Google Account Variable

I want to track how many times a user solves a puzzle in my game, this is because I will only be allowing so many free goes before they have to pay. What I don't want users doing is uninstalling and reinstalling to reset it. Therefore I believe that I would need to track their google account and say whether they have already done it, I guess this could be solved with a mysql database and storing the user in there once they have reached the limit, then flagging them as paid when they do so. I didn't know if there were an easier way to do this that people have tried before.