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birthday seems unnecessary, but otherwise remember that the HN crowd will complain about FB login far more than the general population would.

Maybe you want to give credits/points/stars as a birthday present(or something like that). But that should be something the user chooses to opt-in.

Why not create implicit temporary accounts (tied to some token in a cookie) that will automatically be make permanent when the user registers with email or Facebook? People can play the game and not lose their game history when they register.

Just give me a unique url I can send my friends. You can provide a link for me to send post it to Facebook without my having to log in.

As someone else pointed out, the HN crowd cares more about our privacy than the public at large. Respecting people's privacy isn't about how many people care about it.