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?**

Hint: When she speaks for the first time, if Ali doesn't know whose number is bigger, which number must she

nothave?