My game center is not working.

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If you can't sign in to Game Center or stay connected, follow these steps:

1. Make sure that you're connected to the Internet.
2. If you can't create an account or sign in to Game Center from within a game, try to create an account or sign in using the Game Center app.
3. Sign out of Game Center, then sign back in. If you can't sign in to Game Center with an Apple ID, reset your password or use another email address. To manage your Apple ID, go to My Apple ID.
4. If you're using a Wi-Fi connection, make sure that your Wi-Fi router is configured for Game Center.
5. If you can't sign in on a Mac, continue with the steps to resolve sign in issues.

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from > Using Game Center on your Mac or iOS device - Apple Support

Some have had success in going to the settings, game center and signing in that way.

Whatever is going on is on Apple's end and there's no ETA on a fix.

Maybe, maybe not. My iOS7 iPad is having issues with game center continually logging me out, so I don't think the issue is iOS related, rather something on Apple's end. So it may be something that resolves itself.

It does seem to be an issue that cropped up with the release of iOS9, but since it's not just iOS9 devices that are having issues (in fact my iPad is having issues, my iPod which is on iOS9 not so much) one could guess that the issue is not just tied with iOS9.

and yeah, patience is what we all need to do.

For me the work around of signing in via the settings is working. for others, they just have to not play game center enabled games.

It does seem to be an issue that cropped up with the release of iOS9, but since it's not just iOS9 devices that are having issues (in fact my iPad is having issues, my iPod which is on iOS9 not so much) one could guess that the issue is not just tied with iOS9.

and yeah, patience is what we all need to do.

For me the work around of signing in via the settings is working. for others, they just have to not play game center enabled games.