Thanks for using Apple Support Communities.
When you open Game Center, you can sign in to your Game Center account using your Apple ID.
In rare situations, when you try to sign in to Game Center with your Apple ID you may not be able to.
For more information on this, take a look at this article:
FaceTime, Game Center, Messages: Troubleshooting sign in issues
Using Game Center
- Verify that you are connected to the Internet.
- If you are unable to create or sign in to your Game Center account from within a game, try creating or signing in to your account using the Game Center app.
- Try signing out of your Game Center account, and then sign back in.
- When using a Wi-Fi connection, verify that your Wi-Fi router is configured for Game Center.
- When using a Cellular Data connection on iOS, incoming calls will interrupt your game session. If your Cellular Data network does not support simultaneous voice and data, answering the call may end the current game session.
Best of luck,