73 of 100: I Know What You're Thinking But Do You Know What I'm Thinking?

When Ali first speaks, she does not have 1 or 9, as otherwise, she would know whose number is bigger(no bigger number than 9, no smaller number than 1). Zoe, knowing this, must not have 2 or 8 when she first speaks or she would know whose number is bigger. Ali knows Zoe does not have 1,2,8 or 9, otherwise, Zoe would have known whose number was bigger, so when Ali says that she still does not know whose number is bigger, she does not have 3 or 7. Zoe knows Ali can only have the numbers 4, 5 or 6, and the only one Zoe can have at this point to know hers is bigger is 6, the biggest.

[The problem statement has been changed from "What's Zoe's number?" to "What's Zoe's smallest possible number?" In that case, the answer is 6 only.]

Zoe can have either 6 or 7. So, there are two possible options.

Proof:

(1) Ali confirms that she doesn't have 1 or 9.

(2) Zoe confirms she doesn't have 1, 2, 8 or 9.

(3) Ali confirms that she doesn't have 1, 2, 3, 7, 8 or 9. Therefore, she can only have 4, 5 or 6.

(4) Zoe knows that Ali can only have 4, 5 or 6. So, Zoe must have 6 or 7.

1. Ali did not have 1 or 9, leaving 2 or 8 as good enough for Zoe to come to a decision.
2. Zoe did not have 2 or 8, leaving 3 or 7 as good enough for Ali.
3. Ali did not have 3 or 7, leaving 4 or 6 as good enough for Zoe to make a declaration.
4. If Zoe had 4, she would have said Ali's number is bigger. But she says her number is bigger, which means she had 6 or 7.
   Answer : 6 (smallest possible number for Zoe to come to a conclusion).
   

From first statement, it is clear that Ali did not receive 9, or else he would have known. Now, second statement suggests Zoe did not receive 8 as well by the same logic. Now, third statement suggests Ali did not receive 7 as well by the same logic. Now, last statement suggests Zoe received 6, so she was able to know that she had the bigger number

1. Ali don't know $\to$ A is between [2, 8] (because, if he had 1 or 9 he would know who had the bigger number).
2. Zoe don't know $\to$ Z is between [3, 7] (same as above, and Zoe knows that Ali is between [2, 8].
3. Ali still don't know $\to$ A is between [4, 6] (same as above etc.)
4. Zoe knows $\to$ knowing amounts to his number being either 4 (A's lesser possibility) or 6 (A's greater possibility). As he knows that he has the bigger number, it must be A's greater possibility. Therefore, Z = 6.