Solving the Rubik's Cube Systematically

The Strategy of Eight Corners views the face-centers as mobile. The iron cage formed by corners and edges is regarded as the rest frame in the previous sections. When all other pieces have been settled, the face-centers may not be in their correct locations relative to the colored faces.

Most cubists treats face-center movements as pretty patterns without giving them any serious thoughts. Indeed, only two set of movements are possible.

FM4

FM4 ≡ [M,E²] = M⊕E²⊕M^-1⊕E²

{(F,B), (U,D)} exchanges two pairs of opposing face-centers

To understand FM4, use two ways to look at it:

In the subview of the R and L faces, M is not doing anything, and the commutator is simply E²⊕E² ≡ I
In the subview of the M edges, E is not doing anything, and the commutator is simply M⊕M^-1 ≡ I
Therefore, only the M face-centers are affected. E² swaps one pair of M face-centers. By forming a commutator with M, two different pairs of M face-centers are swapped.

FM6

FM6 ≡ [M,E] = M⊕E⊕M^-1⊕E^-1

{(U,F,L), (B,R,D)} cycles two sets of three face-centers

The three face-centers marked with white Xs move in a cycle (clockwise with respect to UFL), and the other three face-centers marked with black X move in another cycle (counter-clockwise with respect to DRB).

In the subview of the corners and edges, M and E are not doing anything.

In the subview of the face-centers, FM6 is not unlike CM1R:

The first M dips the U face-center down to the E slice.
An E moves the L face-center into the M slice.
Therefore, when we continue with M^-1, we are taking the L face-center back to the top, instead of the original U face-center.
The final E^-1 completes the commutator, and we have moved the L face-center to the U face.
Similar to CM1R, this is a 3-cycle {(U,F,L), ...}.
Since opposing face-centers always stay on the opposite sides of a cube, the other three face-centers are moved in a symmetric manner.
We have accounted for the movement of all six face-centers. Since FM6 does not affect corners and edges, {(U,F,L), (B,R,D)} is the complete description of the algorithm.

Face-Centers