Other Variants

Step 1: FB on the left.

Step 2: 2x2x2.

Step 3: Orient all edges while placing the DF and FR edges.

Step 4: End with one of several possible sub-variants.

The currently suggested and developed sub-variant uses option select to orient the five corners so that there are two corners twisted or the DFR corner is solved and the U layer corners are all oriented. This leads to a final step of TTLL+, TTLL-, or PLL. The best alg for the first step is used depending on the situation and which one to use is already pre-determined. There is no extra thinking time for the option select because recognition is the same for every case.

EOFE OS Algorithm Sheet (Includes TTLL+ and TTLL-)

Additional sub-variants:

• CLS (104 algs) > PLL (21 algs)

• CO (23 algs) > TTLL (93 algs)

• CPLS (26 algs) > 2GLL (84 algs)

• Permute U edges + solve 2 corners > L3C

• Orient 3 corners + permute U edges > Edges solved TTLL subset or edges solved Twisted TTLL subset ("Zeroing")

• Orient 3 corners > Twisted TTLL

• Insert DFR corner while separating L/R stickers of the corners to L/R (104 algs) > Solve with a subset of ZBLL (81 algs) - This makes for an alternate way to produce the Polar variant.

The EOFE variant was proposed by Vincent Trang.

Step 1: FB on the left.

Step 2: Solve the right side 1x2x2 at dbR.

Step 3: Solve the DB edge while orienting all remaining edges.

Step 4: Solve the front square (Last Extended Slot, LXS) using a single algorithm. This step is 116 cases.

Step 5: Finish the last layer using ZBLL. OCLL + PLL is a great choice as well. The ultimate goal is ZBLL.

EODB Algorithms

LXS Algorithms

Beginner Progression Plan (developed by voidrx):

Beginner:

1. Solve the 1x2x3 on the left just as in Step 1 above.

2. Solve the NSB, being the 2x2x2 in dbr. This is the same step that is in most Nautilus variants. So check out those and the Blockbuilding Examples page for a guide.

3. Use one of 11 algorithms to solve EO.

4. Intuitively solve the remaining front square of the first two layers (LXS). You can either solve the DF edge then the right side pair. Or solve the right side pair then the DF edge.

5. Use an algorithm to orient the U layer corners.

6. Use one of two algorithms to permute the U layer corners.

7. Use one of four algorithms to permute the last four edges. This is commonly known as EPLL.

Total number of algorithms: 18

Intermediate:

1. Solve the 1x2x3 on the left just as in Step 1 above.

2. Solve the right side square (dbR) and place an oriented U layer edge in FR.

3. Use one of 51 algorithms to solve EO and the DB edge.

4. Intuitively solve LXS.

5. Use one of 42 algorithms to solve the U layer corners. This is commonly known as COLL. Also, learn full PLL so that you can take advantage of CO skips.

6. Use one of four algorithms to permute the last four edges. This is commonly known as EPLL.

Total number of algorithms: 114

Alternate EO Methods:

The EODB step above requires the user to either align the M slice so that the U layer center is on the U layer. Or the user can orient the centers during the final move of placing the dbR square. This makes it so that either the U or D center is on the U layer. Then the user can learn an alternate set of EODB algs for when the D center is on the U layer. EODB also requires mid-solve EO recognition of seven edges while finding the DB edge. This, in combination with possibly needing to learn the two sets of EODB algorithms, makes EODB feel imperfect.

One alternate that is being looked into is to solve the DB edge after FB and add one of the dbR square edges. Then create the pair that goes with the right side edge and orient the remaining six edges while inserting this pair. To extend this further the pair can be set up to being one move away from being paired then orient all edges while pairing and inserting. It may even be a good option to solve the DB edge, set up a pair, then orient all edges while placing that pair and solving the remaining edge that goes with the pair.

Another EO option, proposed by Cubing Forever, is to solve the dBr square instead after FB. Then do EO plus solve either the DF or DR edge. If the DR edge is solved during EO, the same LXS square algorithms can be used above. If the DF edge is solved during EO, then the square will be on the right and all algorithms will be purely R U based. This variant would likely actually be classified as a variant of the MI1 method.

A final option is to solve the shell as normal, set up a pair on the U layer, then orient all edges while inserting the pair and solving the DF edge.

EO: Use any EO option.

Step 2: Insert right side pair while performing CP.

Step 3: Solve 2GLL plus the DF edge.

Note: At this time the ergonomics for the algorithms for this variant are unknown. It may end up being that this variant works best with an empty square on the R layer. This would likely have more algorithms that use only R and U moves. One suggestion, by Vincent Trang, is to always solve the DFr pair (as is an option in the L5E variant). This would leave the FR edge unsolved and lead to 2GLL+1 algorithms that are more frequently R, U based.

EO (Optional)

Step 2: Solve all corners. There are many methods available for this step. CLL and L5C are the two obvious ones.

CXCOLL Algorithms

Step 3: Solve remaining edges. The user can solve the edges in various ways as described below:

• C15: Solve a single edge then use L5E. May not be the most efficient method and the Nautilus CLL > L5E variant would make more sense.

• C24: Solve two edges then L4E. Depending on which edges are solved in each step, this could be a good strategy.

• C33: Solve any three edges then L3E. This could also be very good. If edges aren't oriented the first step would have a lot of algorithms. But if used, one good way may be to solve the three M-slice edges first. This would leave an easy L/R edges 3-cycle. If all edges are oriented, the opposite way is likely best. To solve the three L/R edges first, then have a 3-4 move M-slice permutation. These strategies also work well if pseudo blockbuilding was used in the first step. For example, any M layer edge can be at DB and recognition won't be impacted when solving these first three edges and also doesn't impact recognition for L3E. There are other sets of three edges that would be great for the first three edges, such as the three on the F layer or the DF edge, FR edge, and any U layer edge. So this gives the user a lot of options to find their preference.

• C42: Solve any four edges then L2E. The first step would be a lot of algorithms. Not recommended at this time.

• C6: Solve all edges at once. If all edges were oriented in previous steps, this subvariant would be at its easiest point.

The recommendation for this variant is to use a variety of strategies. Each one depending on the edge situation after the corners are solved. After trying several edge solving methods and thinking about ergonomics and final edge end states, the currently recommended route is to default to two basic strategies. The first, proposed by Melkor on the Discord server, is to use an algorithm to solve the FR edge while orienting the remaining five edges then finish with L5EP. The second, proposed by Chemnitz on the Discord server, is to use an algorithm to solve the FR + DF edges then finish with ELL. Below are the docs for the first steps of each. L5EP and L5E can be found in the L5E document.

EOFR

FR + DF

L5E Document (contains L5EP and ELL)

EO: Use any EO option.

Step 2: Insert right side pair while orienting the L/R colors of the corners to face L/R and separating the L/R edges to L/R. This creates a DR state on the left and right side.

Step 3: Permute all remaining pieces on L/R. The M-slice edges can potentially be permuted intuitively during recognition or performance of the algorithm.

Advancement: An even more advanced version of this variant would be to place any oriented pair during Step 2. Then permute all remaining pieces using a larger algorithm set.

Note: This is a method that solves many of the problems with DR methods.

• No need to find a method for permuting the last eight corners or last eight edges.

• There are only three edges on the M-slice. This means there is no possibility for the dots permutation case.

• Great lookahead. The edges and corners aren't all over the cube. Everything is visible.

• Better ergonomics for the final steps. Even with the half-turns that are inherent in DR.

EO (Optional)

Step 2: Insert right side pair.

Step 3: CLL+2. This step solves CLL plus any two edges in one algorithm. If EO was performed before this step, then COLL+2 can be used.

Step 4: L3E.

Note: CLL+2 is an algorithm set that would require the application of the Cycle Union System. CLL+1 was developed using the system by creating tables and simulated case cycling in Excel. For CLL+2 it would be a lot of work to develop in that way so a program for the union system would make it much easier.

Step 2: Solve the remaining F2L square (LXS). This is LXS without edges oriented. Can be done intuitively or by using 467 algs.

Step 3: Solve the last layer using your preferred method.

EO (Optional)

Step 2 - Transformation:

a. Create any remaining pair, align it with the 2x2x2 and perform an R or R' move depending on whether the 2x2x2 is at dfr or dbr.

b. Align any oriented edge above the edge within the 2x2x2 and perform an M/M' move to place that edge on the D layer.

Step 3: LL and undo transformation. At this time, the easiest LL method to use would be CLL then ELL. Due to the easy recognition, NMLL may also be a great choice if all edges are oriented relative to the transformation.

Example:

Scramble: B D2 U2 B L2 B F D2 B D2 U' B2 R D U L' B' L2

1x2x3: R U' r' F' B2 U' B2

2x2x2: M' U r U r U M2

Transformed Block: U2 M' U M U' R U2 R' (In this sequence, U2 M' U M creates a pair and simultaneously places an oriented edge at DF.)

Transformed L5C/CLL: U R U2 R' U' R U' R2 U2 R U R' U R

Transformed L5E/ELL: U M' U' M' U2 M' U M U' M' U2 M U M

Undo Transformation: U R U

Note: L5C and L5E don't both have to be transformed. The user can just use TL5C then end with L5E or just use L5C and end with TL5E. In the case of just TL5C, a pair doesn't have to be created. A single corner is all that is needed.

Original proposal for this variant: https://www.speedsolving.com/threads/the-new-method-substep-concept-idea-thread.40975/post-1391229

For each of the variants, pseudo techniques can be used to further reduce move-count. In particular, placing any edge at DB and building any right side 1x2x2 is a natural fit for the L, M, R concept of the method. It also works really well for recognition for many of the variants. Additionally, dbl as built in step 1 and dFr can be built in the form of three squares. These squares can be completely matching or non-matching. This isn't necessary, but it allows for pseudo blockbuilding to reduce the move-count of the method. As far as recognition when pseudo is used, there are some algorithm sets which have had recognition developed for pseudo use. NMCLL is one that is very fast for CLL. When edges are oriented, NMLL is also a great choice.

There are many other variants that were considered but not highlighted due to not being very unique. The LL, ZBLL, and LSLL variants aren't very unique themselves because other methods such as CFOP and ZZ end with those steps. But those variants are included above only because they appear to have potential or because they provide unique blockbuilding opportunities while solving dFr. The primary goal is to develop variants that are natural for the shape of the method. However, if there is a variant outside of that criteria that has potential then it is worth developing. Some considered variants:

Step 2: dFR + CO

Step 3: DF edge + EO

Step 4: PLL

Note: Just another way to get to PLL. Not very interesting.

Step 2: Use any EO method

Step 3: dFR + CO

Step 4: HKPLL

Note: Both not unique and, at this point in time at least, the algorithms for HKPLL aren't very good.