Bind the second to last PAIR using PMs
Instructions for use
For the convenience of explanation, use 
to represent the corner 
that does not consider the twisting orientation, and use 
to represent the corner 
that does not consider the twisting orientation, and use 
to represent the edge 
that does not consider the twisting orientation.
Here we use grey PAIR 
to represents bound PAIR, and DarkSlateBlue PAIR 
to represents Move PAIR.
The location of the Move PAIR does not mean that there must be a PAIR there, but it must include all the bound PAIRs and cannot be dismantled.
The color on a cubie is dark, it means that the cubie is covered by a Move PAIR.
Preparation
Here F face is orange or red, U face is yellow or white, and R face is blue or green. EO has been done.
As shown on the left, the entire 2x2x3 black area is placed at the position of FBLD, and the grey PAIR represents the bound PAIR and that set of Parallel bound PAIR is placed on the layer R.
These 4 corners 
and the edge do not belong to any bound PAIRs and are placed on the R, U layer.
In addition, the edge 
and the edge 
do not belong to any bound PAIRs.
The relative positions of the corner and the corner on the U layer are as shown on the right.
The following discussion on this page is only limited to the R and U layers. All rotations are only discussed in the rotations limited to R and U.
Goal
Just rotate R,U and make the two corner "Same Orientation" while simultaneously turning the corner and the edge into a PAIR.
This core skill I call “SLP” (binding the Second-to-Last PAIR) and give it the nickname “catching the bird in mid-air”.
Static representation of a PM
The "Static Representation" of an IPM is using an illustration of the last figure before breaking a set of Parallel Move PAIRs.
The "Static Representation" of a PTM is using an illustration of the last figure before forming a new set of Parallel Move PAIRs.
We use gold cubes to indicate that this figure represents a PM. Here are some examples to help you understand.
RU-mirror
Use the cube with only U and R layers to create plane mirror symmetry. The mirror plane is as shown on the left, which is called "RU-mirror".
Regardless of the twist orientation, any position and rotation on the U layer will have a RU-mirror on the R layer, and vice versa. As in the example on the right, the two moves are RU-mirror of each other.
A set of Parallel bound PAIRs is initially placed on the R layer, as shown in the left.
We use the dark cyan cube to indicate all positions that will do R,U rotation without breaking that set of Parallel bound PAIRs, as well as including their RU-mirrors.
For a PM whose Move PAIRs contains that set of Parallel bound PAIRs on the R layer, we can get its Static Representation, but it no longer indicates which Move PAIR is the bound PAIR. The RU-mirror of that PM also uses that same illustration. And we use cyan cubes to indicate that this figure represents a PM and its RU-mirror.
Here are some examples to help you understand.
Some positions of
can be shown on the right.
In addition,
or
or
Connected or disconnected
The corner and the edge are said to be connected if their relative positions are as shown on the left.
Otherwise, The corner and the edge are said to be disconnected.
On this page, when the word "disconnected" is mentioned, it means that the corner and the edge to be disconnected.
Similarly, if we only mention the word "connected", it means that the corner and the edge to be connected.
Mirror or Tradtional
There is a move that initially has a Parallel bound PAIR on the R layer, but when it ends, the same Parallel bound PAIR appears on the U layer, this move is called a "Mirror Move". One example is shown on the left.
If there is a move that initially has a Parallel bound PAIR on layer R, but the same Parallel bound PAIR still appears on layer R at the end, this move is called a "Traditional Move". One example is shown on the right.
Moreover, after the above-mentioned layer R and layer U are interchanged, this move also uses the same name.
How to use two PMs to perform a Mirror Move or a Traditional Move?
Assume that the two bound PAIRs are PAIR A and PAIR B. That two bound PAIRs initially form a set of Parallel PAIRs and are on the R layer.
After the first PM, PAIR A has its Parallel Area on the R layer, while PAIR B has its Parallel Area on the U layer.
- The second PM is an IPM. If you do an IPM to interchange PAIR A, you will get a Mirror Move. If you do an IPM to interchange PAIR B, you will get a Traditional Move.
- The second PM is a PTM. If the Parallel Move PAIR for the PTM at the beginning consists of PAIR A, and the single Move PAIR is PAIR B, you will get a Mirror Move. If the Parallel Move PAIR for the PTM at the beginning consists PAIR B, and the single PAIR is PAIR A, you will get a Traditional Move.
Moreover, the above skills are also feasible after the U layer and R layer are interchanged.
A move within two PMs
Here is an example as shown below to illustrate the move within two PM and the timing of use.
mro number is the relative twist orientation of two corners used for PM's operation, which can be +1, 0, −1. This move can only be used when assigned the same number as this.
If mro number is blank, it means that this move does not consider the relative twist orientation of that two corners.
The entire move is composed of PM and cannot be equated to no turn at all, and the first PM is the PM shown in the illustration.
- If nIPM is blank, it means that the move is only the PM shown in the illustration.
- If nIPM is not blank, it means that this move is composed of two PMs, and the second PM does necessarily the same Move PAIR as the first PM.
- 0IPM means that both PMs are PTMs.
- 1IPM means that one of the two PMs is IPM, that is, the entire move is PTM+IPM or IPM+PTM.
- 2IPM means that both PMs are IPMs.
All the bind new PAIR before the Parallel bound PAIRs appears again are listed here, call it the "Fracture List", which can be divided into three categories: mdtA, mdtB, mdtC.
Fracture List
| List of mdtA | |
|---|---|
|
+1
0
1IPM |
|
|
+1
|
|
+1
0
1IPM |
|
|
+1
0IPM 
0
1IPM 
−1
1IPM |
|
|
+1
2IPM 
−1
1IPM |
| List of mdtB | |
|---|---|
|
|
+1
1IPM 
0
0IPM
−1
1IPM |
|
|
0
−1
|
|
|
0
−1
|
|
|
0
1IPM MT exc. |
|
|
+1
1IPM |
| List of mdtC | |
|---|---|
|
|
+1
2IPM 
0
|
|
|
0
0IPM |
|
|
+1
2IPM
0
2IPM MT exc. |
|
|
0
0IPM |
|
|
+1
2IPM |
Twin Mirror Moves for transformation
One twin "Mirror Moves" PTM + PTM for transformation is provided here.
Name each of the following twin Mirror Moves for ease of use
Nikki
Nikki'
Nikki and Nikki' are collectively referred to as "Nikki Twins".
Here we specify the colors at some positions to facilitate the use of the table below.
If the edge is at position 
, either Nikki or Nikki' will cause the corner and the edge to be disconnected.
If the edge is at position 
, either Nikki or Nikki' will cause the corner and the edge to be connected.
If the edge is at position 
, then Nikki will cause the corner and the edge to be disconnected, and Nikki' will cause that corner and that edge to be connected
If the edge is at position 
, then Nikki will cause the corner and the edge to be connected, and Nikki' will cause that corner and that edge to be disconnected
|
Name of
U layer initial |
U layer initial of
Nikki |
U layer initial of
Nikki' |
Effect of
Fracture List and mro number |
connected or
disconnected after the mirror moves |
|---|---|---|---|---|
| Ni0 |
|
|
mdtB ↔ mdtC
+1 ↔ −1 |
|
| Ni1 |
|
|
||
| Ni2 |
|
|
mdtC ↔ mdtA
+1 ↔ 0 |
|
| Ni3 |
|
|
mdtA ↔ mdtB
+1 ↔ −1 |
|
Position and position are collectively referred to as "DC" position.
Position and position are collectively referred to as "SC" position.
Examples of Nikki Twins
Nikki of Ni0
Nikki' of Ni3
One more twin Mirror Moves for transformation
Provide another twin Mirror Move with IPM in the second PM.
Name each of the following twin Mirror Moves for ease of use
Tine
Tine'
Tine and Tine' are collectively referred to as "Tine Twins".
Here are just some of the results you'll need
| Mirror Move |
U layer initial
called "Ti2" |
Effect of
Fracture List and mro number |
|---|---|---|
| Tine |
|
mdtA ↔ mdtB
+1 ↔ −1 |
| Tine' |
|
mdtB ↔ mdtC
+1 ↔ −1 |
It's enough to tell you what the results are when the initial position is shown on the left.
The result is that either Tine or Tine' will cause the corner and the edge to be disconnected.
Although it is Tine Twins, not Nikki Twins, it is still said here that it belongs to the “SC” position.
Examples of Tine Twins
Tine of Ti2
Tine' of Ti2
How to use?
Of course, the following is for when the goal has not been achieved.
On the U layer, observe the facelet of the corner on Orientation Windows, as shown in pink on the right.
We can get the mro number.
Here we use the symbol τ to represent the mro number we used at the beginning.
According to the facelet of the corner on Orientation Windows.
- If it is green, Fracture List use mdtA.
- If it is yellow, Fracture List use mdtB.
- If it is red, Fracture List use mdtC.
Here we use the mdt0 to represent which Fracture List we used at the beginning.
In mdt0 list, based on the relative positions of corner and edge , as well as the number τ, find a matching situation.
There is a matching situation in mdt0 list
The first PM can be done directly according to the PM represented by the Static Presentation of that situation.
If the entire move consists of two PMs, i.e., nIPM is not blank, then the entire move will be either a Mirror Move or a Traditional Move, and the following rules are to be met.
- When nIPM is 0IPM, the entire move needs to be a Mirror Move.
- When nIPM is 1IPM, the entire move needs to be a Traditional Move, except for one particular case, which needs to be a Mirror Move. And that particular case is marked with “MT exc.”.
- When nIPM is 2IPM, the entire move needs to be a Mirror Move, except for one particular case, which needs to be a Traditional Move. And that particular case is marked with “MT exc.”.
When the second PM is an IPM, do that IPM on which layer?
If that edge is between two Parallel Move PAIRs for IPM on the U layer as shown on the right, do that IPM on U.
If that edge is not covered by any Move PAIRs for IPM and not between any two Parallel Move PAIRs for IPM, as shown on the left. If the bound PAIR to be interchanged belongs to the Parallel Move PAIR on the U layer, then do that IPM on U.
Moreover, the above skills are similar for the R layer.
There is no matching situation in mdt0 list
The twin Mirror Moves provided above are only used when your situation is not in mdt0 list.
According to the rules given below, only use U face rotation to determine the initial position of the corner on the U layer. After using one of the Mirror Moves provided above, and you will get a cube that appears in the Fracture List.
Q. How to choose which twin Mirror Move to use?
A. With one exceptions,
- if mdt0 is mdtC and τ is −1, then use the Nikki Twins.
-
Otherwise,
- If it is as shown on the right, use Ti2 of the Tine Twins.
- Otherwise, use the Nikki Twins.
The one exception is that if it is as shown on the right and mdt0 is mdtB and τ is +1, use the Nikki Twins.
Q. How to determine the initial position of corner on the U layer using only U face rotation?
A. With three exceptions,
- if the number τ is −1, then the selected position of that corner will make Fracture List using mdtA.
- If the number τ is +1, then the selected position of that corner will make Fracture List using mdtB.
- If the number τ is 0, then the selected position of that corner will make the mro number still 0 and Fracture List not using mdtA.
- Among Ni0 and Ni1, select Ni0.
The three exceptions are listed below. The below are all mdt0 is mdtC and τ is −1.
these two use Ni0 directly.
use Ni1 directly.
When using the Tine Twins, you will only use it when the edge is at the “SC” position, and it will be usable to you after you have selected it in accordance with the rules in the above paragraph.
Q. When using the Nikki Twins, After determining the initial position of corner on the U layer, which one should I select between the Nikki Twins?
-
A. When the edge is at the "SC" position, either Nikki or Nikki' can be selected.
-
Otherwise, the edge is at the "DC" position
-
if mdt0 is mdtC and τ is −1, select the Mirror Move that will cause the corner and the edge to be connected.
-
Otherwise, select the Mirror Move that will cause the corner and the edge to be disconnected.
-
if mdt0 is mdtC and τ is −1, select the Mirror Move that will cause the corner
Q. What will be the nIPM in Fracture List of the resulting cube after applying the above rules?
- A. if mdt0 is mdtC and τ is −1, nIPM will be 1.
- Otherwise, nIPM will be blank or 0.
Some examples
it is mdtA and mro number is +1.
You can find it in Fracture List,
the entire move is just one IPM.
it is mdtB and mro number is −1.
You can find it in Fracture List,
the entire move is PTM + IPM,
and let it be a Traditional Move.
it is mdtB and mro number is −1, not in Fracture List.
To become mdtA, use Ni3.
To become disconnected, select Nikki in Ni3.
After Nikki of Ni3, mdtB → mdtA, and −1 → +1,
then you can find it in the Fracture List. The nIPM is blank.
it is mdtC and mro number is −1, not in Fracture List.
This special case use Ni0 directly.
To become connected, select Nikki in Ni0
After Nikki of Ni0, mdtC → mdtB, and −1 → +1.
then you can find it in the Fracture List. The nIPM is 1IPM.
it is mdtC and mro number is +1, not in Fracture List.
To become mdtB, use Ti2 of Tine'.
After Ti2 of Tine', mdtC → mdtB, and +1 → −1,
then you can find it in the Fracture List. The nIPM is blank.
Predicting which row in the Fracture List
When you do one Mirror Move as above, you take a cube that was not on the Fracture List and make it fall into the Fracture List.
We want to predict which row it will fall in the Fracture List. This reduces the number of observations.
If mdt0 is mdtC and τ is −1
The results for the row in the Fracture List will be shown on the left, with one exception.
become
That one exception is the change shown on the right.
Otherwise
It is assumed that the results are disconnected, because if the results are connected, there will be only one row available in the fracture list, and there is no need to make a prediction.
-
If the edge
is initially at the “DC” position.
The results for the row in the Fracture List will be shown on the left, with one exception.
That one exception is that if τ is +1, the result for the row in the Fracture List is shown on the right.
-
If the edge
is initially at the “SC” position, and the results are disconnected.
If the Nikki Twins is used, the results will be the same for either Nikki or Nikki', and the results for the row in the Fracture List will be the ones shown on the left, regardless of whether the Nikki Twins or a Tine Twins is used, with two exception.
The two exceptions are listed below.
-
become
Whether you use Nikki or Nikki', you will see the same change as shown on the right.
-
If the mro number is 0 and the Nikki Twins is used, the result of using Nikki and using Nikki' will not be the same.
Using Nikki will give you
Using Nikki' will give you
-