When performing edge movements, view the eight corners as if it is an iron cage. Feel the strain when you twist them from their original positions. All algorithms end by relieving this strain, therefore preserving the corners. Remember the paradigm is more important than the sequences themselves.
There will be more and more slice movements (M, C, S), because the three slices flows (like rivers) without affecting the corners. After all, every edge lies on a slice; while slices consist only of edges and centers, and no corners. On the other hand, there is no movement that move corners but not edges. This asymmetry leads to the observation that edges are easier to settle than corners, and the logical conclusion that we should settle all the corner before any of the edges.
Edge movements are usually commutators of the form [X,Y] = X-1ÅYÅXÅY-1, where X does not affect corners. This ensures that the corners are unaffected by [X,Y]. In the sub-view of corners, X º I, the identity algorithm. So, as far as the corners are concerned, the whole commutator
Another possibility is the conjugate XÅYÅX-1, where Y does not affect corners. In the sub-view of corners, the conjugate equals XÅX-1 º I.
The Eight Corners strategy settles the edges in this order:First, we settle the position of three U edges (without worrying about their orientation) without affecting the other corners or edges. We would first consider cases where the piece we want to fit into U comes from the D face.
|
Movement Dn moves a D edge to another, preserving the U face. Commutating with C, we have:
EM1F º [C-1,D2] = C-1ÅD2ÅCÅD2
EM1F moves FD to FU. Note the similarity with CM1R. C-1 dips UF to FD of the D face, at the same time, FD is moved to DB (this corresponds to the behavior of RUF, RFD, and RDB in R-1 of CM1R). Then D2 moves DB to DF. C move DF back to FU, and D2 completes the commutator on restoring the D corners. |
|
Similar to EM1F, we have EM1RD º [C-1D-1] = C-1ÅD-1ÅCÅD
moving RD to UF. |
conjugating with D, EM1R º DÅEM1RDÅD-1 moves FD to UF. This makes EM1R behaves the same as EM1F.
|
Hence, we have EF1D º EM1F-1ÅEM1R
flips UF. If you have played out the animation, you can hardly miss the last two silly moves (DD-1 º I). It is what happens when you becomes too systematic. The last two moves in EM1R negates each other. In actual life, you won't be doing that — just skip them. |
EM1 and EF1 affects only one piece on the U face. That is, EM1 is a mono-move, EF1 is a mono-flip (flipping a single edge without movement). Mono-changes always form useful commutators.
|
Conjugating EM1 with U gives:
EM3R º [EM1F,U] = EM1FÅUÅEM1F-1ÅU-1
EM3R is analogous to CM3R, but we never use it. We shall see much simpler ways to move three edges. |
|
Conjugating EF1D with U gives:
EF2D º [EF1D,U] = EF1DÅUÅEF1D-1ÅU-1
flips UF and UR. With EF2, you should find no difficulty in flipping all the edges correctly. |
Now we continue with the case where the piece we want to fit into U face comes from the M slice.
EM1 algorithms thus far only moves D edges to U face. If the required edge is in the U face, we may use some EM1-1 to move it down to the D face, where EM1 can be applied. If the required edge is in the M slice, instead of finding a suitable conjugate, there is another set of EM1 algorithms moving RB to UF or FU.
Noting that Mn moves an M edge to another, preserving the U and D pieces, we have
|
EM1BR º [F,M] = FÅMÅF-1ÅM-1
moves RB to FU, is analogous to EM1R. The feeling is the same. First, the FU edge is moved into the M slice where the movement M will operate, while all other U pieces are away from the M slice. The M move places the required piece. The remaining two moves complete the commutator. Usually, we forget about the M centers and the M edges, and do away with the trailing M-1. As we shall see, centers are the easiet pieces to move. |
Notice that, as a side effect, EM1B moves UF to FD. That is why we only fix 3 of the 4 edges in the first phase. After positioning 3 U edges, we turn the cube over, ending up with 3 D edges correct. Leaving the unfixed edge at FD, we proceed with EM1Bs to fix both edges at the same time.
|
Similarly, we have its near mirror image:
EM1BL º [F-1,M2] = F-1ÅM2ÅFÅM2
moves RB to UF. |
|
The orientation of RB as operated by EM1BL is different from that by EM1BR because the RB is now entering the F face in the other direction. Therefore:
EF1B º EM1BR-1ÅEM1BL
like EF1D flips a single edge FU. |
|
and hence conjugating with U gives
EF2B º [EF1B,U] = EF1BÅUÅEF1B-1ÅU-1
like EF2D flips two edges with no other changes. |
Now that we have completed the EM1, we are capable of settling settle the three U edges. Then we turn the cube upside down, making sure that the fourth (ie not-yet-settled) edge is placed at FD. Then uses EM1B to settle the three (now) U edges. It is important to leave the forth edge at FD, because EM1B will move away whatever was at that position.
If you need to move the edge already at FD to FU, then you can't use EM1B (which only moves an edge from the M slice). You have to use the side effect of the commutator EM1BR-1 to move it to UF. In Moving Edges Part II, we would handle the remaining edges.