We follow the conversation step by step:
- “Hi Ann, I don't know when Christine's birthday is, but I also know that you don't know!”
Bo can indeed not know what the birthday is, because with every month more than one day is possible. But she also says she knows that Ann doesn't know either. How can she know that?
Suppose Christine had said to Bo that the the month is April and to Ann that the day is 19. Then Ann would know. The same goes for May: if Ann had been given the day 18, she would have known. So Christine did neither April nor May.
Because Ann says that Bo doesn't know either, Bo will conclude that the month must be July or August.
- “That's right, Bo, but now I know”
Ann knows now. Then it's not July 14 or August 14. Chistine told Ann the day is 16, so it is July 16, but you and Bo don't know that yet!
- “That's funny, because now I also know!”
Okay, now Bo knows. Suppose Christine had told her the month is August. Then she wouldn't know, because it could be August 15th or 17th.
So now we also know the month is July.
Christine's birthday is on July 16.