iPad won't update to iOS 10

My iPad will not recognize iOS 10. I have updated both my iPhones (6's) successfully and easily. However, when I click "Check for Updates" on my iPad, I get a message that version 9.3.5 is the "current version." It does not produce an error message. It is an iPad 3. I've never had problems updating before. I recently did a restore from backup because it refused to play video on one of my apps. The restore fixed that problem, but did not affect my inability to update the iOS. I have tried installing wirelessly and wired to iTunes on two different computers. It still says 9.3.5 is the latest version. I have shut down and restarted the iPad several times. No luck. I've tried on several different networks. No luck.