This article explains how to log into your Game Center account using a Mac. When you sign up for an iCloud account, a Game Center profile is automatically created as well. You can only log into one account per device.
Steps
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Step 1. Click on the Apple menu
The icon looks like an apple and is located in the upper left corner of the screen. A drop-down menu will open.
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Step 2. Click on System Preferences
It is the second option and is found under "About This Mac".
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Step 3. Click on Internet Accounts
The icon is represented by a blue circle with a white "@" inside.
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Step 4. Click on +
Click on the "+" button at the bottom left of the account window.
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Step 5. Scroll down and click Add another account
It's in the box on the right, at the bottom of the list of accounts you can add.
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Step 6. Click on Game Center Account
It is located next to the icon that depicts circles of different colors and sizes.
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Step 7. Enter your Apple ID and password
The Apple ID is usually created the first time you set up a Mac or iPhone.
If you don't have an Apple ID, click on "iCloud" in the "Internet Accounts" section. Then, click on "Create Apple ID"
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Step 8. Click Login
This will log you into your Game Center account.