In the Strategy of Eight Corners, we start with the eight corners. Among these corners, we first try to settle the position of the four U corners (without worrying about their orientation). Then we would turn the cube upside down and process the other four. Finally, we twist the corners, as necessary.
You can choose any of the six faces to be your U. I like to choose yellow because it is bright but not as shiny as white. Remember: you have to look at the U face most of the time.
Moving the first four corners is rather trivial, but let's be formal and find an algorithm CM1R that moves RDF to URF without disturbing other U pieces.
Movement Dn (where n = -1, 1, or 2) moves one D corner to another, preserving the U face. Conjugating with R-1, we have:
For easy viewing, the top two layers are already completed in the cube here. In reality, they are likely to be scrambled. When you are fixing the first corner, nothing is already in place. Not even the face-centers!
Concentrate on the piece at RDF (marked with black Xs): we want to move this yellow-green-red piece upwards to the position URF (marked with white Xs). Also notice how the other three U (yellow) corners are preserved. Get the feeling of R-1 dipping the URF corner into the area affected by D. The essence is in finding a conjugate where only URF (the target location), but no other U pieces, is dipped into the D face. Such conjugations are easier to find if you concentrate your attention on the correct pieces (forming a sub-view, concentrating only on these five pieces: UFL, ULB, UBR, URL, and RFD), instead of always worrying about the whole cube.
Symmetrically, we have
doing a similar job.
Being the mirror image of CM1R, CM1F moves FRD to URF.
This time, the cube has only the other three U corners colored for your viewing. This is more or less the actual situation when you are setting the fourth U corner.
Since CM1R and CM1F are mirror images, they twist corners in opposite ways. Hence, we try
{(URF)+, ...}
CT1 twists URF clockwise without moving it, or disturbing any other pieces on the U face. That is why we settle the position of the pieces before worrying about their orientation.
In the Scientific American article I mentioned earlier, the CT1 is called a quarkscrew, because twisting a corner is identified with a quark in the particle physics analogy.
CM1 and CT1 affects only one piece on the U face.
Any algorithm that affects only one piece on a face is called a mono-change.
CM1R moves RDF (the red-yellow-green cubelet) to URF. In other words, URF (the yellow-orange-green cubelet) is saved somewhere in the D face. U moves another piece (the green-white-red cubelet currently at UBR) in place of UFR, where the inverse of CM1R is going to take place. Therefore, the saved piece is restored to URF, which will be brought back to UBR by U-1. The net result is the corner movement:
That is the beauty of commutators. Similarly:
CT1-1 restores the picture of the D face and E, and twists back (URF)-. The U and U-1 make the whole thing work: URF is twisted clockwise, but the corner that is twisted anticlockwise is UBR instead of URF.
Therefore CT2 ≡ {(URF)+,(UBR)-}, to be precise. Nothing else is disturbed!
We have already studied a lot of algorithms. One point must be stressed: what is important is not the algorithms themselves, but the feeling of how they work, and the logic of how they are built.
With CM3 and CT2, you should have no trouble in settling all the corners and orienting them correctly.
Although CM3 and CT2 only operate on specified pieces, you can make them operate on any corner of your choice by:
Physical Movement
For example: look at the B face and call it F.
In this way, you can operate on any set of corners whose spatial relationship is the same as that in the algorithm.
For instance, you can flip any two adjacent corners using CT2 or CT2-1.
Conjugations
For example, to twist URF and ULB,
use L to move ULB to FLU,
where CT2 can operate, and use L-1 afterwards to complete the conjugation.
If you happen to have good memory, you can try to remember this commutator:
CM4 exchanges two pairs of corners. It exchanges the two white-green corners and the two orange-blue corners.
As you may have noticed, BLD is on the far side of the cube. It is not normally visible. To look at BLD, use the mouse the drag the cube in order to rotate it.
Without CM4, you can use two CM3s to do the same job (four times as many movements!).
CM43 exchanges two pairs of corners as CM4, and it does not disturb any other pieces.
According to the Eight Corners approach, we start with simple movements to settle three U corners, then use CM1 to settle the fourth. One particular situation may seem to be uncatered for. Frequently, you find a pair of corners swapped, but there is no algorithm to swap only two corners. If this happens, perform a single face movement on a face containing both corners. This will lead to either a configuration with 3 or 4 corners out of place, and they can be settled accordingly.
When the four U corners are in the correct position, turn the cube upside down and fix the location of the remaining four corners. Finally, twist the corners using CT2, as necessary.
Usually, we would select suitable algorithms (for example, choosing either CM1R or CM1F) when settling the first four corners so that their orientations are also correct. In this way, the number of CT2 execution required is reduced.
Continue with Moving Edges Part I
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