Getting "Missing Security Answer" when changing Game Centre nickname
Apple's troubleshooting article has details on how to proceed with troubleshooting Game Center.


http://support.apple.com/en-us/TS3970


Why not start there and ask a second, more detailed question if one of the steps has you confused or stuck. Without more details edited in to your question, it's hard to guess where you are stuck in the process.

It might be as simple as you following the steps above and then entering the correct "answer" to your changed security question from https://appleid.apple.com