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Welcome to the all-new iMore Forums! - We're still moving some things around, so you may see a few quirks here and there, but we're working on getting things fully completed as soon as possible. For now, take a look around, and if you run into any major issues, please let us know in this thread!
Like as Nikki said, the standalone app was removed in iOS 10. It still exists as a service though, and compatible games should still allow you to log into your Game Center account within the game itself.