As a physics student, I found that I could solve most of the problems simply by looking at derivations and listening carefully to my reactions to the equations. A soft voice inside me would say, “No, that term just doesn’t seem right. Go and find out what went wrong there.” Or, “Ah, these terms hang together and the result feels right. It must be okay.” And it almost always worked out. My piano teacher would do the same when playing an unfamiliar piece of music. She could play it just by making sure it sounded right.
Were these just party tricks? Or was a more fundamental process going on?