Corner Movements

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.

You can choose any of the six faces to be your U. However, I like to choose white and yellow because they are bright colors. Remember: you have to look at the U face most of the time.

Moving the first four corners is rather trivial. Let's be formal and find an algorithm CM1 that moves RFD to RFU without disturbing other U pieces.

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Movement Dn moves one D corner to another, preserving the U face. Conjugating with R-1, we have: CM1R º R-1D-1R CM1R belongs to the class CM1. R is just a suffix to distinguish it from others of the same class. For easy viewing, the top two layers is already completed in the cube here. In reality, they aren't. When you are fixing the first corner, nothing is already in place. Not even the center facelets!

Concentrate on the piece at RFD: we want to move this red-white-green piece upwards to the position RFU. Also notice how the other three U (red) corners are preserved. Get the feeling of R^-1 dipping the RFU corner into the area affected by D. The essence is in finding a conjugate where only RFU (the target location), but not 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, concentration only on these five pieces), instead of always worrying about the whole cube.

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Symmetrically, we have CM1F º FDF-1 (= FDÅF-1, not (FDF)-1) doing a similar job. This time, the cube has only the other 3 U corners are colored for your viewing. This is more or less the actual situation when you are setting the fourth U corner.

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Since CM1R and CM1F are mirror images, they twist corners in opposite ways. Hence, we try CT1 º CM1R-1ÅCM1F (using the form X-1Y), it twists URF clockwise without moving it. Notice that all the other pieces on the U face are left unaffected. That is why we settle the position of the pieces before worrying about their orientation.

CM1 and CT1 affects only one piece on the U face. 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.

Any algorithm that affects only one piece on a face is called a mono-change. CM1 is a mono-move (moving a piece outside the face, replaced with another piece in its place), CT1 is a mono-twist (twisting a single corner without movement). All mono-changes form useful commutators with Uⁿ:

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CM3R º [CM1R,U] = CM1RÅUÅCM1R-1ÅU-1 CM3R moves RFD (the red-green-white cublet) to UFR. In other words, UFR (the white-green-cyan cubelet) is saved somewhere in the D face. U moves another piece (the red-green-yellow cubelet currently at URB) in place of UFR where the inverse of CM1 is going to take place. Therefore, the saved piece is restored to UFR, which will be brought back to URB by U-1. The net result is the corner movement: CM3R {(UFR, URB, RFD)} All other pieces are intact!

That is the beauty of commutators. Similarly:

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CT2 º [CT1,U] = CT1ÅUÅCT1-1ÅU-1 CT1 twists (RFU)+, but both lower slices are disturbed. CT1-1 restores the picture of all lower slices, and twists (RFU)-. The U and U-1 make the whole thing works: RFU is twisted clockwise, but the corner that is twisted anticlockwise is RBU instead of RFU. Therefore CT2 º {(RFU)+,(RBU)-}, 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 have no trouble in settling all the corners and orienting them correctly.

Although CM3 and CT2 only operate on specified pieces, you can operate them 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 flip URF and UBL, use L to move UBL to FUL, 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:

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CM4 º [R-1,D] = R-1D-1RD {(FRD,FUR)-, (RBD,BLD)+, other edge movements...} CM4 exchanges two pairs of corners. It exchanges the two white-green corners and the two orange=blue corners. Without CM4, you can use two CM3s to do the same job (four times as much movements!).

Moreover, the edge movements in CM4 has a period of three. Therefore:

exchanges two pairs of corners without affecting anything else, nice but hardly useful because we have not started to settle edges.

According to the Eight Corners approach, we start with simple movements to settle three U corners, use CM1 to settle the fourth. Then turn the cube upside down and use CM3 and CM4 to settle the four (now) U corners. Finally, CT2 is used to twist the corners.

One particular situation may seem to be uncatered for. Frequently, you find a pair of corners swapped, but there is no algorithm to swap two corners. When it appears, 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 can be settled accordingly.

Usually, we would select suitable algorithms (for example, choosing among CM1R or CM1F) when settling the first four corners so that their orientation are also correct. In this way, the number of CT2 required is reduced.