Reason for this that you have not activated your facebook account on your device.
To activate the facebook account on your device follow these steps:
1. Go to settings
2. Scroll to middle to and choose facebook
3. Fill form for email and password and press sign in.
Now you will be able to like game on game center.
Operating system i used at the moment: iOS 6.0.1
Other searches for the post:
- game center facebook like unavailable
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