# What Percentage of Configurations of the Cube Are Acceptable Scrambles?



## Christopher Mowla (Jun 17, 2012)

I did some research on cycle classes, and I have come up with a simple formula which can be used as a tool to aid us in determining the percentage of possible configurations which are/are not acceptable scrambles.

Before I explain what I have done so far in its entirety, I'll give an example.

Not that I have heard this to be official by any means, but let's pretend that the following restriction was made standard.

Suppose that a 3x3x3 scramble is acceptable by a certain cuber, if at most 2 corners and 3 edges may be in their correct locations (but all possible orientations are allowed). Then the total percentage of configurations of the 3x3x3 (regular/6 color) cube which can be an acceptable scramble is 90.23%.

On the other hand, if we still didn't care about orientations, if no pieces were allowed to be in their correct locations (whether twisted/flipped or not, since there is no restriction on the orientations of the pieces in this example), then only 13.5336% of the possible 43252003274489856000 configurations would qualify as authentic scrambles.

*Motivation*
This thread is not meant to be a discussion of "authentic scrambles" (it's not about that at all. It's about looking at the cycle classes at an intense level. I definitely am new to this material, and so I hope members who know something pertaining to this subject would chime in sometime ), but also to help those new to the concept of what the numbers inside of the 43252003274489856000 really mean (including myself).

Knowing about this can also add to common conversation with non-cubers and cubers alike about how many configurations of the Rubik's Cube there are (this thread isn't about teaching that because there's an older thread all about that already). Specifically, we can say, "yeah there might technically be a zillion different possible configurations, but of those those configurations, only X% of those configurations are a really messed up cube."

For example, for big cubes, say a 50x50x50, not only can we do patterns on the centers (just like we can with every cube size), but we can literally draw a picture of whatever we want on it (of course, we do this with commutator moves, not that we literally draw on the cube stickers, LOL) which can fit in a 48x48 "pixel" space, because we can move the oblique and X-centers in any fashion we wish. Any picture that you can think of which can fit in that 48x48 square IS one of the possible scrambles for that cube center. That means your favorite movie star, your school mascot, or whatever you can think of. (If you have never thought of it that way, take a moment and ponder on it! )

*The Formula*
Here's My formula: (which I created by merging a few existing ideas in mathematical communities in order to make the formula as simple as I could--and to avoid using an algorithm or program)

\( \frac{n!\text{Round}\left[ k\left( k-2 \right)!e^{-1} \right]}{k\left( k-2 \right)!\left( n-k \right)!} \)[Link]
, where "Round[]" means to round the number in the brackets to the nearest integer, and e is the exponential approximately equal to 2.72.

*Cycle Classes*

There are 21 cycle classes for corners, 76 cycle classes for (middle/central) edges, and an amazing number of 1574 cycle classes for each orbit of wing edges and non-fixed center pieces on big cubes.

All possible cycle classes can be found by just by writing down all possible ways to add integers (excluding the number 1) to get other integers.

Here's the complete list of 1574 cycle classes for 24 objects (you can just look up through "8 Pieces" to and visualize them being for the corners, as well as look up through "12 Pieces" to visualize them being middle (central) edges.)



Spoiler: Permutation Cycle Classes



2 Pieces (1,1)
{{2}}

3 Pieces (1,2)
{{3}}

4 Pieces (2,4)
{{4},{2,2}}

5 Pieces (2,6)
{{5},{3,2}}

6 Pieces (4,10)
{{6},{4,2},{3,3},{2,2,2}}

7 Pieces (4,14)
{{7},{5,2},{4,3},{3,2,2}}

8 Pieces (7,21)
{{8},{6,2},{5,3},{4,4},{4,2,2},{3,3,2},{2,2,2,2}}

9 Pieces (8,29)
{{9},{7,2},{6,3},{5,4},{5,2,2},{4,3,2},{3,3,3},{3,2,2,2}}

10 Pieces (12,41)
{{10},{8,2},{7,3},{6,4},{6,2,2},{5,5},{5,3,2},{4,4,2},{4,3,3},{4,2,2,2},{3,3,2,2},{2,2,2,2,2}}

11 Pieces (14,55)
{{11},{9,2},{8,3},{7,4},{7,2,2},{6,5},{6,3,2},{5,4,2},{5,3,3},{5,2,2,2},{4,4,3},{4,3,2,2},{3,3,3,2},{3,2,2,2,2}}

12 Pieces (21,76)
{{12},{10,2},{9,3},{8,4},{8,2,2},{7,5},{7,3,2},{6,6},{6,4,2},{6,3,3},{6,2,2,2},{5,5,2},{5,4,3},{5,3,2,2},{4,4,4},{4,4,2,2},{4,3,3,2},{4,2,2,2,2},{3,3,3,3},{3,3,2,2,2},{2,2,2,2,2,2}}

13 Pieces (24,100)
{{13},{11,2},{10,3},{9,4},{9,2,2},{8,5},{8,3,2},{7,6},{7,4,2},{7,3,3},{7,2,2,2},{6,5,2},{6,4,3},{6,3,2,2},{5,5,3},{5,4,4},{5,4,2,2},{5,3,3,2},{5,2,2,2,2},{4,4,3,2},{4,3,3,3},{4,3,2,2,2},{3,3,3,2,2},{3,2,2,2,2,2}}

14 Pieces (34,134)
{{14},{12,2},{11,3},{10,4},{10,2,2},{9,5},{9,3,2},{8,6},{8,4,2},{8,3,3},{8,2,2,2},{7,7},{7,5,2},{7,4,3},{7,3,2,2},{6,6,2},{6,5,3},{6,4,4},{6,4,2,2},{6,3,3,2},{6,2,2,2,2},{5,5,4},{5,5,2,2},{5,4,3,2},{5,3,3,3},{5,3,2,2,2},{4,4,4,2},{4,4,3,3},{4,4,2,2,2},{4,3,3,2,2},{4,2,2,2,2,2},{3,3,3,3,2},{3,3,2,2,2,2},{2,2,2,2,2,2,2}}

15 Pieces (41,175)
{{15},{13,2},{12,3},{11,4},{11,2,2},{10,5},{10,3,2},{9,6},{9,4,2},{9,3,3},{9,2,2,2},{8,7},{8,5,2},{8,4,3},{8,3,2,2},{7,6,2},{7,5,3},{7,4,4},{7,4,2,2},{7,3,3,2},{7,2,2,2,2},{6,6,3},{6,5,4},{6,5,2,2},{6,4,3,2},{6,3,3,3},{6,3,2,2,2},{5,5,5},{5,5,3,2},{5,4,4,2},{5,4,3,3},{5,4,2,2,2},{5,3,3,2,2},{5,2,2,2,2,2},{4,4,4,3},{4,4,3,2,2},{4,3,3,3,2},{4,3,2,2,2,2},{3,3,3,3,3},{3,3,3,2,2,2},{3,2,2,2,2,2,2}}

16 Pieces (55,230)
{{16},{14,2},{13,3},{12,4},{12,2,2},{11,5},{11,3,2},{10,6},{10,4,2},{10,3,3},{10,2,2,2},{9,7},{9,5,2},{9,4,3},{9,3,2,2},{8,8},{8,6,2},{8,5,3},{8,4,4},{8,4,2,2},{8,3,3,2},{8,2,2,2,2},{7,7,2},{7,6,3},{7,5,4},{7,5,2,2},{7,4,3,2},{7,3,3,3},{7,3,2,2,2},{6,6,4},{6,6,2,2},{6,5,5},{6,5,3,2},{6,4,4,2},{6,4,3,3},{6,4,2,2,2},{6,3,3,2,2},{6,2,2,2,2,2},{5,5,4,2},{5,5,3,3},{5,5,2,2,2},{5,4,4,3},{5,4,3,2,2},{5,3,3,3,2},{5,3,2,2,2,2},{4,4,4,4},{4,4,4,2,2},{4,4,3,3,2},{4,4,2,2,2,2},{4,3,3,3,3},{4,3,3,2,2,2},{4,2,2,2,2,2,2},{3,3,3,3,2,2},{3,3,2,2,2,2,2},{2,2,2,2,2,2,2,2}}

17 Pieces (66,296)
{{17},{15,2},{14,3},{13,4},{13,2,2},{12,5},{12,3,2},{11,6},{11,4,2},{11,3,3},{11,2,2,2},{10,7},{10,5,2},{10,4,3},{10,3,2,2},{9,8},{9,6,2},{9,5,3},{9,4,4},{9,4,2,2},{9,3,3,2},{9,2,2,2,2},{8,7,2},{8,6,3},{8,5,4},{8,5,2,2},{8,4,3,2},{8,3,3,3},{8,3,2,2,2},{7,7,3},{7,6,4},{7,6,2,2},{7,5,5},{7,5,3,2},{7,4,4,2},{7,4,3,3},{7,4,2,2,2},{7,3,3,2,2},{7,2,2,2,2,2},{6,6,5},{6,6,3,2},{6,5,4,2},{6,5,3,3},{6,5,2,2,2},{6,4,4,3},{6,4,3,2,2},{6,3,3,3,2},{6,3,2,2,2,2},{5,5,5,2},{5,5,4,3},{5,5,3,2,2},{5,4,4,4},{5,4,4,2,2},{5,4,3,3,2},{5,4,2,2,2,2},{5,3,3,3,3},{5,3,3,2,2,2},{5,2,2,2,2,2,2},{4,4,4,3,2},{4,4,3,3,3},{4,4,3,2,2,2},{4,3,3,3,2,2},{4,3,2,2,2,2,2},{3,3,3,3,3,2},{3,3,3,2,2,2,2},{3,2,2,2,2,2,2,2}}

18 Pieces (88,384)
{{18},{16,2},{15,3},{14,4},{14,2,2},{13,5},{13,3,2},{12,6},{12,4,2},{12,3,3},{12,2,2,2},{11,7},{11,5,2},{11,4,3},{11,3,2,2},{10,8},{10,6,2},{10,5,3},{10,4,4},{10,4,2,2},{10,3,3,2},{10,2,2,2,2},{9,9},{9,7,2},{9,6,3},{9,5,4},{9,5,2,2},{9,4,3,2},{9,3,3,3},{9,3,2,2,2},{8,8,2},{8,7,3},{8,6,4},{8,6,2,2},{8,5,5},{8,5,3,2},{8,4,4,2},{8,4,3,3},{8,4,2,2,2},{8,3,3,2,2},{8,2,2,2,2,2},{7,7,4},{7,7,2,2},{7,6,5},{7,6,3,2},{7,5,4,2},{7,5,3,3},{7,5,2,2,2},{7,4,4,3},{7,4,3,2,2},{7,3,3,3,2},{7,3,2,2,2,2},{6,6,6},{6,6,4,2},{6,6,3,3},{6,6,2,2,2},{6,5,5,2},{6,5,4,3},{6,5,3,2,2},{6,4,4,4},{6,4,4,2,2},{6,4,3,3,2},{6,4,2,2,2,2},{6,3,3,3,3},{6,3,3,2,2,2},{6,2,2,2,2,2,2},{5,5,5,3},{5,5,4,4},{5,5,4,2,2},{5,5,3,3,2},{5,5,2,2,2,2},{5,4,4,3,2},{5,4,3,3,3},{5,4,3,2,2,2},{5,3,3,3,2,2},{5,3,2,2,2,2,2},{4,4,4,4,2},{4,4,4,3,3},{4,4,4,2,2,2},{4,4,3,3,2,2},{4,4,2,2,2,2,2},{4,3,3,3,3,2},{4,3,3,2,2,2,2},{4,2,2,2,2,2,2,2},{3,3,3,3,3,3},{3,3,3,3,2,2,2},{3,3,2,2,2,2,2,2},{2,2,2,2,2,2,2,2,2}}

19 Pieces (105,489)
{{19},{17,2},{16,3},{15,4},{15,2,2},{14,5},{14,3,2},{13,6},{13,4,2},{13,3,3},{13,2,2,2},{12,7},{12,5,2},{12,4,3},{12,3,2,2},{11,8},{11,6,2},{11,5,3},{11,4,4},{11,4,2,2},{11,3,3,2},{11,2,2,2,2},{10,9},{10,7,2},{10,6,3},{10,5,4},{10,5,2,2},{10,4,3,2},{10,3,3,3},{10,3,2,2,2},{9,8,2},{9,7,3},{9,6,4},{9,6,2,2},{9,5,5},{9,5,3,2},{9,4,4,2},{9,4,3,3},{9,4,2,2,2},{9,3,3,2,2},{9,2,2,2,2,2},{8,8,3},{8,7,4},{8,7,2,2},{8,6,5},{8,6,3,2},{8,5,4,2},{8,5,3,3},{8,5,2,2,2},{8,4,4,3},{8,4,3,2,2},{8,3,3,3,2},{8,3,2,2,2,2},{7,7,5},{7,7,3,2},{7,6,6},{7,6,4,2},{7,6,3,3},{7,6,2,2,2},{7,5,5,2},{7,5,4,3},{7,5,3,2,2},{7,4,4,4},{7,4,4,2,2},{7,4,3,3,2},{7,4,2,2,2,2},{7,3,3,3,3},{7,3,3,2,2,2},{7,2,2,2,2,2,2},{6,6,5,2},{6,6,4,3},{6,6,3,2,2},{6,5,5,3},{6,5,4,4},{6,5,4,2,2},{6,5,3,3,2},{6,5,2,2,2,2},{6,4,4,3,2},{6,4,3,3,3},{6,4,3,2,2,2},{6,3,3,3,2,2},{6,3,2,2,2,2,2},{5,5,5,4},{5,5,5,2,2},{5,5,4,3,2},{5,5,3,3,3},{5,5,3,2,2,2},{5,4,4,4,2},{5,4,4,3,3},{5,4,4,2,2,2},{5,4,3,3,2,2},{5,4,2,2,2,2,2},{5,3,3,3,3,2},{5,3,3,2,2,2,2},{5,2,2,2,2,2,2,2},{4,4,4,4,3},{4,4,4,3,2,2},{4,4,3,3,3,2},{4,4,3,2,2,2,2},{4,3,3,3,3,3},{4,3,3,3,2,2,2},{4,3,2,2,2,2,2,2},{3,3,3,3,3,2,2},{3,3,3,2,2,2,2,2},{3,2,2,2,2,2,2,2,2}}

20 Pieces (137,626)
{{20},{18,2},{17,3},{16,4},{16,2,2},{15,5},{15,3,2},{14,6},{14,4,2},{14,3,3},{14,2,2,2},{13,7},{13,5,2},{13,4,3},{13,3,2,2},{12,8},{12,6,2},{12,5,3},{12,4,4},{12,4,2,2},{12,3,3,2},{12,2,2,2,2},{11,9},{11,7,2},{11,6,3},{11,5,4},{11,5,2,2},{11,4,3,2},{11,3,3,3},{11,3,2,2,2},{10,10},{10,8,2},{10,7,3},{10,6,4},{10,6,2,2},{10,5,5},{10,5,3,2},{10,4,4,2},{10,4,3,3},{10,4,2,2,2},{10,3,3,2,2},{10,2,2,2,2,2},{9,9,2},{9,8,3},{9,7,4},{9,7,2,2},{9,6,5},{9,6,3,2},{9,5,4,2},{9,5,3,3},{9,5,2,2,2},{9,4,4,3},{9,4,3,2,2},{9,3,3,3,2},{9,3,2,2,2,2},{8,8,4},{8,8,2,2},{8,7,5},{8,7,3,2},{8,6,6},{8,6,4,2},{8,6,3,3},{8,6,2,2,2},{8,5,5,2},{8,5,4,3},{8,5,3,2,2},{8,4,4,4},{8,4,4,2,2},{8,4,3,3,2},{8,4,2,2,2,2},{8,3,3,3,3},{8,3,3,2,2,2},{8,2,2,2,2,2,2},{7,7,6},{7,7,4,2},{7,7,3,3},{7,7,2,2,2},{7,6,5,2},{7,6,4,3},{7,6,3,2,2},{7,5,5,3},{7,5,4,4},{7,5,4,2,2},{7,5,3,3,2},{7,5,2,2,2,2},{7,4,4,3,2},{7,4,3,3,3},{7,4,3,2,2,2},{7,3,3,3,2,2},{7,3,2,2,2,2,2},{6,6,6,2},{6,6,5,3},{6,6,4,4},{6,6,4,2,2},{6,6,3,3,2},{6,6,2,2,2,2},{6,5,5,4},{6,5,5,2,2},{6,5,4,3,2},{6,5,3,3,3},{6,5,3,2,2,2},{6,4,4,4,2},{6,4,4,3,3},{6,4,4,2,2,2},{6,4,3,3,2,2},{6,4,2,2,2,2,2},{6,3,3,3,3,2},{6,3,3,2,2,2,2},{6,2,2,2,2,2,2,2},{5,5,5,5},{5,5,5,3,2},{5,5,4,4,2},{5,5,4,3,3},{5,5,4,2,2,2},{5,5,3,3,2,2},{5,5,2,2,2,2,2},{5,4,4,4,3},{5,4,4,3,2,2},{5,4,3,3,3,2},{5,4,3,2,2,2,2},{5,3,3,3,3,3},{5,3,3,3,2,2,2},{5,3,2,2,2,2,2,2},{4,4,4,4,4},{4,4,4,4,2,2},{4,4,4,3,3,2},{4,4,4,2,2,2,2},{4,4,3,3,3,3},{4,4,3,3,2,2,2},{4,4,2,2,2,2,2,2},{4,3,3,3,3,2,2},{4,3,3,2,2,2,2,2},{4,2,2,2,2,2,2,2,2},{3,3,3,3,3,3,2},{3,3,3,3,2,2,2,2},{3,3,2,2,2,2,2,2,2},{2,2,2,2,2,2,2,2,2,2}}

21 Pieces (165,791)
{{21},{19,2},{18,3},{17,4},{17,2,2},{16,5},{16,3,2},{15,6},{15,4,2},{15,3,3},{15,2,2,2},{14,7},{14,5,2},{14,4,3},{14,3,2,2},{13,8},{13,6,2},{13,5,3},{13,4,4},{13,4,2,2},{13,3,3,2},{13,2,2,2,2},{12,9},{12,7,2},{12,6,3},{12,5,4},{12,5,2,2},{12,4,3,2},{12,3,3,3},{12,3,2,2,2},{11,10},{11,8,2},{11,7,3},{11,6,4},{11,6,2,2},{11,5,5},{11,5,3,2},{11,4,4,2},{11,4,3,3},{11,4,2,2,2},{11,3,3,2,2},{11,2,2,2,2,2},{10,9,2},{10,8,3},{10,7,4},{10,7,2,2},{10,6,5},{10,6,3,2},{10,5,4,2},{10,5,3,3},{10,5,2,2,2},{10,4,4,3},{10,4,3,2,2},{10,3,3,3,2},{10,3,2,2,2,2},{9,9,3},{9,8,4},{9,8,2,2},{9,7,5},{9,7,3,2},{9,6,6},{9,6,4,2},{9,6,3,3},{9,6,2,2,2},{9,5,5,2},{9,5,4,3},{9,5,3,2,2},{9,4,4,4},{9,4,4,2,2},{9,4,3,3,2},{9,4,2,2,2,2},{9,3,3,3,3},{9,3,3,2,2,2},{9,2,2,2,2,2,2},{8,8,5},{8,8,3,2},{8,7,6},{8,7,4,2},{8,7,3,3},{8,7,2,2,2},{8,6,5,2},{8,6,4,3},{8,6,3,2,2},{8,5,5,3},{8,5,4,4},{8,5,4,2,2},{8,5,3,3,2},{8,5,2,2,2,2},{8,4,4,3,2},{8,4,3,3,3},{8,4,3,2,2,2},{8,3,3,3,2,2},{8,3,2,2,2,2,2},{7,7,7},{7,7,5,2},{7,7,4,3},{7,7,3,2,2},{7,6,6,2},{7,6,5,3},{7,6,4,4},{7,6,4,2,2},{7,6,3,3,2},{7,6,2,2,2,2},{7,5,5,4},{7,5,5,2,2},{7,5,4,3,2},{7,5,3,3,3},{7,5,3,2,2,2},{7,4,4,4,2},{7,4,4,3,3},{7,4,4,2,2,2},{7,4,3,3,2,2},{7,4,2,2,2,2,2},{7,3,3,3,3,2},{7,3,3,2,2,2,2},{7,2,2,2,2,2,2,2},{6,6,6,3},{6,6,5,4},{6,6,5,2,2},{6,6,4,3,2},{6,6,3,3,3},{6,6,3,2,2,2},{6,5,5,5},{6,5,5,3,2},{6,5,4,4,2},{6,5,4,3,3},{6,5,4,2,2,2},{6,5,3,3,2,2},{6,5,2,2,2,2,2},{6,4,4,4,3},{6,4,4,3,2,2},{6,4,3,3,3,2},{6,4,3,2,2,2,2},{6,3,3,3,3,3},{6,3,3,3,2,2,2},{6,3,2,2,2,2,2,2},{5,5,5,4,2},{5,5,5,3,3},{5,5,5,2,2,2},{5,5,4,4,3},{5,5,4,3,2,2},{5,5,3,3,3,2},{5,5,3,2,2,2,2},{5,4,4,4,4},{5,4,4,4,2,2},{5,4,4,3,3,2},{5,4,4,2,2,2,2},{5,4,3,3,3,3},{5,4,3,3,2,2,2},{5,4,2,2,2,2,2,2},{5,3,3,3,3,2,2},{5,3,3,2,2,2,2,2},{5,2,2,2,2,2,2,2,2},{4,4,4,4,3,2},{4,4,4,3,3,3},{4,4,4,3,2,2,2},{4,4,3,3,3,2,2},{4,4,3,2,2,2,2,2},{4,3,3,3,3,3,2},{4,3,3,3,2,2,2,2},{4,3,2,2,2,2,2,2,2},{3,3,3,3,3,3,3},{3,3,3,3,3,2,2,2},{3,3,3,2,2,2,2,2,2},{3,2,2,2,2,2,2,2,2,2}}

22 Pieces (210,1001)
{{22},{20,2},{19,3},{18,4},{18,2,2},{17,5},{17,3,2},{16,6},{16,4,2},{16,3,3},{16,2,2,2},{15,7},{15,5,2},{15,4,3},{15,3,2,2},{14,8},{14,6,2},{14,5,3},{14,4,4},{14,4,2,2},{14,3,3,2},{14,2,2,2,2},{13,9},{13,7,2},{13,6,3},{13,5,4},{13,5,2,2},{13,4,3,2},{13,3,3,3},{13,3,2,2,2},{12,10},{12,8,2},{12,7,3},{12,6,4},{12,6,2,2},{12,5,5},{12,5,3,2},{12,4,4,2},{12,4,3,3},{12,4,2,2,2},{12,3,3,2,2},{12,2,2,2,2,2},{11,11},{11,9,2},{11,8,3},{11,7,4},{11,7,2,2},{11,6,5},{11,6,3,2},{11,5,4,2},{11,5,3,3},{11,5,2,2,2},{11,4,4,3},{11,4,3,2,2},{11,3,3,3,2},{11,3,2,2,2,2},{10,10,2},{10,9,3},{10,8,4},{10,8,2,2},{10,7,5},{10,7,3,2},{10,6,6},{10,6,4,2},{10,6,3,3},{10,6,2,2,2},{10,5,5,2},{10,5,4,3},{10,5,3,2,2},{10,4,4,4},{10,4,4,2,2},{10,4,3,3,2},{10,4,2,2,2,2},{10,3,3,3,3},{10,3,3,2,2,2},{10,2,2,2,2,2,2},{9,9,4},{9,9,2,2},{9,8,5},{9,8,3,2},{9,7,6},{9,7,4,2},{9,7,3,3},{9,7,2,2,2},{9,6,5,2},{9,6,4,3},{9,6,3,2,2},{9,5,5,3},{9,5,4,4},{9,5,4,2,2},{9,5,3,3,2},{9,5,2,2,2,2},{9,4,4,3,2},{9,4,3,3,3},{9,4,3,2,2,2},{9,3,3,3,2,2},{9,3,2,2,2,2,2},{8,8,6},{8,8,4,2},{8,8,3,3},{8,8,2,2,2},{8,7,7},{8,7,5,2},{8,7,4,3},{8,7,3,2,2},{8,6,6,2},{8,6,5,3},{8,6,4,4},{8,6,4,2,2},{8,6,3,3,2},{8,6,2,2,2,2},{8,5,5,4},{8,5,5,2,2},{8,5,4,3,2},{8,5,3,3,3},{8,5,3,2,2,2},{8,4,4,4,2},{8,4,4,3,3},{8,4,4,2,2,2},{8,4,3,3,2,2},{8,4,2,2,2,2,2},{8,3,3,3,3,2},{8,3,3,2,2,2,2},{8,2,2,2,2,2,2,2},{7,7,6,2},{7,7,5,3},{7,7,4,4},{7,7,4,2,2},{7,7,3,3,2},{7,7,2,2,2,2},{7,6,6,3},{7,6,5,4},{7,6,5,2,2},{7,6,4,3,2},{7,6,3,3,3},{7,6,3,2,2,2},{7,5,5,5},{7,5,5,3,2},{7,5,4,4,2},{7,5,4,3,3},{7,5,4,2,2,2},{7,5,3,3,2,2},{7,5,2,2,2,2,2},{7,4,4,4,3},{7,4,4,3,2,2},{7,4,3,3,3,2},{7,4,3,2,2,2,2},{7,3,3,3,3,3},{7,3,3,3,2,2,2},{7,3,2,2,2,2,2,2},{6,6,6,4},{6,6,6,2,2},{6,6,5,5},{6,6,5,3,2},{6,6,4,4,2},{6,6,4,3,3},{6,6,4,2,2,2},{6,6,3,3,2,2},{6,6,2,2,2,2,2},{6,5,5,4,2},{6,5,5,3,3},{6,5,5,2,2,2},{6,5,4,4,3},{6,5,4,3,2,2},{6,5,3,3,3,2},{6,5,3,2,2,2,2},{6,4,4,4,4},{6,4,4,4,2,2},{6,4,4,3,3,2},{6,4,4,2,2,2,2},{6,4,3,3,3,3},{6,4,3,3,2,2,2},{6,4,2,2,2,2,2,2},{6,3,3,3,3,2,2},{6,3,3,2,2,2,2,2},{6,2,2,2,2,2,2,2,2},{5,5,5,5,2},{5,5,5,4,3},{5,5,5,3,2,2},{5,5,4,4,4},{5,5,4,4,2,2},{5,5,4,3,3,2},{5,5,4,2,2,2,2},{5,5,3,3,3,3},{5,5,3,3,2,2,2},{5,5,2,2,2,2,2,2},{5,4,4,4,3,2},{5,4,4,3,3,3},{5,4,4,3,2,2,2},{5,4,3,3,3,2,2},{5,4,3,2,2,2,2,2},{5,3,3,3,3,3,2},{5,3,3,3,2,2,2,2},{5,3,2,2,2,2,2,2,2},{4,4,4,4,4,2},{4,4,4,4,3,3},{4,4,4,4,2,2,2},{4,4,4,3,3,2,2},{4,4,4,2,2,2,2,2},{4,4,3,3,3,3,2},{4,4,3,3,2,2,2,2},{4,4,2,2,2,2,2,2,2},{4,3,3,3,3,3,3},{4,3,3,3,3,2,2,2},{4,3,3,2,2,2,2,2,2},{4,2,2,2,2,2,2,2,2,2},{3,3,3,3,3,3,2,2},{3,3,3,3,2,2,2,2,2},{3,3,2,2,2,2,2,2,2,2},{2,2,2,2,2,2,2,2,2,2,2}}

23 Pieces (253,1254)
{{23},{21,2},{20,3},{19,4},{19,2,2},{18,5},{18,3,2},{17,6},{17,4,2},{17,3,3},{17,2,2,2},{16,7},{16,5,2},{16,4,3},{16,3,2,2},{15,8},{15,6,2},{15,5,3},{15,4,4},{15,4,2,2},{15,3,3,2},{15,2,2,2,2},{14,9},{14,7,2},{14,6,3},{14,5,4},{14,5,2,2},{14,4,3,2},{14,3,3,3},{14,3,2,2,2},{13,10},{13,8,2},{13,7,3},{13,6,4},{13,6,2,2},{13,5,5},{13,5,3,2},{13,4,4,2},{13,4,3,3},{13,4,2,2,2},{13,3,3,2,2},{13,2,2,2,2,2},{12,11},{12,9,2},{12,8,3},{12,7,4},{12,7,2,2},{12,6,5},{12,6,3,2},{12,5,4,2},{12,5,3,3},{12,5,2,2,2},{12,4,4,3},{12,4,3,2,2},{12,3,3,3,2},{12,3,2,2,2,2},{11,10,2},{11,9,3},{11,8,4},{11,8,2,2},{11,7,5},{11,7,3,2},{11,6,6},{11,6,4,2},{11,6,3,3},{11,6,2,2,2},{11,5,5,2},{11,5,4,3},{11,5,3,2,2},{11,4,4,4},{11,4,4,2,2},{11,4,3,3,2},{11,4,2,2,2,2},{11,3,3,3,3},{11,3,3,2,2,2},{11,2,2,2,2,2,2},{10,10,3},{10,9,4},{10,9,2,2},{10,8,5},{10,8,3,2},{10,7,6},{10,7,4,2},{10,7,3,3},{10,7,2,2,2},{10,6,5,2},{10,6,4,3},{10,6,3,2,2},{10,5,5,3},{10,5,4,4},{10,5,4,2,2},{10,5,3,3,2},{10,5,2,2,2,2},{10,4,4,3,2},{10,4,3,3,3},{10,4,3,2,2,2},{10,3,3,3,2,2},{10,3,2,2,2,2,2},{9,9,5},{9,9,3,2},{9,8,6},{9,8,4,2},{9,8,3,3},{9,8,2,2,2},{9,7,7},{9,7,5,2},{9,7,4,3},{9,7,3,2,2},{9,6,6,2},{9,6,5,3},{9,6,4,4},{9,6,4,2,2},{9,6,3,3,2},{9,6,2,2,2,2},{9,5,5,4},{9,5,5,2,2},{9,5,4,3,2},{9,5,3,3,3},{9,5,3,2,2,2},{9,4,4,4,2},{9,4,4,3,3},{9,4,4,2,2,2},{9,4,3,3,2,2},{9,4,2,2,2,2,2},{9,3,3,3,3,2},{9,3,3,2,2,2,2},{9,2,2,2,2,2,2,2},{8,8,7},{8,8,5,2},{8,8,4,3},{8,8,3,2,2},{8,7,6,2},{8,7,5,3},{8,7,4,4},{8,7,4,2,2},{8,7,3,3,2},{8,7,2,2,2,2},{8,6,6,3},{8,6,5,4},{8,6,5,2,2},{8,6,4,3,2},{8,6,3,3,3},{8,6,3,2,2,2},{8,5,5,5},{8,5,5,3,2},{8,5,4,4,2},{8,5,4,3,3},{8,5,4,2,2,2},{8,5,3,3,2,2},{8,5,2,2,2,2,2},{8,4,4,4,3},{8,4,4,3,2,2},{8,4,3,3,3,2},{8,4,3,2,2,2,2},{8,3,3,3,3,3},{8,3,3,3,2,2,2},{8,3,2,2,2,2,2,2},{7,7,7,2},{7,7,6,3},{7,7,5,4},{7,7,5,2,2},{7,7,4,3,2},{7,7,3,3,3},{7,7,3,2,2,2},{7,6,6,4},{7,6,6,2,2},{7,6,5,5},{7,6,5,3,2},{7,6,4,4,2},{7,6,4,3,3},{7,6,4,2,2,2},{7,6,3,3,2,2},{7,6,2,2,2,2,2},{7,5,5,4,2},{7,5,5,3,3},{7,5,5,2,2,2},{7,5,4,4,3},{7,5,4,3,2,2},{7,5,3,3,3,2},{7,5,3,2,2,2,2},{7,4,4,4,4},{7,4,4,4,2,2},{7,4,4,3,3,2},{7,4,4,2,2,2,2},{7,4,3,3,3,3},{7,4,3,3,2,2,2},{7,4,2,2,2,2,2,2},{7,3,3,3,3,2,2},{7,3,3,2,2,2,2,2},{7,2,2,2,2,2,2,2,2},{6,6,6,5},{6,6,6,3,2},{6,6,5,4,2},{6,6,5,3,3},{6,6,5,2,2,2},{6,6,4,4,3},{6,6,4,3,2,2},{6,6,3,3,3,2},{6,6,3,2,2,2,2},{6,5,5,5,2},{6,5,5,4,3},{6,5,5,3,2,2},{6,5,4,4,4},{6,5,4,4,2,2},{6,5,4,3,3,2},{6,5,4,2,2,2,2},{6,5,3,3,3,3},{6,5,3,3,2,2,2},{6,5,2,2,2,2,2,2},{6,4,4,4,3,2},{6,4,4,3,3,3},{6,4,4,3,2,2,2},{6,4,3,3,3,2,2},{6,4,3,2,2,2,2,2},{6,3,3,3,3,3,2},{6,3,3,3,2,2,2,2},{6,3,2,2,2,2,2,2,2},{5,5,5,5,3},{5,5,5,4,4},{5,5,5,4,2,2},{5,5,5,3,3,2},{5,5,5,2,2,2,2},{5,5,4,4,3,2},{5,5,4,3,3,3},{5,5,4,3,2,2,2},{5,5,3,3,3,2,2},{5,5,3,2,2,2,2,2},{5,4,4,4,4,2},{5,4,4,4,3,3},{5,4,4,4,2,2,2},{5,4,4,3,3,2,2},{5,4,4,2,2,2,2,2},{5,4,3,3,3,3,2},{5,4,3,3,2,2,2,2},{5,4,2,2,2,2,2,2,2},{5,3,3,3,3,3,3},{5,3,3,3,3,2,2,2},{5,3,3,2,2,2,2,2,2},{5,2,2,2,2,2,2,2,2,2},{4,4,4,4,4,3},{4,4,4,4,3,2,2},{4,4,4,3,3,3,2},{4,4,4,3,2,2,2,2},{4,4,3,3,3,3,3},{4,4,3,3,3,2,2,2},{4,4,3,2,2,2,2,2,2},{4,3,3,3,3,3,2,2},{4,3,3,3,2,2,2,2,2},{4,3,2,2,2,2,2,2,2,2},{3,3,3,3,3,3,3,2},{3,3,3,3,3,2,2,2,2},{3,3,3,2,2,2,2,2,2,2},{3,2,2,2,2,2,2,2,2,2,2}}

24 Pieces (320,1574)
{{24},{22,2},{21,3},{20,4},{20,2,2},{19,5},{19,3,2},{18,6},{18,4,2},{18,3,3},{18,2,2,2},{17,7},{17,5,2},{17,4,3},{17,3,2,2},{16,8},{16,6,2},{16,5,3},{16,4,4},{16,4,2,2},{16,3,3,2},{16,2,2,2,2},{15,9},{15,7,2},{15,6,3},{15,5,4},{15,5,2,2},{15,4,3,2},{15,3,3,3},{15,3,2,2,2},{14,10},{14,8,2},{14,7,3},{14,6,4},{14,6,2,2},{14,5,5},{14,5,3,2},{14,4,4,2},{14,4,3,3},{14,4,2,2,2},{14,3,3,2,2},{14,2,2,2,2,2},{13,11},{13,9,2},{13,8,3},{13,7,4},{13,7,2,2},{13,6,5},{13,6,3,2},{13,5,4,2},{13,5,3,3},{13,5,2,2,2},{13,4,4,3},{13,4,3,2,2},{13,3,3,3,2},{13,3,2,2,2,2},{12,12},{12,10,2},{12,9,3},{12,8,4},{12,8,2,2},{12,7,5},{12,7,3,2},{12,6,6},{12,6,4,2},{12,6,3,3},{12,6,2,2,2},{12,5,5,2},{12,5,4,3},{12,5,3,2,2},{12,4,4,4},{12,4,4,2,2},{12,4,3,3,2},{12,4,2,2,2,2},{12,3,3,3,3},{12,3,3,2,2,2},{12,2,2,2,2,2,2},{11,11,2},{11,10,3},{11,9,4},{11,9,2,2},{11,8,5},{11,8,3,2},{11,7,6},{11,7,4,2},{11,7,3,3},{11,7,2,2,2},{11,6,5,2},{11,6,4,3},{11,6,3,2,2},{11,5,5,3},{11,5,4,4},{11,5,4,2,2},{11,5,3,3,2},{11,5,2,2,2,2},{11,4,4,3,2},{11,4,3,3,3},{11,4,3,2,2,2},{11,3,3,3,2,2},{11,3,2,2,2,2,2},{10,10,4},{10,10,2,2},{10,9,5},{10,9,3,2},{10,8,6},{10,8,4,2},{10,8,3,3},{10,8,2,2,2},{10,7,7},{10,7,5,2},{10,7,4,3},{10,7,3,2,2},{10,6,6,2},{10,6,5,3},{10,6,4,4},{10,6,4,2,2},{10,6,3,3,2},{10,6,2,2,2,2},{10,5,5,4},{10,5,5,2,2},{10,5,4,3,2},{10,5,3,3,3},{10,5,3,2,2,2},{10,4,4,4,2},{10,4,4,3,3},{10,4,4,2,2,2},{10,4,3,3,2,2},{10,4,2,2,2,2,2},{10,3,3,3,3,2},{10,3,3,2,2,2,2},{10,2,2,2,2,2,2,2},{9,9,6},{9,9,4,2},{9,9,3,3},{9,9,2,2,2},{9,8,7},{9,8,5,2},{9,8,4,3},{9,8,3,2,2},{9,7,6,2},{9,7,5,3},{9,7,4,4},{9,7,4,2,2},{9,7,3,3,2},{9,7,2,2,2,2},{9,6,6,3},{9,6,5,4},{9,6,5,2,2},{9,6,4,3,2},{9,6,3,3,3},{9,6,3,2,2,2},{9,5,5,5},{9,5,5,3,2},{9,5,4,4,2},{9,5,4,3,3},{9,5,4,2,2,2},{9,5,3,3,2,2},{9,5,2,2,2,2,2},{9,4,4,4,3},{9,4,4,3,2,2},{9,4,3,3,3,2},{9,4,3,2,2,2,2},{9,3,3,3,3,3},{9,3,3,3,2,2,2},{9,3,2,2,2,2,2,2},{8,8,8},{8,8,6,2},{8,8,5,3},{8,8,4,4},{8,8,4,2,2},{8,8,3,3,2},{8,8,2,2,2,2},{8,7,7,2},{8,7,6,3},{8,7,5,4},{8,7,5,2,2},{8,7,4,3,2},{8,7,3,3,3},{8,7,3,2,2,2},{8,6,6,4},{8,6,6,2,2},{8,6,5,5},{8,6,5,3,2},{8,6,4,4,2},{8,6,4,3,3},{8,6,4,2,2,2},{8,6,3,3,2,2},{8,6,2,2,2,2,2},{8,5,5,4,2},{8,5,5,3,3},{8,5,5,2,2,2},{8,5,4,4,3},{8,5,4,3,2,2},{8,5,3,3,3,2},{8,5,3,2,2,2,2},{8,4,4,4,4},{8,4,4,4,2,2},{8,4,4,3,3,2},{8,4,4,2,2,2,2},{8,4,3,3,3,3},{8,4,3,3,2,2,2},{8,4,2,2,2,2,2,2},{8,3,3,3,3,2,2},{8,3,3,2,2,2,2,2},{8,2,2,2,2,2,2,2,2},{7,7,7,3},{7,7,6,4},{7,7,6,2,2},{7,7,5,5},{7,7,5,3,2},{7,7,4,4,2},{7,7,4,3,3},{7,7,4,2,2,2},{7,7,3,3,2,2},{7,7,2,2,2,2,2},{7,6,6,5},{7,6,6,3,2},{7,6,5,4,2},{7,6,5,3,3},{7,6,5,2,2,2},{7,6,4,4,3},{7,6,4,3,2,2},{7,6,3,3,3,2},{7,6,3,2,2,2,2},{7,5,5,5,2},{7,5,5,4,3},{7,5,5,3,2,2},{7,5,4,4,4},{7,5,4,4,2,2},{7,5,4,3,3,2},{7,5,4,2,2,2,2},{7,5,3,3,3,3},{7,5,3,3,2,2,2},{7,5,2,2,2,2,2,2},{7,4,4,4,3,2},{7,4,4,3,3,3},{7,4,4,3,2,2,2},{7,4,3,3,3,2,2},{7,4,3,2,2,2,2,2},{7,3,3,3,3,3,2},{7,3,3,3,2,2,2,2},{7,3,2,2,2,2,2,2,2},{6,6,6,6},{6,6,6,4,2},{6,6,6,3,3},{6,6,6,2,2,2},{6,6,5,5,2},{6,6,5,4,3},{6,6,5,3,2,2},{6,6,4,4,4},{6,6,4,4,2,2},{6,6,4,3,3,2},{6,6,4,2,2,2,2},{6,6,3,3,3,3},{6,6,3,3,2,2,2},{6,6,2,2,2,2,2,2},{6,5,5,5,3},{6,5,5,4,4},{6,5,5,4,2,2},{6,5,5,3,3,2},{6,5,5,2,2,2,2},{6,5,4,4,3,2},{6,5,4,3,3,3},{6,5,4,3,2,2,2},{6,5,3,3,3,2,2},{6,5,3,2,2,2,2,2},{6,4,4,4,4,2},{6,4,4,4,3,3},{6,4,4,4,2,2,2},{6,4,4,3,3,2,2},{6,4,4,2,2,2,2,2},{6,4,3,3,3,3,2},{6,4,3,3,2,2,2,2},{6,4,2,2,2,2,2,2,2},{6,3,3,3,3,3,3},{6,3,3,3,3,2,2,2},{6,3,3,2,2,2,2,2,2},{6,2,2,2,2,2,2,2,2,2},{5,5,5,5,4},{5,5,5,5,2,2},{5,5,5,4,3,2},{5,5,5,3,3,3},{5,5,5,3,2,2,2},{5,5,4,4,4,2},{5,5,4,4,3,3},{5,5,4,4,2,2,2},{5,5,4,3,3,2,2},{5,5,4,2,2,2,2,2},{5,5,3,3,3,3,2},{5,5,3,3,2,2,2,2},{5,5,2,2,2,2,2,2,2},{5,4,4,4,4,3},{5,4,4,4,3,2,2},{5,4,4,3,3,3,2},{5,4,4,3,2,2,2,2},{5,4,3,3,3,3,3},{5,4,3,3,3,2,2,2},{5,4,3,2,2,2,2,2,2},{5,3,3,3,3,3,2,2},{5,3,3,3,2,2,2,2,2},{5,3,2,2,2,2,2,2,2,2},{4,4,4,4,4,4},{4,4,4,4,4,2,2},{4,4,4,4,3,3,2},{4,4,4,4,2,2,2,2},{4,4,4,3,3,3,3},{4,4,4,3,3,2,2,2},{4,4,4,2,2,2,2,2,2},{4,4,3,3,3,3,2,2},{4,4,3,3,2,2,2,2,2},{4,4,2,2,2,2,2,2,2,2},{4,3,3,3,3,3,3,2},{4,3,3,3,3,2,2,2,2},{4,3,3,2,2,2,2,2,2,2},{4,2,2,2,2,2,2,2,2,2,2},{3,3,3,3,3,3,3,3},{3,3,3,3,3,3,2,2,2},{3,3,3,3,2,2,2,2,2,2},{3,3,2,2,2,2,2,2,2,2,2},{2,2,2,2,2,2,2,2,2,2,2,2}}



Next to each piece number group of cycle classes in the spoiler above I have an ordered pair of the form: (total number of cycle classes for each time, cumulative number of cycle classes).

This was generated by Mathematica. For those interested in the code to generate these, just do the following one at a time for each of the 24 different object sizes (k will be the only code to adjust. You give it a value of 2 and continue through 24 to get the results above).

```
IntegerPartitions[[I]k[/I], All, Range[2, 24]]
```

Examples:
{9,9,4,2} means two 9-cycles, one 4-cycle, and one 2-cycle.

{2} means one 2-cycle, {3,3,2,2} means two 3-cycles and two 2-cycles, {2,2,2,2,2,2} means six 2-cycles, etc.

An n-cycle is said to have length n if n different pieces (whether its corners or edges on the 3x3x3, or wing edges or non-fixed centers in big cubes) involved in the swap swap so that, when the algorithm causing cycle(s) is executed n times, then all of the pieces will be restored to their original locations, and not until then. The pieces may not be oriented correctly (assuming that you execute an algorithm to a solved cube), but they will be in their correct locations once the algorithm has been repeated the nth time.

A common misconception to beginners (including myself when I started) is that PLL Parity is a "2-cycle" because it swaps two dedges. Even if you realize that it actually swaps 4 pieces, you still might make it to be a 4-cycle. However, it is a two 2-cycle (meaning two SEPARATE/disjoint 2-cycle swaps).




​
But regular PLLs (the F,J,N,R,T,V,and Y Permutations) are 2-cycles, because they just involve two pieces from each piece type.






​ 

* About my Formula*
What my formula does is it counts the total number of cycle classes for each of the 24 groups of piece numbers. 

What do I mean by "number?" A U Perm is a 3-cycle of edges. R2 U' R' U' R U R U R U' R. We can add a setup move R to it and get a different 3-cycle of edges R' U' R' U' R U R U R U'. So just by this comparison, we know that there are at least two different 3-cycles of edges which reach different portions on the cube. In fact, there are 440 different 3-cycles of edges (cases like the above in which only three edges are cycled, while the rest of the cube is solved).

To arrive at that number, we do the calculation:
\( \frac{12!}{\left( 3^{1}\times 1! \right)\left( 1^{12-3}\times \left( 12-3 \right)! \right)}=440 \).

(I originally learned about cycle classes from here. So it might be good to look at, but I hope I make more sense!)

We have 12! because on a 3x3x3 cube (or any odd cube which has middle/central edges), there are twelve middle (central) edges: the total number of permutations than can exist in 12 pieces is 12!, which is (12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 479001600.

If we were analyzing wing edges, for example, we would use 24! instead, since there are 24 wing edges (referring to big cube edges). Suppose we wanted to calculate how many possible variations of the cycle class {19,3,2} there are for wing edges (this cycle class is under "24 Pieces in the spoiler"). 
We would do the calculation:

\( \frac{24!}{\left( 19^{1}\times 1! \right)\left( 3^{1}\times 1! \right)\left( 2^{1}\times 1! \right)\left( 1^{24-24}\left( 24-24 \right)! \right)}=\frac{24!}{\left( 19 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right)}= \) 5442529839765258240000 (since 0! = 1).

Another example. For a cycle class which involves more than one of the same cycle, such as {10,3,3,2,2} (under "20 Pieces" in the spoiler above), we would do the following (you should be able to see a pattern emerging, comparing the calculation below with the calculations for the previous examples)

\( \frac{24!}{\left( 10^{1}\times 1! \right)\left( 3^{2}\times 2! \right)\left( 2^{2}\times 2! \right)\left( 1^{24-20}\left( 24-20 \right)! \right)}= \) 17952789402003456000.

And in general, we have

\( \left( \text{Number of pieces the piece type has} \right)! \) divided by:

\( \left( \left( \text{cycle a} \right)^{\#\text{of occurrences of cycle a}}\times \left( \#\text{of occurrences of cycle a} \right)! \right) \) times
\( \left( \left( \text{cycle b} \right)^{\#\text{of occurrences of cycle b}}\times \left( \#\text{of occurrences of cycle b} \right)! \right) \) times
\( \left( \left( \text{cycle c} \right)^{\#\text{of occurrences of cycle c}}\times \left( \#\text{of occurrences of cycle c} \right)! \right) \) times...
\( \left( \left( \text{1-cycles} \right)^{\#\text{untouched}}\times \left( \#\text{untouched} \right)! \right) \)
("# Untouched" means the amount of pieces which were not touched by any of the cycles multiplied in the denominator. They can be thought of as 1-cycles, where they are cycled with themselves and thus do not change position (possibly they could change orientation, but that still won't affect their 1-cycle status).

Since 3 objects only have one cycle class (3-cycles) (and 2 objects only have one cycle class: 2-cycles), this is where my formula comes in. In the case for middle/central edges of the 3x3x3 (or any other odd cube size), we would choose _n_ = 12 since there are twelve middle edges. We would choose _k_ = 3, since we want to know the total number of all cycle classes under 3 objects (and, as I have said already, 3-cycles is the only cycle class under 3 objects, and that's the only reason we can use my formula to do the following calculation). [Link]

Now I just mentioned that we must choose _n_ = 12, but we can choose _n_ = 7, for example, if we wish to see how many possible cycle classes (of some number of pieces). For example, we can still use my formula for 3-cycles (because it is the only cycle class under "3 Pieces"), and so if we let _n_ = 7 (let's say for Petrus Method or something similar) and still let _k_ = 3, my formula yields 70 as the answer.

The only reason I have said that we "must use 8 (for the corners), 12 (for the middle edges), or 24 (for the wing edges, and all non-fixed center orbits for big cubes) is because at the beginning of my post I mentioned acceptable scrambles (the entire cube).

So what's so useful about my formula anyway? Well, I have calculated the number of cycle classes for all twenty-four objects (not using my formula at all, which is what I came up with after the fact!



Spoiler: Number of Cycle Classes



2 Pieces
(n!/((n-2)!2))

3 Pieces
(n!/((n-3)!(3)))

4 Pieces
(n!/((n-4)!(4)))+n!/((n-4)!(2^2*2!))

5 Pieces
(n!/((n-5)!(5)))+n!/((n-5)!(3)(2))

6 Pieces
(n!/((n-6)!(6)))+n!/((n-6)!(4)(2))+n!/((n-6)!(3^2*2!))+n!/((n-6)!(2^3*3!))

7 Pieces
(n!/(((n-7)!7)))+n!/((n-7)!(5)(2))+n!/((n-7)!(4)(3))+n!/((n-7)!(3)(2^2*2!))

8 Pieces
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9 Pieces
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10 Pieces
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11 Pieces
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12 Pieces
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13 Pieces
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14 Pieces
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15 Pieces
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16 Pieces
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17 Pieces
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18 Pieces
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19 Pieces
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20 Pieces
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21 Pieces
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22 Pieces
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23 Pieces
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24 Pieces
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/((n-24)!(8)(6)(3^2*2!)(2^2*2!))+n!/((n-24)!(8)(6)(2^5*5!))+n!/((n-24)!(8)(5^2*2!)(4)(2))+n!/((n-24)!(8)(5^2*2!)(3^2*2!))+n!/((n-24)!(8)(5^2*2!)(2^3*3!))+n!/((n-24)!(8)(5)(4^2*2!)(3))+n!/((n-24)!(8)(5)(4)(3)(2^2*2!))+n!/((n-24)!(8)(5)(3^3*3!)(2))+n!/((n-24)!(8)(5)(3)(2^4*4!))+n!/((n-24)!(8)(4^4*4!))+n!/((n-24)!(8)(4^3*3!)(2^2*2!))+n!/((n-24)!(8)(4^2*2!)(3^2*2!)(2))+n!/((n-24)!(8)(4^2*2!)(2^4*4!))+n!/((n-24)!(8)(4)(3^4*4!))+n!/((n-24)!(8)(4)(3^2*2!)(2^3*3!))+n!/((n-24)!(8)(4)(2^6*6!))+n!/((n-24)!(8)(3^4*4!)(2^2*2!))+n!/((n-24)!(8)(3^2*2!)(2^5*5!))+n!/((n-24)!(8)(2^8*8!))+n!/((n-24)!(7^3*3!)(3))+n!/((n-24)!(7^2*2!)(6)(4))+n!/((n-24)!(7^2*2!)(6)(2^2*2!))+n!/((n-24)!(7^2*2!)(5^2*2!))+n!/((n-24)!(7^2*2!)(5)(3)(2))+n!/((n-24)!(7^2*2!)(4^2*2!)(2))+n!/((n-24)!(7^2*2!)(4)(3^2*2!))+n!/((n-24)!(7^2*2!)(4)(2^3*3!))+n!/((n-24)!(7^2*2!)(3^2*2!)(2^2*2!))+n!/((n-24)!(7^2*2!)(2^5*5!))+n!/((n-24)!(7)(6^2*2!)(5))+n!/((n-24)!(7)(6^2*2!)(3)(2))+n!/((n-24)!(7)(6)(5)(4)(2))+n!/((n-24)!(7)(6)(5)(3^2*2!))+n!/((n-24)!(7)(6)(5)(2^3*3!))+n!/((n-24)!(7)(6)(4^2*2!)(3))+n!/((n-24)!(7)(6)(4)(3)(2^2*2!))+n!/((n-24)!(7)(6)(3^3*3!)(2))+n!/((n-24)!(7)(6)(3)(2^4*4!))+n!/((n-24)!(7)(5^3*3!)(2))+n!/((n-24)!(7)(5^2*2!)(4)(3))+n!/((n-24)!(7)(5^2*2!)(3)(2^2*2!))+n!/((n-24)!(7)(5)(4^3*3!))+n!/((n-24)!(7)(5)(4^2*2!)(2^2*2!))+n!/((n-24)!(7)(5)(4)(3^2*2!)(2))+n!/((n-24)!(7)(5)(4)(2^4*4!))+n!/((n-24)!(7)(5)(3^4*4!))+n!/((n-24)!(7)(5)(3^2*2!)(2^3*3!))+n!/((n-24)!(7)(5)(2^6*6!))+n!/((n-24)!(7)(4^3*3!)(3)(2))+n!/((n-24)!(7)(4^2*2!)(3^3*3!))+n!/((n-24)!(7)(4^2*2!)(3)(2^3*3!))+n!/((n-24)!(7)(4)(3^3*3!)(2^2*2!))+n!/((n-24)!(7)(4)(3)(2^5*5!))+n!/((n-24)!(7)(3^5*5!)(2))+n!/((n-24)!(7)(3^3*3!)(2^4*4!))+n!/((n-24)!(7)(3)(2^7*7!))+n!/((n-24)!(6^4*4!))+n!/((n-24)!(6^3*3!)(4)(2))+n!/((n-24)!(6^3*3!)(3^2*2!))+n!/((n-24)!(6^3*3!)(2^3*3!))+n!/((n-24)!(6^2*2!)(5^2*2!)(2))+n!/((n-24)!(6^2*2!)(5)(4)(3))+n!/((n-24)!(6^2*2!)(5)(3)(2^2*2!))+n!/((n-24)!(6^2*2!)(4^3*3!))+n!/((n-24)!(6^2*2!)(4^2*2!)(2^2*2!))+n!/((n-24)!(6^2*2!)(4)(3^2*2!)(2))+n!/((n-24)!(6^2*2!)(4)(2^4*4!))+n!/((n-24)!(6^2*2!)(3^4*4!))+n!/((n-24)!(6^2*2!)(3^2*2!)(2^3*3!))+n!/((n-24)!(6^2*2!)(2^6*6!))+n!/((n-24)!(6)(5^3*3!)(3))+n!/((n-24)!(6)(5^2*2!)(4^2*2!))+n!/((n-24)!(6)(5^2*2!)(4)(2^2*2!))+n!/((n-24)!(6)(5^2*2!)(3^2*2!)(2))+n!/((n-24)!(6)(5^2*2!)(2^4*4!))+n!/((n-24)!(6)(5)(4^2*2!)(3)(2))+n!/((n-24)!(6)(5)(4)(3^3*3!))+n!/((n-24)!(6)(5)(4)(3)(2^3*3!))+n!/((n-24)!(6)(5)(3^3*3!)(2^2*2!))+n!/((n-24)!(6)(5)(3)(2^5*5!))+n!/((n-24)!(6)(4^4*4!)(2))+n!/((n-24)!(6)(4^3*3!)(3^2*2!))+n!/((n-24)!(6)(4^3*3!)(2^3*3!))+n!/((n-24)!(6)(4^2*2!)(3^2*2!)(2^2*2!))+n!/((n-24)!(6)(4^2*2!)(2^5*5!))+n!/((n-24)!(6)(4)(3^4*4!)(2))+n!/((n-24)!(6)(4)(3^2*2!)(2^4*4!))+n!/((n-24)!(6)(4)(2^7*7!))+n!/((n-24)!(6)(3^6*6!))+n!/((n-24)!(6)(3^4*4!)(2^3*3!))+n!/((n-24)!(6)(3^2*2!)(2^6*6!))+n!/((n-24)!(6)(2^9*9!))+n!/((n-24)!(5^4*4!)(4))+n!/((n-24)!(5^4*4!)(2^2*2!))+n!/((n-24)!(5^3*3!)(4)(3)(2))+n!/((n-24)!(5^3*3!)(3^3*3!))+n!/((n-24)!(5^3*3!)(3)(2^3*3!))+n!/((n-24)!(5^2*2!)(4^3*3!)(2))+n!/((n-24)!(5^2*2!)(4^2*2!)(3^2*2!))+n!/((n-24)!(5^2*2!)(4^2*2!)(2^3*3!))+n!/((n-24)!(5^2*2!)(4)(3^2*2!)(2^2*2!))+n!/((n-24)!(5^2*2!)(4)(2^5*5!))+n!/((n-24)!(5^2*2!)(3^4*4!)(2))+n!/((n-24)!(5^2*2!)(3^2*2!)(2^4*4!))+n!/((n-24)!(5^2*2!)(2^7*7!))+n!/((n-24)!(5)(4^4*4!)(3))+n!/((n-24)!(5)(4^3*3!)(3)(2^2*2!))+n!/((n-24)!(5)(4^2*2!)(3^3*3!)(2))+n!/((n-24)!(5)(4^2*2!)(3)(2^4*4!))+n!/((n-24)!(5)(4)(3^5*5!))+n!/((n-24)!(5)(4)(3^3*3!)(2^3*3!))+n!/((n-24)!(5)(4)(3)(2^6*6!))+n!/((n-24)!(5)(3^5*5!)(2^2*2!))+n!/((n-24)!(5)(3^3*3!)(2^5*5!))+n!/((n-24)!(5)(3)(2^8*8!))+n!/((n-24)!(4^6*6!))+n!/((n-24)!(4^5*5!)(2^2*2!))+n!/((n-24)!(4^4*4!)(3^2*2!)(2))+n!/((n-24)!(4^4*4!)(2^4*4!))+n!/((n-24)!(4^3*3!)(3^4*4!))+n!/((n-24)!(4^3*3!)(3^2*2!)(2^3*3!))+n!/((n-24)!(4^3*3!)(2^6*6!))+n!/((n-24)!(4^2*2!)(3^4*4!)(2^2*2!))+n!/((n-24)!(4^2*2!)(3^2*2!)(2^5*5!))+n!/((n-24)!(4^2*2!)(2^8*8!))+n!/((n-24)!(4)(3^6*6!)(2))+n!/((n-24)!(4)(3^4*4!)(2^4*4!))+n!/((n-24)!(4)(3^2*2!)(2^7*7!))+n!/((n-24)!(4)(2^10*10!))+n!/((n-24)!(3^8*8!))+n!/((n-24)!(3^6*6!)(2^3*3!))+n!/((n-24)!(3^4*4!)(2^6*6!))+n!/((n-24)!(3^2*2!)(2^9*9!))+n!/((n-24)!(2^12*12!))


The results in the spoiler are based directly off of the calculations which were done in previous examples. I have left _n_ as a variable so that you can substitute your desired number for _n_, since, for example, "2 Pieces" through "8 Pieces" can be used for corners if you substitute _n _= 8 (or less), "2 Pieces" through "12 Pieces" may be used for middle edges (but "13 Pieces" and up can only be used for calculations regarding big cube parts).

Also, just in case you are wondering how I managed to do all of those calculations (in the spoiler above).


Spoiler



I just started with the cycle classes data (in the first big spoiler). I copied and pasted it into WordPad (but you can use Word or any search and replace routine), and I did the following search and "replace all" in the following order:

, = )(
{ = {(
} = )}
})({ = )+x!/((x-_)!
{({ = x!/((x-_)!
})} = )

To the left of the equal sign was the original text, and to the right is what it was replaced with.

For the substitution with the underscore "_", this is for the very first cycle class in each of the main 24 blocks of data. After that substitution is made, I manually went and replaced the underscore with the corresponding number to the "piece number block of data" I was looking at. For example, that substitution is the "n!/((n-24)!" in the 24th block of data, so the underscore for that one is 24.

After I did those substitutions, I then did (in this order):
(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2) = (2^12*12!)
(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2) = (2^11*11!)
...
(2)(2) = (2^2*2!)
(3)(3)(3)(3)(3)(3)(3)(3) = (3^8*8!)
(3)(3)(3)(3)(3)(3)(3) = (3^7*7!)
...
...
(12)(12) = (12^2*2!)

These match the first cycle class count calculations I did.


So anyway, my formula gives the total number for each block. I didn't want to leave you guys with just those calculations as the main "tool" you can use. My formula tells you, for example, the sum of all of the data in each main block. For example, the sum of the entire "16 Pieces" block of data.



Spoiler



16 Pieces
(n!/((n-16)!(16)))+n!/((n-16)!(14)(2))+n!/((n-16)!(13)(3))+n!/((n-16)!(12)(4))+n!/((n-16)!(12)(2^2*2!))+n!/((n-16)!(11)(5))+n!/((n-16)!(11)(3)(2))+n!/((n-16)!(10)(6))+n!/((n-16)!(10)(4)(2))+n!/((n-16)!(10)(3^2*2!))+n!/((n-16)!(10)(2^3*3!))+n!/((n-16)!(9)(7))+n!/((n-16)!(9)(5)(2))+n!/((n-16)!(9)(4)(3))+n!/((n-16)!(9)(3)(2^2*2!))+n!/((n-16)!(8^2*2!))+n!/((n-16)!(8)(6)(2))+n!/((n-16)!(8)(5)(3))+n!/((n-16)!(8)(4^2*2!))+n!/((n-16)!(8)(4)(2^2*2!))+n!/((n-16)!(8)(3^2*2!)(2))+n!/((n-16)!(8)(2^4*4!))+n!/((n-16)!(7^2*2!)(2))+n!/((n-16)!(7)(6)(3))+n!/((n-16)!(7)(5)(4))+n!/((n-16)!(7)(5)(2^2*2!))+n!/((n-16)!(7)(4)(3)(2))+n!/((n-16)!(7)(3^3*3!))+n!/((n-16)!(7)(3)(2^3*3!))+n!/((n-16)!(6^2*2!)(4))+n!/((n-16)!(6^2*2!)(2^2*2!))+n!/((n-16)!(6)(5^2*2!))+n!/((n-16)!(6)(5)(3)(2))+n!/((n-16)!(6)(4^2*2!)(2))+n!/((n-16)!(6)(4)(3^2*2!))+n!/((n-16)!(6)(4)(2^3*3!))+n!/((n-16)!(6)(3^2*2!)(2^2*2!))+n!/((n-16)!(6)(2^5*5!))+n!/((n-16)!(5^2*2!)(4)(2))+n!/((n-16)!(5^2*2!)(3^2*2!))+n!/((n-16)!(5^2*2!)(2^3*3!))+n!/((n-16)!(5)(4^2*2!)(3))+n!/((n-16)!(5)(4)(3)(2^2*2!))+n!/((n-16)!(5)(3^3*3!)(2))+n!/((n-16)!(5)(3)(2^4*4!))+n!/((n-16)!(4^4*4!))+n!/((n-16)!(4^3*3!)(2^2*2!))+n!/((n-16)!(4^2*2!)(3^2*2!)(2))+n!/((n-16)!(4^2*2!)(2^4*4!))+n!/((n-16)!(4)(3^4*4!))+n!/((n-16)!(4)(3^2*2!)(2^3*3!))+n!/((n-16)!(4)(2^6*6!))+n!/((n-16)!(3^4*4!)(2^2*2!))+n!/((n-16)!(3^2*2!)(2^5*5!))+n!/((n-16)!(2^8*8!))


This is really what is most useful for getting a quick glimpse of what percentage all cycle classes for one piece number (in this case, 16 pieces) take up with all 24! (or 12! or 8!, depending on which piece type you are investigating) permutations.

That is, *the sum of all of the data in the spoiler will add up to 24! if you substitute n = 24. *This alone gave me a glimpse of how large 24! really is!

Now, let's say you want to do all of this work for 60! (the number of higher order minx wing edges). Well, we could do this and end up with 10 times the data we had for 24!, or we could just use my formula which WolframAlpha can compute all values of 60 objects in a few seconds! [Link!]

Of course, on minxes, odd permutations cannot exist in the wing edges, so this computation wouldn't be so useful since half of the 60! permutations are odd. But here is what we can definitely use my formula for.

For a scramble of the entire cube,

*The Corners*[Link] *The Middle Edges* [Link] *The wing edges and moveable centers (big cube parts)* [Link]

And of course, we can choose other values for _k_ and _n_ for only portions of the cube.

*How I found that formula?*


Spoiler: Formula Derivation



All I did was copy the calculations in the last big spoiler (which has _n_ as an input variable) into Mathematica, and I got the following (well I rewrote a few of them to show a pattern):



```
2 Pieces([I]k [/I]= 2)
[math]\frac{n!}{2(n-2)!}[/math]

3 Pieces([I]k [/I]= 3)
[math]\frac{n!}{3(n-3)!}[/math]

4 Pieces ([I]k [/I]= 4)
[math]\frac{3}{2}\frac{n!}{4\left( n-4 \right)!}[/math]

5 Pieces ([I]k [/I]= 5)
[math]\frac{11}{6}\frac{n!}{5\left( n-5 \right)!}[/math]

6 Pieces ([I]k [/I]= 6)
[math]\frac{53}{24}\frac{n!}{6\left( n-6 \right)!}[/math]

7 Pieces ([I]k [/I]= 7)
[math]\frac{309}{120}\frac{n!}{7\left( n-7 \right)}[/math]

8 Pieces ([I]k [/I]= 8)
[math]\frac{2119}{720}\frac{n!}{8\left( n-8 \right)!}[/math]

9 Pieces ([I]k [/I]= 9)
[math]\frac{16687}{5040}\frac{n!}{9\left( n-9 \right)!}[/math]

10 Pieces ([I]k [/I]= 10)
[math]\frac{148329}{40320}\frac{n!}{10\left( n-10 \right)!}[/math]

11 Pieces ([I]k [/I]= 11)
[math]\frac{1468457}{362880}\frac{n!}{11\left( n-11 \right)!}[/math]
...
[math]\frac{\text{some  }\!\!\#\!\!\text{ }}{\left( k-2 \right)!}\frac{n!}{k\left( n-k \right)!}[/math]
```
So the only tricky part was determining how to express the sequence {1, 1, 3, 11, 53, 309, 2119, 16687, 148329, 1468457,...}.
Luckily Mathematica recommended this page, from which I searched google and found this PDF which had a compact formula.

So I did a lot of work to create a formula (from doing all of those computations which I have shown in the spoiler). From there, Mathematica simplified them into the form I showed in the code box above. Then Mathematica forwarded me to a page which had a name for that sequence of numbers.

\( \left\{ \frac{\left( n+2 \right)n!}{\exp \left( 1 \right)} \right\} \)

Now, you may also find the numbers to that sequence by taking the kth derivative of the following function and then substituting zero for _x_ to obtain a number in the sequence each differentiation (this is a result I also found on the first page Mathematica forwarded me to. *Note that I had to adjust the k on the right hand side so that the two would be equal for the same value of k.*):

\( \left. \frac{d^{k}}{dx^{k}}\left[ \frac{e^{-x}}{\left( x-1 \right)^{2}} \right] \right|_{x=0}^{{}}=\text{Round}\left[ \left( k+2 \right)\left( k \right)!e^{-1} \right]\text{ for }k\ge 0,\text{ }k\in \mathbb{Z} \)


Now, I'm going to show my calculations for the very first results in this post about the percent of the configurations which satisfied the pretend "restrictions" I had for examples. These were for the 3x3x3, and so we start with the 3x3x3 permutation formula:

\( \left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)= \) 43252003274489856000 total possible configurations. (I'm aware of cuBerBruce mentioning that the number of speedsolving elements is 24 times this, but we are calculating percentages, so I don't think even that will make any difference).

All we do is edit the factorial portions.



cmowla said:


> Suppose that a 3x3x3 scramble is acceptable by a certain cuber, if at most 2 corners and 3 edges may be in their correct locations (but all possible orientations are allowed). Then the total percentage of configurations of the 3x3x3 (regular/6 color) cube which can be an acceptable scramble is 67.6658%.


If you click the *corner* and edge *links* that I put in this post previously, we need to find the number of permutations in which involves 6, 7 and 8 corner pieces and 9, 10,11,and 12 edges pieces and replace the factorials of the original formula with them.


\( \left( 7420+14832+14833 \right)3^{7} \)
\( +\left( 2936912088107426+176214840+176214841 \right)2^{10} \)
= 39026375885497512960 allowed configurations. So,

\( \frac{\text{39026375885497512960}}{\text{43252003274489856000}}\times 100= \) 90.23%.

AND



cmowla said:


> On the other hand, if we still didn't care about orientations, if no pieces were allowed to be in their correct locations (whether twisted/flipped or not, since there is no restriction on the orientations of the pieces in this example), then only 13.5336% of the possible 43252003274489856000 configurations would qualify as authentic scrambles.


For this, we just retrieve (well, we have the numbers for them now since we did the previous example) the number of cycle classes for 12 edge pieces and 8 corner pieces and replace the factorials now with just them:

\( \left( 14833\times 3^{7} \right)\left( 176214841\times 2^{10} \right)= \) 5853561946973604864. So we have

\( \frac{\text{5853561946973604864}}{\text{43252003274489856000}}\times 100= \) 13.5336% allowed permutations.

*Final Note about the 3x3x3 examples.*
I could be wrong, but I think it's very acceptable to even cut the percentages (calculated above) in half because:

Since half of all permutations are even, and half are odd, then we should safely be able to assume that we can get all odd permutation scrambles by applying one outer layer quarter turn move to even permutation scrambles. If you think about it, the scramble does not look much different at all with just one quarter turn move difference.


*What Makes an Authentic Big Cube Scramble* (How many pieces are allowed to be in their correct locations)?

I'm not at all sure what convention I should take to determine the percentage of all possible configurations which qualifies as a good scramble for big cubes, but here's what I have come up with so far:

 


No more than 1/3 of each composite center solved (I'm speaking of the average because some centers might begin to be more solved and others not solved at all.) 
No more than 3 wing edges (in each orbit) solved. 

Both of the above are from any frame of reference on the big even cube.


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