When performing edge movements, view the eight corners as if it was 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 frequent use of slice movements (E, M, S), because the three slices flow (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 moves 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 corners before any of the edges.
Edge movements are usually commutators of the form [X,Y] = X⊕Y⊕X-1⊕Y-1, where X does not affect corners. This ensures that the corners are unaffected by [X,Y]. In the sub-view of corners, X ≡ I, the Identity Algorithm. If you are not familiar with the Identity Algorithm, you may want to revisit this section.
So, as far as the corners are concerned, the whole commutator
Another possibility is the conjugate X⊕Y⊕X-1, where Y does not affect corners. In the sub-view of corners, the conjugate equals X⊕X-1 ≡ 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 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 M, we have:
EM1F moves FD to FU. Note the similarity with CM1R:
Similar to EM1F, we have its (near) mirror image
moving RD to UF.
conjugating with D,
If you have played out the animation, you can hardly miss the last two silly moves (DD-1 ≡ I). It is what happens when you become too systematic. The last two moves in EM1R negate each other. In actual life, you won't be doing that — just skip them.
Hence, we have
flips UF. Notice that EF1D only conserves the U edges. It disturbs the D edges in complicated ways.
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 is analogous to CM3R, but we never use it. We shall see much simpler ways to move three edges.
Conjugating EF1D with U gives:
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 E slice.
EM1 algorithms thus far only move 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 E slice, instead of finding a suitable conjugate, there is another set of EM1 algorithms moving RB to UF or FU.
Noting that En moves an E edge to another, preserving the U and D pieces, we have
moves RB to FU. This is analogous to EM1R: the feeling is the same. First, the FU edge is moved into the E slice where the movement E will operate, while all other U pieces are away from the E slice. The E-1 move places the required piece. The remaining two moves complete the commutator. Usually, we forget about the E centers and the E edges, and do away with the trailing E. As we shall see, centers are the easiest pieces to move.
The sequence listed ends with an E. Since we have not started to fix the E edge, you can ignore the last E and save one move.
Notice that among the U and D edges, EM1BR only affects FU and FD. We'll talk about it again below, when we are completing the U and D edges.
From EM1BR, we have its near mirror image:
moves RB to UF.
Again, you may ignore the trailing E2.
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:
like EF1D flips a single edge FU (but disturbs the edges on the E and D faces).
Notice that by joining EM1BR-1 to EM1BL, you have two F-1 side by side, which is equivalent to F2.
You can also trim the leading and trailing E moves too, because we have not yet started to fix the slice.
and hence conjugating with U gives
like EF2D, EF2B flips two edges with no other changes.
While both achieve the same result, EF2D is better because it is more convenient to make an M turn than an E turn.
Or the shorter version based on EM1B Simplfied instead.
Saved four moves compared with EF2B.
What if the piece you need is already on the U face? Usually you just ignore that edge and handle other edges first. Since the piece is occupying another cubelet's location, sooner or later it will be expelled from the U face. Or you can use EM1R-1 or EM1F-1 to move the piece away from the U face, then you can use the method above to move it back to the correct position on the U face.
After fixing three of the four U edges, turn the cube over, so that what was D is now U.
Remember we have already fixed 3 of the 4 (now) D edges. Keep the not-yet-fixed D edge at FD. Why? Because EM1BR and EM1BL only affects FD on the D. You will be using EM1BR and EM1BL to move the E edges to U. By keep the not-yet-fixed D edges at FD, you will not disturb the three D edges that you have already fixed.
Continue with Moving Edges Part II
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