I have found a way to fix the iOS 5/6 Game Center!

DISCLAIMER: I am not responsible for any damage caused to your device. This will work on a CoolBooter OS.

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I have posted this on r/LegacyJailbreak on Reddit, but I'm posting it here so more people can see my fix and follow it themselves. 

If you are using iOS 5, steps 2, 3, 5, 6 and 7 do not apply to your device. Otherwise, if you are on iOS 6, follow every step.

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1. Add the source http://repo.thireus.com and install the tweak "iOS SSL Kill Switch". Enable the tweak in Settings after installing it.

2. Go to var/mobile/Library/Preferences/ and back up the file "com.apple.purplebuddy", doesn't matter where you back it up to.

3. Once you have backed it up, delete it from var/mobile/Library/Preferences.

4. Reboot iOS. (Boot into iOS 6 via CoolBooter if using CoolBooter, you will lose your custom boot logo if you had one)

5. You should be loaded into iOS 6 Setup. Skip signing in. (icloud, imessage, and other apple ID features will still work)

6. Choose if you wanna send diagnostics or not.

7. You will be done with iOS Setup, and everything will be normal, your apps will still be there, and you'll still be jailbroken if you were jailbroken.

8. Open Game Center to reveal the moment of truth.

Note: This also works on iOS 5, and you don't need to do the purplebuddy reset thing on iOS 5, just the SSL kill. On iOS 4, Game Center naturally works.