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2x2 method idea

elrog

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I have come up with a 2x2 method that is probably more of a begginer method. It would take 20 algorithms. This method would take 3 steps with the first being the most complex.

Step 1: partially orient corners and seperate top and bottom corners

Step2: orient corners

Step 3:pBL

In the first step, you will make sure that none of your top and bottom colored stickers are on the left and right sides. I'll call corners with the top/bottom color on the left and right sides bad, while corners that don't have the top/bottom color on the left/right sides will be good.

To keep a corner in a good position, you may do L, R, U2, F2, D2, and B2 moves, thus, if you do a 90 degree turn of the U, D, F, or B face, you will change the good corners
in that layer to bad and vice-versa. Doing this is much like solving edge permutation on the 3x3. You do have to keep in mind that any corner with the top/bottom colors on the top/bottom will not change to bad if you do a U or D turn, and like-wise with F and B while having the top/bottom color on the front/back sides.

You should also remember that you should always make the right and left sides be the sides with fewer top/bottom colors on them. You will always be able to get no more than 2 top/bottom stickers on the left/right sides. Seperating the top and bottom colors out afterwards is very simple and should be fairly easy to begginers. On average, I'd say step 1 takes about 4-6 moves after some practice getting used to it.


Due to the restrictions already placed on corner orientation, orienting all 8 corners takes only 15 algorithms. I have not created algorithms for these, but they shouldn't be to hard to come up with. If this method seems interesting to anyone I may put the work into generating them. After you do this, you apply a PBL algorithm of which there are 5 of.

For a more advanced version of this method, PBL and orientation could be combined into one step, but this would take a large number of algorithms because it would restrict your ability to AUF creating more PBL cases. Another similar mathod that is more advanced would be to seperate out the top/bottom corners while making sure that they have an "even" permutation in the top and bottom layers. You could then orient all of the corners with 36 algorithms, and finish with PBL. I would be very surprised if this hasn't been thought of before.

If a 3x3 varient of this method were to be made, it would most likely include solving edge orientation before restricting the corner orientation, and solving something else at the same time as PBL.
 
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elrog

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I have generated the orientation algs now. Surprisingly it only took a few hours. I have organized them into cases with an even (E) or uneven (U) permutation for the top and bottom layers. Any case with an even or uneven orientation in the top, must also have the same in the bottom. It is also impossible to have an uneven permutation in one layer while the other layers orientation is comlete.

The headlights case should always have the headlights on front (I will call this E1). The T case should always have the 2 unoriented corners on the right side (Iwill call this E2). With the H case, it doesn't matter (This doesn't mean you can do this without first meeting the orientation requirements of step 1) (I will call this E3).

There are also 3 unveven cases for a single layer. The first has all corners oriented except the UFR corner which should have the top/bottom color on the front (I will call this U1). The second case would have the UFL corner with the top/bottom color on the front, and the URB corner with the top/bottom color on back (I will call this U2). The third case will have headlights on front and the URL corner with the top/bottom color on back and the UFR corner is correctly oriented (I will call this U3).

I will refer to cases by the case on the top over the case on the bottom. Example: E1/E2 This would have the headlights case on top and the T case on bottom. For cases that are on the bottom, the case should be oriented the same way that it is in the top (the headlight on front, the T's unoriented corners on the right, case U1's unoriented corner in the front and on the right and so on). If all of the corners are oriented correctly, the case will be case 0.

Case / algorithm / alternate alg if any

E1/E1 / (X) U R2 U2 R'
E1/E2 / (X) U2 R U2 R' U2
E3/E1 / F R' U2 R' F2 R'
E2/E2 / R' F2 U F2'
E2/E3 / L F R' U R' U F' R' / R' U2 L' U' R' D' L D'
E3/E3 / F2 U2

U1/U1 / D L D F' L' / D R F D' F'
U2/U1 / R' U R U R'
U1/U3 / L' F2 L U' R2 D' R2
U2/U2 / (X') L2 U' R2 U2 R' U2
U3/U2 / R2 U2 F' R' U' F'
U3/U3 / L' F2 U L' F L' F' / L' F2 U R' D L' D'

E1/0 / F R U R' U' F'
E2/0 / R U R' U' L' U R U'
E3/0 / R2 U2' R' U2' R2

I did not inlude mirror cases. It may also be necessary to do a cube rotation to set a case up. The average move count of the algorithms is 5.8666. Hope you like it.
 
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elrog

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Well, I can recognize the first 2 steps without much trouble, but I don't think I'd be able to predict the PBL afterwards. Of course, I can't do this with any other method either. Recognition of the second step is actually fairly well despite orientating at all 8 corners, because you don't have to see all of the corners to tell what case you have.
 

elrog

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To deal with multiples of cases, I will add ' to the end of a case for the mirror case. Example: E2'/0 This would be a T orientation case with the two unoriented corners on the left side as opposed to E2/0 which has the two unoriented corners on the right side. I will try to avoid this when possible for simplicity.

I've come up with another variant of this method. Rather than separate the corners, then orient, you can orient them first, then separate. If you orient first, you can convert one orientation case to another by using cube rotations or L or R moves. I have put these cases into groups that can be converted to one another easily while preserving the restricted orientation after step 1. Because these can all be converted to one another easily, it is only necessary to learn a single algorithm in each group. Here are the groups:

(U3/U2 , U2/U1) - Group 1

(E3/E1 , E1/E2 , E1/0 , E2/E3 , U1/U1 , U3/U3 , E2/0 , E2/E2 , U2/U2 , U1/U3 , E3/0) - Group 2

(E3/E3) - Group 3

I did not include 0/0 in any of the groups, though they may be easily convertible, because this would link all of the cases into 1 Group, and you won't be going from the solved case to any other case.

There are 3 groups, but you don't necessarily need 3 algorithms. Many cases are trivial such as E3/E3 and E2/E2. Because it is easy to convert from one case to another with practice, you can narrow this down to only 1 algorithm. That one algorithm is for the U2/U1 case in Group 1. Because group 2 is pretty large, not all cases can be easily converted to any other case, but there are a number of easy cases you can convert to that are within Group 2. Here are some trivial cases that are good to convert to, and their solutions:

Group 1 cases:
None

Group 2 cases:
E2/E2 – R , U3/U1 - R U2 R' , U2/U2 - F2 R , E3/0 – R2 U2 R

Group 3 cases:
E3/E3 - x

Note that any case (except the solved case or E3/E3) goes through E2/E2 at some point because it is 1 move away from solved. Here are some conversions from one case to another in group 2:

E3/E1 -> U3/U1 - L U2

E1/E2 -> U3/U1 - B2

E1/0 -> E3/E1 - x' y2

E2/E3 -> E1/0 - x'

U1/U1 -> U1/U3 - U2 L U2

U3/U3 -> E2/0

E2/0 -> U3/U1 - B2 L’ D2

Converting from case to case can be done intuivively. I can normally get everything oriented in 6-7 moves. You can also seperate out pieces intuitively. You would then finish with PBL. For a more advanced version, you could do both the seperation and PBL in one step. Heres a thread I found on this earlier:

http://www.speedsolving.com/forum/showthread.php?35203-OSPA(New-2x2-Method)

I would like to know how everyone thinks this would compare to Guimond orientation.

EDIT: As I use this method more (the intuitive approach for orienting corners), I have come to like 2x2s more and more. I always just solved them before like corners on a 3x3, which was really slow. I have been using a 4x4 for this because I do not actually have a 2x2, and it is a Rubiks brand. It cannot corner cut at all, thugh it turns nicely. For this and other reasons, I have not been trying to solve quickly, nor have I tried to see how much I can solve from inspection. I do seem to be averaging a few less moves the more I use the method. I also think it would be very possible to 2 look tusing this method. So far, I've had 4, 7, 8, and 9 move solve. I did atleast 30 moves to mix them up and was not looking at the cube while mixing it, so they were fully scrambled. I went and did some giumond solves looking at the algs, and I find that my method is about the same, just an intuitive version of it. I also don't picture the corners moving in the same way as a guimon user would, but I realize that they do.

I have been making a list of cases for orientating all of the corners when they are seperated and have an even orientation throughout the top and bottom layers. This would take 35 algorithms or 27 without mirrors. Over half of the algorithms for my method as stated in my first post will be used in this making it a good buildup for it. You may also get the top and bottom layers in an even permutation and use my intuitive appoach for orienting them not worrying about seperating colors, and use 28 algorithms (20 without mirrors) to solve for cases that have a top/bottom color on a corner on the left/right sides. This provides a more advanced method for each of the varients letting you build up more if you wish to.

EDIT 2: I went and generated some random cases for orienting all 8 corners once they have been put into an even permutation throughout the top and bottom layers (not caring about sorting corners), and I found that it wouldn't be much less moves that just orienting corners intuitively or with Guimond. One advantage I think it may have though, Is being able to solve more in 1 look. It never takes more than 1 move to get an even permutation throughout the top/bottom layers. I think that this would work well with doing PBL and seperation in one step.

One difference in all of the varients of my method an Guimong is the need to be color nuetral. In Guimond, you need to be color nuetral to find a side with 3 opposite colors on it. With my method, color nuetralitly can make it harder to tell which corners are oriented incorrectly for step 1. One thing that a color nuetral solver may do different with my method is, rather than use a set color and look for a left and right side with no more than 2 top/bottom colors on it, they might just call whatever the right and left side is when the pick it up the right and left sides throughout the solve. They would just need to look at those 2 sides and find which 2 opposite colors there are no more than 2 of on those sides.
 
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