Game Center nickname password question:

My daughters each have their own nicknames for Game Center on their iPod Touches. Each nickname has its own password. My youngest can't remember her password.

Since I updated her iOS, I have to log her back into her Game Center account. Is there a way to retrieve her nickname password without resetting my Apple ID password? Since they are minors and don't have email addresses, they are all on my Apple ID. Resetting my Apple ID doesn't seem to work, and it's the only thing Apple seems to want to do. I can't find a way to contact Game Center directly.

Any other ideas?