combinations of Rubik cube

Question

A Rubick's Cube has owl heads on it, which can be misoriented. How many (times) MORE combinations are there of this cube vs. one that has blank stickers?

Can someone give me some hints? Thanks

Answer 1

Hint: Imagine fixing the center of the cube, and rotating only the six faces. The six center cubies (one in each face) do not move, only rotate. This determines the orientations of the eight non-center cubies on the top layer.

Answer 2

The answer is $4^6/2$ times more combinations than a regular Rubik's Cube. This is because the first 5 centers can be in any orientation, but the last center piece is restricted to only 2 possibilities instead of 4.

First notice that a quarter-turn is an ODD permutation:

$$(1234) = (41)(42)(43).$$

And since a solved (regular) cube is an EVEN permutation, that means any permutation that affects just the center pieces must also be even. So the MOST number of extra cube positions is $4^6/2$. Now we just need to show that all those positions are possible. It turns out they are.

By construction, it is possible to rotate a center piece 180 degrees:

$$(URLU^2R'L')^2.$$

This rotates the top center piece 180 degrees. To rotate the top center clockwise 90 degrees and the left center counterclockwise 90 degrees, do:

$$URL'F'BDU'L'UD'FB'R'L.$$