There's other answers here that suggest it's enough to merely pass though a correct state, without having executed all of the operations. If that's valid, then

There's other answers here that suggest it's enough to merely pass though a correct state, without having executed all of the operations. If that's valid, then the wording of the puzzle should state this more forcefully. Even then, the last sentence can't be satisfied. Imagine if you were competing against the machine, it could simply always select moves that left one corner in the down position. From my reading of this puzzle, you can't sample the board in order to capture the orientation of the caps. Hence, I don't believe there's a way you could reach an outcome. Also, the base case doesn't work. Imagine there was one cap. You can choose to flip it or not flip it. Can you get everything in the up position without sampling the board? "Can you devise a sequence that ensures they all face up? Down?" No. Garbage in, garbage out. The only thing I can think of that might be useful at the moment: repeatedly hammering some combination of sides and diagonals, causing some statistical pattern to emerge. But without knowing the origin states, not sure how to use that. There's a similar problem here: http://gurmeet.net/puzzles/tumblers-on-a-rotating-table/ http://gurmeet.net/puzzles/tumblers-on-a-rotating-table/