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Try different sequences of buttons on a blank slate and see what combinations of lights you get. Can you find a sequence which gives you the inverse of another combination you already know how to get to?
For example, let's say you know which buttons to push to light the corners and only the corners. As long as you remember this set of buttons (again, order doesn't matter) you'll know what to push if you want all four corner lights to toggle on/off.
Now, let's say later on, you find a set of buttons which gives you everything *but* the corner lights. What does this tell you? And what can you do with that information?