A2

A2 is a modular speedsolving method for 2x2x2. The eight corners are reduced to four pairs then the pairs are placed into position. It is a modular method in that, for the first step, new pair types can continue be added to the speedsolver’s knowledge base. A major advantage is that if the speedsolver already knows CLL, EG, or any other applicable set, there are no new algorithms to learn.
Contents
Steps
1. First Two Pairs (F2P): In this step, two pairs are constructed and placed on the D layer. These pairs, and even the individual corners with them, do not need to be correctly permuted or oriented and the corners that make up the pairs do not need to all be from the D layer. These pairs can be a mixture of corners from all around the cube.
2. Last Two Pairs (L2P): This step completes the four pairs and places them all into position. The last four corners on the U layer are solved in a way that forms two complete pairs. At the same time, the pairs on the D layer are also completely formed.
History
In 2010, James Straughan discovered an NMCLL recognition method that works for speedsolving. Then in 2012, he completed a relationship table showing how CLL cases are transformed after a turn is performed when there is an oriented corner on the U layer[1]. These two creations were later combined into a 2x2x2 method making use of matching and nonmatching pairs. These pairs were set up in CLL and EG form. In 2020, work was completed on expanding options for how the first two pairs can be formed. The name A2 is inspired by Thom Barlow’s K4 method for 4x4x4.
First Two Pairs
This is a general technique of forming the first four corners, a way of thinking about the those corners as two pairs. Really A2 is the application of pseudo solving in combination with Transformation to the first four corners. The pair combination technique doesn’t have to be followed. The solver could simply put together the first four corners however is desired. But the two pair technique is recommended because it helps the speedsolver to understand and quickly solve the first step. It isn't necessary to memorize the table. Once it is understood how the pairs connect, the solver can easily locate the next two corners. There will already be formed pairs in every scramble. A good progression is to first learn the pair combinations marked No Swap, then move on to Adjacent and Opposite swaps, and, finally, the more advanced forms that contain twisted corners. No Swap is the equivalent of a nonmatching CLL solve and Adjacent and Opposite swaps are the equivalent of nonmatching EG. The more advanced pair forms are in later sections.
Layer/Face (CLL/EG)
A useful technique is to rotate the cube when looking at a pair. In a couple of cases the pairs will become another pair from the table. Pair 7 turns into pair 13 and pair 10 turns into pair 14. The other pairs when flipped either turn into themselves or into a twisted corners case. In every case, flipping the pair also allows for more options for the second pair.
Twisted Corner (TCLL)
This set of pair options makes use of a single misoriented corner, the equivalent of the TCLL method.
Again, flip the pair and it turns into a different pair to give more options for the next pair.
Separated Face (SCLL)
Another option for further expanding the types of pairs one can use is separation. This is like the NMLL last layer method in that the corners on the D layer will be oriented relative to the left and right side of the cube. The left and right side will contain only orange and red for example.
Forming the First Two Pairs
One Color First Pair
 If the first pair has two of the same colors on the D layer:
 If the first pair contains another two colors on the side that are the same, the next pair will also contain two sets of matching colors. One set is a color that will go on the D layer. The other set is the opposite color of the two other matching colors on the side of the first pair.
 If the first pair doesn’t contain two other colors on the side that are the same, then you are simply going to find the other two colors that belong on the D layer.
Two Color First Pair
 If the first pair has two different colors on the D layer, the other pair the solver finds or creates will also contain only these two colors on the D layer. Both of these two colors have opposite colors on the cube. These colors will be avoided when adding the next pair.
 Among the corner choices for the second pair, choose the two corners that have the same colors as the D layer and also have the color that matches the nonopposite color mentioned above.
Just as the solver can easily recognize matching and opposite colors for a CLL layer or an EG face, so too will the solver eventually recognize pair types. It isn’t even necessary to rigidly follow the first pair step. Really the goal is to view the two pairs as a complete form, to not create one pair then create another. It is best to plan and build them together as a whole concept. The two pair combinations are a guide towards accomplishing this.
Last Two Pairs
This step is the same as solving the last four corners in CLL, EG, or any similar method. The NMCLL recognition method has been developed just for this. If a solver has already learned a recognition method and doesn't want to learn the NMCLL recogntion, the standard recognition can be used. There are only two small differences. The U colors and the F/B color recognition depends on the first two pairs. If the D face colors are opposite, then the recognition for nonU colors will be opposite from usual. If the D face colors are neither the same nor opposite, then recognition will correspond with the colors on L/R/F/B.
Below is the normal Sune case. The version on the right has dots to show the U color orientation. The lines show the recognition that will be used for explanation.
Sune with opposite face colors:
Now the bottom layer has two opposite colors. That means the U layer does too and that those will be the U orientation colors (white and yellow). The first of the two images above has the same recognition as the normal Sune. Blue and green are opposite and red and orange are opposite. In the second set of images, the U orientation colors are still white and yellow but the other colors are a little different. Notice that the FDL and FRD colors are opposite because the left and right layer are an R2 away from solved. The line on the image shows red and red. Those are opposite because the two reds at FDL and BDR are opposite from each other. Blue and green will always remain as normal because they are on the left and right side. Top Scramble: (R2) R U2 R' U' R U' R'. Bottom Scramble: (R2) U R U2 R' U' R U' R'.
The images above show Sune with neither matching nor opposite colors on the bottom layer, yellow and orange. The U orientation colors will be the opposite of those two, white and red. More specifically, the U orientation colors will be the two corners with white+green and the two corners with red+blue. The first of the images above shows the orientation with red+blue at UFL, white+green at FUR, white+green at RUB, and red+blue at BUL. Recognition in this case is the same as normal Sune. The second image shows the orientation with white+green at UFL, white+green at FUR, red+blue at RUB, and red+blue at BUL. The first line on the image shows white and red. These two colors are part of the F/B colors at FDL (red), FRD (yellow), BDR (white), and BLD (orange). It will be pretended that red and yellow are the same and white and orange are the same and that those two sets are the opposite of each other. So, the two lines in the image show the opposites white+red and blue+green. Top Scramble: (R) R U2 R' U' R U' R'. Bottom Scramble: (R) U R U2 R' U' R U' R'.
Another big advantage to this step is that transformation can be applied to the algorithms. Meaning, if there is an oriented corner on the U layer, an F/B/L/R or even a simple rotation will change the corner case to a different one. This means that the number of cases in large algorithm sets such as 1LLS can be greatly reduced. Taking advantage of this opportunity can also provide the benefit of fewer moves and the use of more comfortable algorithms. Misoriented corners can be used in this same way, further reducing the effort needed in a solve. This transformation can even be applied in the middle of some algorithms and at the end of the solve for cancellations.
Examples
Layer
 Scramble: F2 U R' F R' U R2 U' F2
 F2P: x U' F'
 L2P: U F R2 U' R2 U' R2 U R2 F' (U' R')
Face
 Scramble: R' U2 R2 F2 U' R F' R F U
 F2P: z y' U L2
 L2P: R U R' U F R U' R2 F' (Normally this algorithm has an R at the end, but it cancels out thanks to the nonmatching corners)
Face
 Scramble: R' U F' U2 F2 U' R2 U' F U'
 F2P: y z U L
 L2P: U' R U R' U R2 U' R2 F R2 F' (U2 R')
Face
 Scramble: F' R2 F U' R F' U R' U'
 F2P: y z'
 L2P: F' R' U' F2 R U' F R' (U' R2)
Twisted Corner (TCLL +)
 Scramble: R2 U' R' U' R2 U F2 U R'
 F2P: z y U' F2 (misoriented corner is at DFR)
 L2P: U2 R U' R2 F R F' (U' F)
3/4 Layer (1LLS)
 Scramble: F2 U' R2 U F' U2 F R' F2
 V: y2 B2 R'
 L3P: R2 U' R2 U R' U2 R U2 (End of first step cancels with beginning of L3P)
3/4 Layer (1LLS with transformation)
 Scramble: R2 F' U R2 U R U2 F' R2
 V: x y2 F'  Creates a pair and places a pseudo corner at DBR. DFR is the corner used for transformation.
 L3P: R2 F R U2 R U' R' U2 F' R (U' R U')
Opposite twisted pairs (ODCLL)
 Scramble: F2 R2 U R' F2 U R2 F' R' U2
 F2P: x'  The two pairs on the D layer belong to opposite layers and aren't oriented relative to each other.
 L2P: U2 F2 R U' R2 F' R2 U' R' U2
Advancement
Additional F2P Types
Because A2 is a modular method, any possible pair formation can be built into the A2 method. In this way, a new pair type wouldn’t be an A2 method variant. It would be considered an A2 addon. Users can continue to add to the types of pairs that they are able to create, TEG (twisty EG) or more than one twisted corner for example, expanding options. Though the further one goes, the more algorithms are needed to be learned and, possibly, the longer the final adjustments become.
The pairs above all follow a rule of having L/R colors that are opposite. It is possible to violate this rule and use an advanced pair type that eventually needs to be both permuted and oriented during the last two pairs algorithm. Example scramble: R U R' U R U R2 U R.
This post details every possible pair type that can be used in A2.
Alternate F2P Base
The two pairs don’t need to be formed together on the D layer. One pair could be at DL and the other at BR in an A2R variant or one pair at DL and the other at UR in an A2R2 variant, often saving a single turn. These would require different algorithms. A further advancement would be to have one pair of any type and another single corner on the D layer and solve the last five corners and the three corners on the D layer using one look last slot. The number of algorithms wouldn't increase from normal CLL, EG, SCLL, or any other method because this would be the application of Transformation as shown in this post. This concept and the same 2x2 method was rediscovered in 2017 by Joseph Briggs and is discussed in the "as a 2x2x2 method" section on the 42 method wiki page[2].