# Maximum Cyclic Order for 3x3x3 - nxnxn supercubes and the Megaminx



## Christopher Mowla (Oct 23, 2019)

I searched this subforum for threads related to the topic on maximum cyclic order for the 3x3x3. One of the best threads that was started was Possible orders of Rubik's Cube positions. Of course, to calculate *maximum* order, we need to be able to calculate all possible orders (which is more precisely what Ravi did in that particular thread, as probably anyone reading this thread already has heard from somewhere that it takes no more than 1260 times to repeat any algorithm to return the cube to the state it was in before you first executed it) and then select the highest result.

For those who have not visited twistypuzzles.com recently, I started a thread strictly on addressing the *maximum* order -- without listing all possible (smaller) orders -- for the megaminx, 3x3x3, and 4x4x4 through nxnxn supercubes.

Maximum Cyclic Order for 3x3x3, Megaminx, & 4x4x4 Supercube

Because my calculations for the 5x5x5 through nxnxn supercubes (which I ended up covering as well) rely on the logic of calculating the order of *any given algorithm* (a related thread to this in this subforum is Repeat Same Moves Many Times), I would also refer interested readers to my write-up on math.stackexchange.

*Further Readings:*
I mention the term "parity state" with regards to the nxnxn supercube. See the "Analysis of the 4x4x4 Rubik's cube and a little bit about C(n,w,c) function" section of bcube's article, the thread Theory about big cube parity, and Chris Hardwick's thread, _The interrelation of piece parities on the n x n x n supercube_, to understand (in full) why I:

Broke up the 3x3x3 maximum order calculations (when an algorithm is *not allowed* to contain M, E,S,x,y,z moves) into *two* parts,
Broke up the 3x3x3 maximum order calculations (when an algorithm *IS allowed* to contain M, E,S,x,y,z moves) into *four* parts,
Broke up the 4x4x4 supercube maximum order into *four* parts (and why I wrote "Corners ODD , X-Centers ODD: Wings EVEN", etc.), and
Only did *one* calculation for the megaminx. (_Hint:_ See my post on how to calculate the number of positions of the _n_ x _n_ x _n_ minx/minx^n.)
in my calculations in the thread, Maximum Cyclic Order for 3x3x3, Megaminx, & 4x4x4 Supercube.


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