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 )
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?
Best Answer
This puzzle is an example of a group action. In this case, our set $X$ is the collection of all possible arrangements of the green and red dots, and the group $G$ that is acting on $X$ is the subgroup of $S_X$ (the group of all permutations of $X$) generated by the three elements corresponding to a turn of the first circle, second circle, and third circle, respectively. The question is then, if $x\in X$ is the current arrangement of the dots, and $y\in X$ is the desired arrangement of the dots, how to find a $g\in G$ such that $gx=y$ (presumably, the puzzle would not be set up in a way where there is no such $g$, i.e., the desired arrangement can actually be reached from the initial arrangement.)