How can I play chess over Game Center in Mountain Lion?
First of all, make sure you have added your systems knows about your Apple ID (go to System Preferences -> Users &amp; Groups -> Your User -> Apple ID).

Then launch Chess; obviously you'll need to have an internet connection.

After a short wait (a second or two in my case, might depend on your internet connection), a badge like the following should appear in the Chess window:



After it appears, you should simply be able to go to Game -> New... and select GameCenter Match in Players.