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In the game Machinarium, there is the following puzzle where the goal is to get all of the green points on the green area by rotating them along any of the 3 circles engraved on the background plate. ( Here is a video showing it in motion )

enter image description here

Is there some theory behind this kind of puzzle?

Edit: And if there is, is there some way to leverage that theory to get a more efficient solution than randomly clicking until a solution is reached?

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