PPW2 method

Way of explanation

The Base 4PAIRs are placed in the shape as shown on the left, where the silver center represents the F face, and the black center represents the U face.

For the convenience of explanation, use to represent Hub Corner that does not consider the twisting orientation, and use to represent Dog Tail Corner that does not consider the twisting orientation.

The so-called two colors are "Opposite Color", which means that the colors of the center that is opposite to each other, that is,  ,   are "Opposite Color",  ,  are "Opposite Color",  ,  are "Opposite Color".

"Similarity and difference of permutation parity" means "the permutation parity of the three corners, the Dog Tail Corner and the Remaining 2Corners, is different or the same as the permutation parity of the Basic 4PAIRs".

"The position of Dog Tail Corner" means "the relative position of Dog Tail Corner and Hub Corner".

Orientation Windows is only used for the other three corners of the non-Hub Corner on the U layer. Use the color on the Orientation Windows to observe the twist orientation of the corner, that is, the magenta observation facelet as shown on the left, and there is no need to pay attention to the twist orientation of the Hub Corner.

"The color of the corner" means "the color of the corner on Orientation Windows".

Goal

• The similarities and differences of permutation parity are the same.
• The color of the Dog Tail Corner on Orientation Windows is red color, and the colors of the Remaining 2Corners on Orientation Windows are Opposite Color or the same color.
• The position of the Dog Tail Corner is as shown on the right.

The meaning of the symbol used in the corners on the U layer

This method observes the twisting orientation of the corners on the U layer, as shown on the left, based on the magenta Orientation Windows used in the non-Hub Corner on the U layer.

The arrow symbol as shown on the left not only represents the change of the position of the corner,
but also represents that the same color is maintained on Orientation Windows after the position of the corner is changed.

→

The red facelet on the corner shown on the right demonstrates how the arrow symbol makes a corner change.

The arrow symbol and the twist symbol as shown on the left are in the same corner position,
which means the corner that was originally in that position not only applies the rule of the arrow symbol, but also applies the original twist symbol rule at that position for the same corner.

→

Shown on the right demonstrates how the arrow symbol and the twist symbol in the same position make a corner change.

A series of Interchaging PAIRs Move

Here is a list of the different positions of Hub Corner on the U layer, and the influence of Interchange PAIRs Move (if you don't understand the term, try one of the move sequences R' D R2 D' R directly). As for the position change of the edge, that is, the influence of Interchange PAIRs Kernel on the edge

Name	Do U face rotation first to make the position of Hub Corner as shown below, then do Interchange PAIRs Move	After Interchange PAIRs Move, whether to change the similarities and differences of permutation parity	After the Interchange PAIRs Move, the change of the corners on the U layer
oi1		Change
oi2		Change
oi3		No change
oi4		No change

Move seqence only include the U turn and the R turn do not change the position of each corner and the position of the Basic 4PAIRs in the end, so this series of move sequences will of course not change the similarities and differences of permutation parity and this series of move seqences can also be regarded as the same pair in the R, U layers do two times Interchange PAIRs Move with different empty pairs to return to the original position. Only two of them are listed here.

Name	Move sequence	The twist change of the corner on the U layer. The position change of the edge.
ei1	R' U' R2 U R2 U' R' U
ei2	R U R2 U' R2' U R U'

Observe the color on Orientation Windows to twist the corners in the same orientation

The above moves are all twisting the three corners in the same orientation on the U layer. You can make good use of the knowledge of twisting the three corners in the same orientation below, and observe with the color on Orientation Windows.
Don't pay attention to the color of Hub Corner in this step.
On the U layer, only Dog Tail corner will have red color and corners with orange color must not be Dog Tail Corner.

There are 4 corners on the U layer, if three of them are twisted in the same orientation, excluding the corner α, as shown on the right where is the corner α. It is referred to as "Same Orientation Twist" here, and it is more precisely called "Same Orientation Twist Excluding the corner α".

The color of Dog Tail Corner is red color, and the color of Remaining 2Corners are Opposite Color or the same color is the completed state.
Here, only do Same Orientation Twist each time to reach the minimum number of times to complete the state, to classify all the twist orientations of corners on the U layer, and divide them into "Completed State", "1 Time Can Be Completed", "Within 1 Time Can Not Be Completed" these three categories. Collectively referred to as "Category of Orientation Times".

Discrimination method of "Category of Orientation Times"

• "Completed State" if and only if there is red color on Orientation Windows, and the colors of the other two facelets are Opposite Color or the same color.
• "1 Time Can Be Completed" if and only if there is no red color on Orientation Windows.
• "Within 1 Time Can Not Be Completed" if and only if there is red color on Orientation Windows, and the colors of the other two facelets are not Opposite Color or the same color.

Under "1 Time Can Be Completed", in the process of doing Same Orientation Twist to Complete State, one corner can be completely untwisted, then this corner is called "No Twist Required Corner".

→

For example of "1 Time Can Be Completed" as shown on the right, is the "No Twist Required Corner".

Under "1 Time Completion", look for the "No Twist Required Corner"

• No orange color on Orientation Windows.
  • There are two facelets on the Orientation Windows that are Opposite Color or the same color, and both belong to Remaining 2Corners, then the Hub Corner is the "No Twist Required Corner"
  • There are only two facelets on the Orientation Windows that are Opposite Color or the same color, and the remaining facelet belong to Remaining 2Corners, then the corner to which this facelet belongs is the "No Twist Required Corner"
• Only one facelet of Orientation Windows is orange color.
  • If the other two facelets on Orientation Windows are not Opposite Color or the same color, the corner with orange color on Orientation Windows is the "No Twist Required Corner".
  • If the other two facelets on Orientation Windows are Opposite Color or the same color, then the corner belonging to Remaining 2Corners among those two corners is the "No Twist Required Corner".
• There are two facelets with orange color on Orientation Windows, then the Hub Corner is the "No Twist Required Corner".

"Category of Orientation Times" for three twist results of Same Orientation Twist

For the four corners of the U layer, select one of the corners, which is corner α, and repeat "Same Orientation Twist Excluding the corner α" to get three twist results

• If the corner α is Dog Tail Corner, the "Category of Orientation Times" of that three twist results would be all the same. If the "Category of Orientation Times" is "1 Time Can Be Completed", even "No Twist Required Corner" would be the same corner.
• If the corner α is not the Dog Tail Corner, and in addition to the corner α and the Dog Tail Corner, the other two corners on the U layer are the corner β and the corner γ. The "Category of Orientation Times" of that three twist results will only be one of the following two
  • "Completed State", "1 Time Can Be Completed" and the No Twist Required Corner is the corner α, "1 Time Can Be Completed" and the No Twist Required Corner is the corner α.
  • "Within 1 Time Can Not Be Completed", "1 Time Can Be Completed" and the No Twist Required Corner is the corner β, "1 Time Can Be Completed" and the No Twist Required Corner is the corner γ.

Doing Same Orientation Twist under "Within 1 Time Can Not Be Completed"

The Remaining 2Corners are corner α, corner β, and the color of these two corners are not Opposite Color or the same color. If (the color of corner α, the color of corner β) is one of (  or  ,   or  ), (  or  ,  ), ( ,   or  ), then it is called that corner β is Twisted Clockwise Than corner α.

Furthermore, adding that the color of Dog Tail Corner is red, then "Category of Orientation Times" will be "Within 1 Time Can Not Be Completed".
Only by doing a counterclockwise "Same Orientation Twist Excluding the corner α" or clockwise "Same Orientation Twist Excluding the corner β", can the later "No Twist Required Corner" be Hub Corner.
If you do counterclockwise "Same Orientation Twist Excluding the Hub Corner", "No Twist Required Corner" will be the corner β. If you do clockwise "Same Orientation Twist Excluding the Hub Corner", "No Twist Required Corner" will be the corner α.

How to use

The above moves show that the similarities and differences of permutation parity will be changed together with the twist orientation of the corners and the position of Dog Tail Corner. Without changing the similarities and differences of permutation parity, the twist orientation of the corners and the position movement of Dog Tail Corner can be done independently. Therefore, the basic elementary level and the most inflexible approach can be

Stage 1

Use the move oi1 or oi2 when the similarities and differences of permutation parity are different.

Stage 2

Use the move ei1 or ei2 to adjust the color of Dog Tail Corner and Remaining 2Corners to be correct.

Stage 3

Use the move oi3 or oi4 to adjust the position of Dog Tail Corner to be correct.

Therefore, only the moves listed above are sufficient if they are only required to be solved.

A more advanced approach is

Stage 1

Use following more advanced approach when the similarities and differences of permutation parity are different.

Stage 2

Use the move ei1 or ei2 with No Twist Required Corner to adjust the color of Dog Tail Corner and Remaining 2Corners to be correct.

Stage 3

Use the move oi3 or oi4 to adjust the position of Dog Tail Corner to be correct.

More advanced approach when the similarities and differences of permutation parity are different

For the convenience of explanation, use to represent No Twist Required Corner, and use to represent a corner that is Hub Corner and at the same time also No Twist Required Corner without considering the twisting orientation.

Using "Category of Orientation Times" of the twist orientations of corners on the U layer

After doing the move below, "Category of Orientation Times" of the twist orientations of corners on the U layer will be "1 Time Can Be Completed" or "Completed State". And you can use the above knowledge to know in advance which corner is No Twist Required Corner.

Completed State

Do the move oi3 or oi4 to make Dog Tail Corner as shown on the right, then do the move oi1.

Within 1 Time Can Not Be Completed

Do the move oi3 or oi4 to make Dog Tail Corner as shown on the right, then do the move oi2.

1 Time Can Be Completed

	Move
	oi1
	oi2
	oi2
	oi1 + oi3
	oi3 + oi2

	Move
	oi1
	oi2
	oi1
	oi4 + oi1
	oi4 + oi1
	oi3 + oi2

Special case

If the similarities and differences of permutation parity are the same, and "Category of Orientation Times" of the twist orientations of corners on the U layer is "Within 1 Time Can Not Be Completed", here is a faster skill.

Stage 1

After doing the move below, "Category of Orientation Times" of the twist orientations of corners on the U layer will be "1 Time Can Be Completed". And you can use the above knowledge to know in advance which corner is No Twist Required Corner.

Stage 2

Use the move ei1 or ei2 with No Twist Required Corner to adjust the color of Dog Tail Corner and Remaining 2Corners to be correct.

	Move	After the move, No Twist Required Corner will have the same result as the following Same Orientation Twist
	oi2 + oi2
	oi1 + oi1