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What if you did phasing, like in ZZ-b, but with CFOP? You would put two opposite LL edges opposite of each other, not necessarily correctly oriented, but in the same orientation. This would reduce the number of LL cases from 3916 to ~600. Phasing is mostly intuitive, and only has 6 cases. Though this is technically a 3LLSLL, I think it is better then other 3LLSLL methods, such as OLL/PLL, as phasing is relatively lightweight. What do you guys think?

What if you did phasing, like in ZZ-b, but with CFOP? You would put two opposite LL edges opposite of each other, not necessarily correctly oriented, but in the same orientation. This would reduce the number of LL cases from 3916 to ~600. Phasing is mostly intuitive, and only has 6 cases. Though this is technically a 3LLSLL, I think it is better then other 3LLSLL methods, such as OLL/PLL, as phasing is relatively lightweight. What do you guys think?

Since the phased edges are in the same orientation, the remaining two edges are in the same orientation as well. Thus there are 4 eo possibilities: f/b unoriented, l/r unoriented, all unoriented, and none unoriented. Thus the total number of algorithms will be exactly 4 times ZZLL, which is 169 algs. Thus this method has 4 * 169 + 6 = 682 algs, compared to ZB's 799. The f2l pair and phasing take less than 1 look, as they are simple and easy to plan through lookahead. Thus in reality, this method takes ~2.5 looks or less, if you understand what I mean. On the other hand, each of the steps of ZB have pretty complicated recognition, and will take 2 looks. Thus I think the total recognition time for my method and the ZB method will not be too different.

Since the phased edges are in the same orientation, the remaining two edges are in the same orientation as well. Thus there are 4 eo possibilities: f/b unoriented, l/r unoriented, all unoriented, and none unoriented. Thus the total number of algorithms will be exactly 4 times ZZLL, which is 169 algs. Thus this method has 4 * 169 + 6 = 682 algs, compared to ZB's 799. The f2l pair and phasing take less than 1 look, as they are simple and easy to plan through lookahead. Thus in reality, this method takes ~2.5 looks or less, if you understand what I mean. On the other hand, each of the steps of ZB have pretty complicated recognition, and will take 2 looks. Thus I think the total recognition time for my method and the ZB method will not be too different.

Okay, I usually wouldn't do this, but my post got drowned out by Muke and his I don't even know what arguments, so I'm gonna copy it here.

New method:
1. Wheels 2 1x2x2 at DBR and DBL.
2. EOEdges EO, then edges.
3. EP Permute UL, UR, FL, FR.
4. F3C First 3 Corners - solve DFR, DFL, UFL - 162 algs.
5. L3C Last 3 Corners - 24 algs, but they are included in F3C.

Spoiler: More

162 algs. Solving CO parity, then F3C + L3C will drop down the number to 54. EOEdges looks harder than it is. First, EO, then you just have to make sure your M-Slice solution solves UL, UR, FL, and FR. there are only 3 possibly permutations for them. Also, maybe instead of making sure crosses are solved (UL, UR, FL, FR), you could leave them unsolved and solve parity with?

Okay, I usually wouldn't do this, but my post got drowned out by Muke and his I don't even know what arguments, so I'm gonna copy it here.

New method:
1. Wheels 2 1x2x2 at DBR and DBL.
2. EOEdges EO, then edges.
3. EP Permute UL, UR, FL, FR.
4. F3C First 3 Corners - solve DFR, DFL, UFL - 162 algs.
5. L3C Last 3 Corners - 24 algs, but they are included in F3C.

Spoiler: More

162 algs. Solving CO parity, then F3C + L3C will drop down the number to 54. EOEdges looks harder than it is. First, EO, then you just have to make sure your M-Slice solution solves UL, UR, FL, and FR. there are only 3 possibly permutations for them. Also, maybe instead of making sure crosses are solved (UL, UR, FL, FR), you could leave them unsolved and solve parity with?

1(a) = > This is the same step as in Roux. Blockbuild a 1x2x3 block on the F face.
1(b) = > Finish the E-slice, placing FR and BR edges in their respective spots.

Step 2: SOAP

2(a) = > Pair two D face corners (white corners) and place them in the FRU and BRU or FRD and BRD positions. These corners do no need to be oriented or permuted correctly relative to the FR and BR edges.
2(b) = > Perform SOAP style corners. First orient all 6 remaining corners and then permute them. This step would require that first block and E-slice edges are preserved.

Step 3: EO + FD edge

3(a) = > Perform EO regularly as in Roux.
3(b) = > Place FD edge finishing the second block.

Step 4: L6EP
= > Perform Last 6 edges as in normal Roux.

(1) = > Step 1 can be planned out in inspection. This would most likely be easier than planning first block and 1x2x2 square in normal Roux.
(2) = > The orientation step of the SOAP Method is mostly 2-Gen which would make them finger-trickable. If the white corners are placed in the FRD and BRD positions (the separation step in SOAP) the orientation step could be done without looking at the D-face corners (unless you skip the separation step all together).
(3) = > The orientation algorithms can be done such that no corners on the D-face are dis-permuted meaning that if you predict/track how U-layer corners are permuted (during the orientation step) this step can be done in one look instead of two.
(4) = > Based on how the second block is solved it may be easier to preform NMLL (yellow corners/FD edge instead of white)
(4) = > The EO and FD placement can be done simultaneously.
(5) = > After performing EO and FD placement you can perform a Rw2 and see all M-slice edges leaving no blindspots.
(6) = > Because you solve 6 corners simultaneously as well as performing EO and FD placement at the same time, this method should be at least as, if not more, efficient than Roux.
(7) = > There aren't that many algs to learn, 56 I think.
(8) = > I'm naming it SOUP (or SOUXP; SOAP+Roux).

Cons:

(1) = > I haven't generated any algs (as I don't really know how to; although I assume you should be able to modify the existing ones with wide moves). Given that you need to preserve the first block and E-slice they may be less efficient (although hopefully the 2-gen will make up for that).
(2) = > SOAP as a 2x2 method hasn't proven to be viable (yet) and incorporating it may not be all that useful/fast.
(3) = > SOAP corners may have difficult look-ahead and may need to be two looked.
(4) = > Step (2) and Step (3) may prove to be more difficult than they're worth (Ya'll can decide that for yourselves).

So I've been thinking about PCMS lately and believe I've come up with an interesting variant.

Overview:

Step 1: Place 2 corner edge pairs on the left face and finish the E-slice.
Step 2: SOAP method for corners (https://www.cubestuff.cf/?soap).
2a: Separate corners to their respected faces.
2b: Orient and permute last 6 corners while preserving the E-slice.
Step 3: Permute centers while placing LD and RD edges.
Step 4: Orient remain 6 edges while placing DB edge.
Step 5: Permute last 5 edges (L5EP).

Pros:

(1) Inspection: It seems realistic to plan out all of step 1 in inspection and possibly step 2a.
(2) Corners: Being that the corners are solved using a modified 2x2 method (SOAP) it should be more efficient. It also should maintain the freedom of the pairs of PCMS.
(3) Algs: There wouldn't be too many algs (SOAP has 58, L5EP has 16) and the SOAP algs are mostly 2-gen.
(4) Because this method borrows from the SOAP method it should be clean af. #CoronaCubing

Cons:

(1) SOAP isn't really a proven 2x2 method so it just might not be that fast.
(2) The potential is very reliant on what you can plan in inspection.
(3) It just may not be worth it to preform corners compared to how PCMS does.
(4) Corners may take 2 looks instead of 1 (which will probably make it not that relevant).

If this has already been preposed, sorry. If not then let me know what you guys think.

Since the phased edges are in the same orientation, the remaining two edges are in the same orientation as well. Thus there are 4 eo possibilities: f/b unoriented, l/r unoriented, all unoriented, and none unoriented. Thus the total number of algorithms will be exactly 4 times ZZLL, which is 169 algs. Thus this method has 4 * 169 + 6 = 682 algs, compared to ZB's 799. The f2l pair and phasing take less than 1 look, as they are simple and easy to plan through lookahead. Thus in reality, this method takes ~2.5 looks or less, if you understand what I mean. On the other hand, each of the steps of ZB have pretty complicated recognition, and will take 2 looks. Thus I think the total recognition time for my method and the ZB method will not be too different.

I think you left out the 4 adjacent mis-oriented cases. F and R, F and L, B and R, B and L. I do not get why phased edges would be in the same orientation, is that part of the phasing process in your method?

1(a) = > This is the same step as in Roux. Blockbuild a 1x2x3 block on the F face.
1(b) = > Finish the E-slice, placing FR and BR edges in their respective spots.

Step 2: SOAP

2(a) = > Pair two D face corners (white corners) and place them in the FRU and BRU or FRD and BRD positions. These corners do no need to be oriented or permuted correctly relative to the FR and BR edges.
2(b) = > Perform SOAP style corners. First orient all 6 remaining corners and then permute them. This step would require that first block and E-slice edges are preserved.

Step 3: EO + FD edge

3(a) = > Perform EO regularly as in Roux.
3(b) = > Place FD edge finishing the second block.

Step 4: L6EP
= > Perform Last 6 edges as in normal Roux.

(1) = > Step 1 can be planned out in inspection. This would most likely be easier than planning first block and 1x2x2 square in normal Roux.
(2) = > The orientation step of the SOAP Method is mostly 2-Gen which would make them finger-trickable. If the white corners are placed in the FRD and BRD positions (the separation step in SOAP) the orientation step could be done without looking at the D-face corners (unless you skip the separation step all together).
(3) = > The orientation algorithms can be done such that no corners on the D-face are dis-permuted meaning that if you predict/track how U-layer corners are permuted (during the orientation step) this step can be done in one look instead of two.
(4) = > Based on how the second block is solved it may be easier to preform NMLL (yellow corners/FD edge instead of white)
(4) = > The EO and FD placement can be done simultaneously.
(5) = > After performing EO and FD placement you can perform a Rw2 and see all M-slice edges leaving no blindspots.
(6) = > Because you solve 6 corners simultaneously as well as performing EO and FD placement at the same time, this method should be at least as, if not more, efficient than Roux.
(7) = > There aren't that many algs to learn, 56 I think.
(8) = > I'm naming it SOUP (or SOUXP; SOAP+Roux).

Cons:

(1) = > I haven't generated any algs (as I don't really know how to; although I assume you should be able to modify the existing ones with wide moves). Given that you need to preserve the first block and E-slice they may be less efficient (although hopefully the 2-gen will make up for that).
(2) = > SOAP as a 2x2 method hasn't proven to be viable (yet) and incorporating it may not be all that useful/fast.
(3) = > SOAP corners may have difficult look-ahead and may need to be two looked.
(4) = > Step (2) and Step (3) may prove to be more difficult than they're worth (Ya'll can decide that for yourselves).

SOAP corner orientation and permutation algs are probably trash and the recognition would be hard for permutation. EO FD is a weird step and would require you do M and S move switching. Why not just solve FD edge as a part of the e slice?
Also, please prove how efficient this method is on this scramble. You can use 1 look soap algs if you want
R2 D L2 D L2 B2 U B2 R2 U B L' D' F2 R' F L2 D' U F

classic roux solution:
z' y' F' D U M' U' r' B 7/7
r U R U M' U' R U' r U R' 11/18
U' L' U R U' L U R 8/26
U' M U' M' U M U M' U2 M' U2 M 12/38
38stm

I think you left out the 4 adjacent mis-oriented cases. F and R, F and L, B and R, B and L. I do not get why phased edges would be in the same orientation, is that part of the phasing process in your method?

SOAP corner orientation and permutation algs are probably trash and the recognition would be hard for permutation. EO FD is a weird step and would require you do M and S move switching. Why not just solve FD edge as a part of the e slice?
Also, please prove how efficient this method is on this scramble. You can use 1 look soap algs if you want
R2 D L2 D L2 B2 U B2 R2 U B L' D' F2 R' F L2 D' U F

classic roux solution:
z' y' F' D U M' U' r' B 7/7
r U R U M' U' R U' r U R' 11/18
U' L' U R U' L U R 8/26
U' M U' M' U M U M' U2 M' U2 M 12/38
38stm

This is a LEOR for big cubes variant:
1. Left block + opposite center
2. D and B centers + DB dedge
3. Last 2 centers
4. Dedge pairing while solving EO
5. Complete EOLine
6. Right block
7. LL

This is a LEOR for big cubes variant:
1. Left block + opposite center
2. D and B centers + DB dedge
3. Last 2 centers
4. Dedge pairing while solving EO
5. Complete EOLine
6. Right block
7. LL