9.
Statement 4: Zoe says, "Now I know that my number is bigger!"
Zoe's certainty indicates that she has determined her number is greater than Ali's. This statement implies that Ali's number cannot be 1 or 2 because if it were, Zoe wouldn't be able to conclude her number is larger. Therefore, Zoe's number is at least 3.
In conclusion, based on the logical deductions, Zoe's smallest possible number is 3.