I purchased a license on one of my devices, but it is not appearing on my other device.

If your purchase was recent (less than a day ago), then allow some time for Google Play and Ultimate To-Do List to synchronize purchase information.  Ultimate To-Do List checks with Google Play once per day to see if a license has been purchased.  However, if the free trial has expired, you can tap on the "Check Again For a License" button to check immediately.

If the license has not been detected after a day, try rebooting your device.  This will likely force Google Play to synchronize its license data.

Also, check to make sure that both of your devices are linked to the same Google account.  You can check this in the Accounts area of your device's settings.  Also, the sync settings for each account should include synchronization of app data.  If you have multiple Google accounts, see this article for additional troubleshooting tips.

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