# How many disconnected graphs of the Rubik's cube exist?

Let us say that a Rubik's cube in a particular configuration is in a particular "state". All other configurations of this cube (other "states"), which can be achieved by rotations of the cube can be thought to be connected to each other... rotations are like walking the edges of the graph where the different states are the vertices.

Now, if our Rubik's cube is of the type where we can peel off the stickers easily and put them back too, then we can swap 2 of the stickers in a corner.

I understand that the cube cannot be solved any more. Our cube has moved to a new state which is disconnected from the previous graph built above. Also, all the states reachable from this new state are also disconnected from the other graph. (otherwise the cube would be solvable). So now we have 2 graphs which are disconnected from each other. A third set of states might be disconnected from both the above states.

For all possible reasonable color assignments ( by reasonable, I mean to preserve the number of tiles for each color=9), how many such disconnected graphs exist for a 3x3x3 Rubik's cube?

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For the related problem where you can move around the subcubes arbitrarily but can't peel off stickers, the answer is 12. The answer is surely much larger for your question, because you're allowing me to combine the stickers in arbitrary ways. – Michael Lugo Oct 1 '10 at 15:48

## 1 Answer

You can calculate this using Burnside's Lemma, http://en.wikipedia.org/wiki/Burnside%27s_lemma, once you know the group of transformations for Rubik's cube.

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