ATCRM

Athefre and tsmosher's Corner Recognition Method (ATCRM) is a universal recognition method for the corners of the last layer. ATCRM can be used to recognize CxLL, NMCLL, CCLL (Conjugated CxLL), and even further into pseudo solving if the user wishes. The basic steps are to first find the orientation of the stickers that belong on the left and right side of the cube then check two pre-determined sticker positions. In a normal solve with matching blocks, the L/R or F/B sticker orientation can be used. Then continue with checking two pre-determined sticker positions. In the two sticker step it is also possible to use the four sticker system from the older NMCLL recognition instead. However, it is recommended to use the two sticker version for speedsolving. Tim Mosher (tsmosher) has created an amazing document that provides algorithms for the cases, an explanation of the recognition, and examples. Click the link below to go to the document and get started learning. Ryan Hudgens (OreKehStrah) has developed a document for OH use. That is also linked below.

View 2H document

View OH document 

You are free to use a different two predetermined stickers to check for each orientation and it would still be ATCRM. If you prefer to check other locations, then change it for any desired cases. The stickers chosen for the document have been optimized to be two that are on the U layer for each orientation where that is possible. Most orientations have the two stickers on the U layer and there aren't many that don't. This means that the two sticker step can be recognized from multiple angles in almost every case. For those few cases where one or both stickers are on the sides, another angle can be learned and it is simple because of the symmetry. If you want to change up a case or few from the document, always choose any one sticker from the two corners that belong on the left side and any one sticker from the two corners that belong on the right.

This page also describes in more detail why this recognition method works. The various levels of pseudo to which the method applies are detailed. If you are just wanting to learn the recognition for speedsolving then I recommend learning from the document linked above. I recommend first learning the recognition method before studying this page to understand what is happening in each step.

In 2006 when I started using Roux, I found the mention of non-matching blocks on Gilles' website. He had a recognition method based on first recognizing the L/R stickers then finding the stickers that should be on the U layer, but the table was incomplete. I completed the table and it was then added to his site. My interest in pseudo continued and so I started using non-matching blocks in Roux. Around that time I also had the realization that xy2 neutrality would probably be best for non-matching blocks. This means solving any of the eight first blocks from the left or right side. Other Roux users at the time were x2y2 neutral. This was at a time when the few people using Roux were solving with white and yellow on the left and right, myself included. It wasn't until a few years later that a guide by Waffo taught with white and yellow on the top and bottom and the Roux community eventually shifted to x2y neutrality.

In 2010 I found a better way of recognizing non-matching corners that involved checking pre-determined sticker locations after finding the L/R stickers. This made non-matching corner recognition viable in speedsolves. In 2021 I started trying to solve the transformed corners recognition problem (as seen in the 42 method and A2). I discovered a recognition method based on the non-matching corner recognition that I had developed for Roux. I talked about this transformed corner recognition method on Discord and Tim Mosher offered to create a document for the recognition method. After a lot of discussion on how the document should look and the rules of the recognition, Tim asked me if I thought it would be possible to make it a three sticker recognition method after finding the L/R sticker orientation. I looked into it and discovered that not only was that possible but that it can be a two sticker recognition system. The two sticker recognition was able to be applied to not only CCLL, but also NMCLL and others. Over the time of the main document being developed, Tim Mosher and I simplified the rules to make the recognition method what it is now.

ATCRM is based on the way that the symmetries of the corners changes when pseudo becomes involved. The primary attribute of the recognition method is its use of the stickers that belong on the left and right sides of the puzzle. These stickers don't change their symmetry when any combination of L or R moves is performed, while all other stickers do change. ATCRM can be used in normal solves and down to various depths of pseudo. This means from normal to non-matching blocks to transformation and even further. Each of these will be described using pseudo depths.

This is a normal solve where there is no relative layer that is offset.

When you perform a U layer turn in a normal solve, the U, F, B, L, and R layer stickers all remain the same. A normal state produces the common 43 possible cases. This works well for a recognition method that starts by finding the orientation of the four U layer stickers. However, it loses its effectiveness once pseudo becomes involved.

ATCRM works exactly as well here in a normal solve as it does in a pseudo solve.

This is when a single layer is offset relative to the U layer corners. The common term for this is non-matching block. The example image below shows an R turn performed.

If you perform an R layer turn on a solved cube to cause the pseudo state, the U layer stickers change. Looking at the image below, the same two yellow stickers are still at UFL and UBL. However, the U layer stickers at UFR and UBR are now blue. You can see that there is also a blue sticker at the FUL position and also a yellow sticker not visible at the BUR position. If the corners are scrambled after this R layer turn is performed, it becomes difficult to know which yellow stickers and which blue stickers are the correct ones when finding the orientation. If you try to make a recognition method where you use the three blue stickers and the green sticker that is at BUL or the three yellow stickers and the white sticker at FUR for the four sticker orientation step, it becomes 162 different patterns to memorize. So how do you easily recognize pseudo corners while at the same time not having to learn a lot of patterns?

If you look at the image again you can see that there are two red stickers on the right and two orange stickers on the left side not visible in the image. This is the same as the solved state image above. These stickers didn't change. They remain the same no matter what R or L move you do. The L/R stickers maintain their symmetry. We can take advantage of this and use these four stickers for our orientation step. In a normal solve there are four U layer stickers that are all the same color and the final case count is 43 including the solved corner state. When using the four L/R stickers for the orientation step, there are two different colors. In the case of the image below, these are orange and red. Because there are two colors it doubles the number of patterns to make 84. There are still only 42 algorithms to memorize, but each algorithm has two different patterns. This occurs because the L/R stickers can be lined up with the left and right layers or the F and B layers. If you do the inverse of an algorithm such as Sune without moving the U layer or doing a U2 first, it will produce one type of L/R sticker orientation. But if you do a U or U' move first then do that same algorithm in reverse, it produces a different L/R orientation. This shows why the number of patterns is doubled but the number of algorithms remains the same. These various L/R sticker orientation cases also provide partial corner permutation information. Because the two orange stickers are supposed to be next to each other and the two red stickers are supposed to be next to each other, some of the corner permutation state can be seen by the position of these stickers. Which leads to, and helps with, the final step of the recognition method.

In the recognition method that first finds the U layer stickers, the next step is to compare all corners against each other and find matching and opposing stickers. ATCRM works differently. It compares the two corners that are supposed to be at UL against each other and the two corners that are supposed to be at UR against each other. This is accomplished by looking at just two stickers in total. One sticker on one corner of each set. As mentioned in the previous paragraph, the L/R sticker orientation provides partial CP information. That's why the user only needs to check one sticker on each corner to know the full case. As mentioned in the recognition method document, there is one rule to be learned for two sticker recognition. When checking the two stickers, for the sticker that is on a corner that belongs on the right side of the puzzle, if that sticker is a U/D color, pretend that it isn't a U/D color. If it isn't a U/D color, pretend that it is a U/D color. This rule applies if the corners are anything other than normal or R2 NMB, meaning R/R' NMB, transformation, or anything further.

To see why two sticker recognition works, and why the U/D color flipping rule is necessary, look at the image again. There are two matching blue stickers at UFR and UBR and two opposing stickers (white and yellow) at FUR and BUR. This has occurred because the R move has brought the white sticker up from the D layer to FUR and placed the yellow U layer sticker at BUR. These are U and D layer stickers. So there is a line of matching stickers along the U layer from UFR to UBR and a line of opposing stickers going from FUR to BUR. These two opposing stickers and the two matching blue stickers are the other half of what makes the two sticker recognition work. You can see the same thing in the two corners on the left side. Two matching stickers along the U layer from UFL to UBL and two opposing stickers going from FUL to BUL.

In ATCRM both the orientation step and the two sticker step are accomplished without the user needing to do any special thinking. Simply check the orientation of the L/R stickers then look at two specific, pre-memorized sticker locations.

This is when the U layer is first offset then the R layer is offset. The term for this is Transformation, which is a sub-concept within conjugation. The example image below shows the turns U R performed.

When the moves U R are performed, there are again two yellow stickers at UL and two opposing stickers going form FUL to BUL. That doesn't change compared to P1. However, we now have a situation where there are two different stickers on the corners at UR. This makes it even more difficult to try to use a recognition method which starts by finding the orientation of the stickers which should be on the U layer. But, if we again use the L/R stickers we still have good recognition and still have just 84 patterns. Finding the L/R orientation in P2 (transformation) involves one new rule. If the sticker on the right side of the corner that was brought to the D layer (in this case the RDF sticker) is an L or R layer sticker (red or orange), recognize as normal by finding the orientation of the orange and red stickers. If the RDL sticker is an F or B layer sticker, find the orientation of the F and B layer stickers on the three U layer corners that are on the U layer. In the image below there is an F/B layer sticker at RDL (green), so we find the orientation of the green and blue stickers on the three U layer corners that are on the U layer. These three corners are at UFL UBL and UBR in the image below. The specific green and blue stickers are at LUF, LUB, and RBU. It isn't necessary to check the other corner, the D layer corner that is at UFR in the image, because the orientation of that one is automatically determined by the other three. This rule is necessary for P2 because when the first U move is performed it sets into place what the L/R stickers will be. If U2 R is performed, the L/R stickers are still orange and red. If U R or U' R is performed, the L/R stickers become blue and green. This is with the exception of the single corner that is brought from the D layer to the U layer after the R move (the one at UFR in the image). That corner will always be red. However, as mentioned, that D layer corner can be ignored.

For the two sticker step, the two corners at UFL and UBL completely match each other so nothing changes. For the two corners at UFR and UBR, we use the rule that was introduced in P1. Just as in the P1 image above, the R turn brought up a white sticker to FUR and a yellow U layer sticker to BUR. We know that these two stickers are opposite but the two stickers at UFR and UBR don't look related at all. But those unrelated stickers don't contain the white or yellow U/D stickers. So in the two sticker step when checking these two corners, we can simply check if the sticker we are looking at has a white/yellow or if it has anything else. If it has white or yellow, it will be one case. If it is not white or yellow it is another case.

This is when there have been three turns of pseudo applied. The order would be an R layer turn, then a U layer turn, then another R layer turn. For P3 the order can't be U R U because the final U would only be an AUF since we are trying to solve the U layer corners. That order would be P2 instead. P3 is technically non-matching blocks + transformation. Which naturally becomes P1 + P2. In the ATCRM document the rules for N, P1, and P2 are described. P3 contains new rules for recognition.

Recognition works very similarly to the usual method. Though there are new rules. For the orientation step, new orientation patterns would need to be learned or the user would work out intuitively the orientation pattern for the current case. For the two sticker step, it would become a three sticker system in most P3 types.

The first image below shows the turns R2 U2 R performed. The orientation pattern here is the same as usual. There are two reds at LUF and LUB and one orange at RBU. The same U/D two sticker rule can be used.

The second image shows the turns R U2 R performed. Again the orientation pattern is the same. We would check a single sticker on one of the two corners on the left side and use the two sticker rule. However, we can't check just a single sticker on the two corners on the right side. This is because there is a green on the U layer and also a green at FUR and one U/D color is on the U layer at UFR and one is on the B layer at BUR. This could be done using two stickers by knowing what corner is being checked. Though that may add additional thinking time to the solve.

The third image shows R U R performed. Here the stickers are very different from usual. The orientation pattern to be checked would again be based on the sticker that is at RDB. If it is an F/B sticker, the orientation would be the U/D stickers of the two red corners that are at UFL and UBR. These specific U/D stickers are at LUF and RBU. The corner at UFR has red also, but we can again ignore that one. Then the other sticker to check for orientation would be the F/B sticker of the remaining corner at UBL. That sticker is at LBU. If the sticker at RDB is a U/D sticker, then we check the F/B stickers of those two red corners at UFL and UBR and the U/D sticker of the corner at UBL. For the two sticker step to complete the recognition, recall that in the recognition document the subcases for each orientation are NN, UU, UN, or NU. For the two corners that belong on the left side, if the sticker you check is an L/R color it is an N. Otherwise it is a U. For the two corners that belong on the right side, if the sticker being checked is an F/B sticker, it is an N. Otherwise it is a U.

There are additional rules at the P3 level depending on the type of P1 + P2 combination that was performed. It may not be worth it to use P3 in speedsolves. Not just recognition-wise, but also because the number of moves that are required to restore the pseudo after the corners have been solved.

ATCRM technically works at all depths. However, the further it goes into pseudo the more the stickers become mixed together and the more the rules for recognition change. This means some kind of advanced tracking would have to be done to keep track of certain stickers to ensure that the CLL case is recognizable. The example image below shows the turns U R U R performed.

There are other possible pseudo states which ARCM supports. These states can often be classified within one of the depths as described above. While ATCRM can be used to identify the corner case in these situations, just like description of P3 mentions, the further into pseudo the solve goes the more moves it takes to undo it if the turns are being undone after an algorithm. For more complex pseudo states it may be best to have the algorithm itself undo the pseudo, as in the A3 concept. Below is an example of a P2 depth which is the setup U R2.

This kind of corner case is easy to recognize using ATCRM but requires a bit more thought in the solve. The stickers that are needed to find the orientation are the R layer colored stickers on the two corners that belong on the right side and the F/B stickers that are on the other two corners. It's kind of like a double version of the typical U/U' R transformed corners where the D layer corner that was brought to the U layer will always have its R layer sticker involved in the orientation. However in this U R2 case, two corners were brought from the D layer to the U layer. After the orientation case is found, the two sticker recognition is the same.

There are many more pseudo examples. Likely just a few are good for speedsolving.