Improved method

Why this section?

So you have followed the method, tried the example, and even roughly know what you are trying to do, but it doesn't help? The basic method DOES work, but is a little unwieldy, so here are the different situations that you can meet and how to deal with them, plus a few videos of the moves.

The moves you need to know to make the method much easier

There are a number of different situations that you meet regularly with this method, and some of them can be done a little more easily with the addition of two new algorithms that greatly simplify the placement of the corners.

The old and new corner moving algorithms with videos

The original corner permuting algorithm given in my method is:

ALG 2 (video): corner permutation
	Ux(R2D' R2D2 R2D R2D' R2D2 R2D R2)Ux'
	Uy(R2D' R2D2 R2D R2D' R2D2 R2D R2)Uy'

but here are two simpler ones that can be used instead to simplify this task:

ALG C1 (video): 1 layer corner permutation on 3 or 4 pieces
	[(Uw)(LD2L')(Uw')-(Ux)(LD2L')(Ux')-(Uw)(LD2L')(Uw')]
	[(Uy)(LD2L')(Uy')-(Uz)(LD2L')(Uz')-(Uy)(LD2L')(Uy')]

ALG C2: (video): 2 layer corner permutation on 3 pieces
	Ux
	Dy(RB'R'BRB'R'BRB'R'B)Dy'
	Dz(RB'R'BRB'R'BRB'R'B)Dz'
	Ux'

Algorithm C1 is used when all the corners are in the U layer, and algorithm C2 is used when one corner is in the U layer and two are in the D layer. To cater for the cases when all are in the U layer or two are in the U layer whilst one is in the D layer you can either flip the cube through 180 degrees before doing the algorithm (which I would do for C1) or mirror through the middle layer (which I would do for C2).

When you move corners you either want to affect 4 corners, or you want to affect 3 corners. Additionally, all those corners are in the same layer or they are in two different layers. In my method the two layers are U and D.

How the algorithms function

ALG C1

This algorithm can be used to affect either 3 or 4 corners in the same layer.

The corner at FUL after Uw and Ux are swapped and the corner at FUL after Uy and Uz are swapped. In the case Uw, Ux, Uy and Uz are all different (U,U2,U'and none) it will swap four corners in the U layer. If only three of them are different then the algorithm only affects three corners and due to cancellation can be rewritten.

The full algorithm is

	[(Uw)(LD2L')(Uw')-(Ux)(LD2L')(Ux')-(Uw)(LD2L')(Uw')]
	[(Uy)(LD2L')(Uy')-(Uz)(LD2L')(Uz')-(Uy)(LD2L')(Uy')]

which when operating on 3 corners is actually

	[(Ux)(LD2L')(Ux')-(Uy)(LD2L')(Uy')-(Ux)(LD2L')(Ux')]
	[(Ux)(LD2L')(Ux')-(Uz)(LD2L')(Uz')-(Ux)(LD2L')(Ux')]

thus cancelling to

	[(Ux)(LD2L')(Ux')-(Uy)(LD2L')(Uy')]
	[(Uz)(LD2L')(Uz')-(Ux)(LD2L')(Ux')]

This algorithm uses FLU as the exchange point. It rotates (abc) by swapping a with b, and then swapping the piece that is in the original place that a was with c. What you do is rotate corner a to FUL using Ux and do the simple (LD2L'), you then move the piece you want to exchange with it (ie corner b) to FUL and do the move again, then rotate the corner slot, etc.

Notice the similarity between this and the corner rotating algorithm. Also, note that the whole algorithm is composed of 6 sections, but only the first two and last two are actually needed in the case where 3 corners are being cycled as the middle two sections cancel out. If you are swapping 2 distinct pairs then there won't be any cancelling available. Remember just because there is cancelling, you don't have to do it, you could do the whole algorithm, but what would be the point?

ALG C2

This works on one corner in U and two corners in D, and uses BUR/BDR as the exchange pair. What you do is rotate Ux at the beginnning so that the single corner is at BUR. Next rotate Dy so that the corner that the single corner moves to is at BDR. Do the simple sequence starting R (note that it is a four move sequence repeated three times), and then move the next corner on the D layer to BDR and do that simple sequence again. Finally reverse the D and U moves.

	Ux
	Dy(RB'R'BRB'R'BRB'R'B)Dy'
	Dz(RB'R'BRB'R'BRB'R'B)Dz'
	Ux'

Viola! You have done a 3-cycle which involves a corner from the U layer and two corners from the D layer. It is easy enough to adjust this so that you do two from the U layer instead. It is also easy enough to adjust this so that you don't need the preliminary Ux. Practise on a solved cube to get the hang of it.

Applying and comparing use of the new corner moving algorithms

Here are a couple of examples to contrast what you do using the algorithm in the basic method and what you do with these new algorithms.

A: corners all started in the U face and finish in the U face

a: a simple 3-cycle (234) of corners in the U-face

(R2D' R2D2 R2D R2D' R2D2 R2D R2) U(R2D' R2D2 R2D R2D' R2D2 R2D R2)U' which uses ALG1 from the method to do (234) as the pair swap (34) followed by the pair swap (23)
or you could use the Jessica last layer algorithm R'FR'B2RF'R'B2U2
or use this much easier sequence ALG C1 which does (234) as the pair swap (23) followed by the pair swap (24) using the FLU as the exchange point. [(U')(LD2L')(U)-(U2)(LD2L')(U2)-(U')(LD2L')(U)] [(U')(LD2L')(U)-(U)(LD2L')(U')-(U')(LD2L')(U)]
or more simply using ALG C1 with cancelling you can do [(U')(LD2L')(U)-(U2)(LD2L')(U2)-(U)(LD2L')(U')-(U')(LD2L')(U)]

b: a double pair swap (12)(34)

ALG 1 gives (R2D' R2D2 R2D R2D' R2D2 R2D R2) U2(R2D' R2D2 R2D R2D' R2D2 R2D R2)U2
you could use a Jessica last layer algorithm (move sequence omitted - you get the idea)
ALG C1 gives [()(LD2L')()-(U')(LD2L')(U2)-()(LD2L')()] [(U)(LD2L')(U')-(U2)(LD2L')(U2)-(U)(LD2L')(U)]

c: a simple 3-cycle (123) in the U-face

ALG 1 gives (R2D' R2D2 R2D R2D' R2D2 R2D R2) U'(R2D' R2D2 R2D R2D' R2D2 R2D R2)U
again you could use a Jessica last layer algorithm
ALG C1 gives [()(LD2L')()-(U)(LD2L')(U')-(U2)(LD2L')(U2)-()(LD2L')()] when you apply cancelling