Solving the Rubik's Cube Systematically

Edge Movements Part III

C edges

At this stage, all U and D edges, with the possible exception of FU/FD, which may need to be exchanged. Physically rotate the cube sideways to view the remaining four E edges as M edges.

last 4 edges

Now that both the the R and L faces (ie the original U and D faces) have been (largely) settled, we shift to a new paradigm. Regard these two faces as two metal plates with the M slice in between. Only this M slice have edges to be solved. They can always be solved with either of the following algorithms using suitable conjugates of Mn.

The remaining cubes on this page have yellow on the R face, as a result of the physical rotation.

Note that Fa is called Z in the standard notation. Refer to this page for the movement names.

EM3

Noting that M moves one M edge to another,

EM3 ≡ [U2,M] = U2⊕M⊕U2⊕M-1
{(UF, DB, UB)}

EM3 is topologically the same as EM1F. At that time, our sole concern was FU. Actually it moves three edges without affecting other things.

Use two different ways to look at EM3:

  1. In the subview of the R and L faces, M ≡ I. Therefore EM3 = U2⊕U2 = I.
  2. In the M slice subview, U2 exchanges UF and UB. M moves another pair of M edges to the U face for U2 to exchange. Since the two exchanges are made on overlapping edges, the net change is the movement of three edges.

Formally, we can also say that U2 is a mono-move on the F face. The commutator EM3 is thus analogous to CM3 and EM3R.

EM4

Similar to EM3, we have

EM4 ≡ [U2,M2] = U2M2U2M2
{(UF, UB), (DF, DB)}

Again, use two ways to look at EM4:

  1. In the subview of the R and L faces, EM4 and EM3 are the same.
  2. In the M slices subview, U2 exchanges UF and UB. M2 moves another pair of M edges to the U face for U2 to exchange. This time, the two exchanges are made on non-overlapping edges, the net change is the exchange of two separate pairs of edges.

Unfinished Business

Use EM3 and EM4 to settle the position of the M edges. In some cases, you would end up with one single pair of swapped edges, which indicates it is time for some unfinished business.

Swap FU/FD

Remember that back in Part II, there is one situation where we left FU and FD swapped and unhandled. We are treating them now.

After the physical rotation at the beginning of Part III, the four E edges have become M edges. FU and FD now lie on the front of the R and L faces (the two pieces marked with black Xs). Use Mn to move the swapped M edges to the back (the two pieces marked with white Xs, on the B face).

After physical rotation X, a conjugate of EM4 will swap the two pairs of edges.

U⊕EM4⊕U-1 = U⊕[U2,M2]⊕U-1
= U⊕(U2⊕M2⊕U2⊕M2)⊕U-1
Simplifies to U-1⊕M2⊕U2⊕M2⊕U-1
{(UR, UL), (DF, DB)}

Continue to Completing the Cube

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