# 4x4 Rubik's cube centers permutations count

About a 4x4 Rubik's cube. We know the count of permutations.

But how about the permutation of the centers only ?

X X X X
X O O X
X O O X
X X X X


There are 6 colored faces containing 4 stickers at centers (marked with "O"). How many permutations do we have ? This is 24 facelets but the answser is not 24! because when we have placed 5 faces, the last face is obviously made with the 4 last facelets of the same color.

How could we calculate the number of the centers position ?

• It is possible to do a lot of 3-cycles involving center facelets alone. I haven't written all the details down, but I'm fairly sure that all the even permutations of the 24 facelets are possible, so the number of permutations is $\dfrac12\,24!$. Commented Feb 21, 2022 at 15:45
• If the centre pieces are distinguishable, then it is $\frac{24!}{2}$ as @JyrkiLahtonen writes. If you consider them to be six sets of indistinguishable pieces, then it's $\frac{24!}{4!^6}$. All this is assuming that the cube has a fixed orientation in space, e.g. by using one of the corner pieces as a reference. Commented Feb 21, 2022 at 15:48
• The 2x2 permutation is "only" 3,674,160 (en.wikipedia.org/wiki/Pocket_Cube) This is about the corners pieces. So I am surprised of the so big number 24! Commented Feb 21, 2022 at 15:52

Using the counting convention described in Jaap's comment it can be shown that we can get all the even permutations of the 24 facelets, giving a total of $$\dfrac12\,24!=310224200866619719680000$$ ways of positioning them.