42

42, Briggs3 (B3 or Briggs for short) started off as an extension to Roux but eventually developed to such an extent it is considered a separate, though related, method.
Contents
Steps
 First Block (FB)
 Solve a 1x2x3 in DL (see Roux)
 Second Block Square and 1 oriented corner (SBsquare+c)
 Solve a 1x2x2 block in BDR
 Orient an LL corner and place in UBR
 Perform an R move
 Conjugated CMLL (also known as L5C reduction or Briggs Last Corners (BLC))
 Last 7 Edges (L7E) usually solve in one of two ways:
 FR first
 Orient remaining edges and solve the FR edge (EO+FR)
 Solve the UL and UR edges (4b in the Roux method)
 Solve the remaining edges (4c in the Roux method)
 ULUR first
 Solve edges which go in ULUR or UFUB (including FR if it is in position) and orient edges
 Solve theremaining edge in the R layer
 Solve the remaining edges (conjugated 4c)
 FR first
Pros
 Low movecount compared to other methods
 Relatively low number of algorithms
 Less complex than many methods with a similar movecounts/algorithm counts
 After the first block is built the rest of the cube can be solved mostly with R, r, M and U moves thus eliminating rotations.
 The blockbuilding and intuitive nature of the method allows for rapid improvements in lookahead and inspection
 Nonlinear blocks are easier to implement as less is solved compared to Roux
 CMLL is one of the best algorithm sets as there are only 42 cases and most algorithms are fast OLLCPs from CFOP
 Relatively easy case recognition
Cons
 Block building can be difficult for a beginner to get used to. The reliance on r and M moves may also be difficult for some people, so much so that cubers who have trouble with M turns should probably not use this as their main method (or better, practice practice practice the M moves). Slice turns can also be slower than using a quarterturn metric.
 Since the Mslice is used often, especially in the final stages, there is a larger chance of a DNF rather than a +2 if the solver misses the second flick in an M2, or if the solver misses the last M move. It is a DNF because M uses both the R and L face in one.
 Multiple cases for each algorithm can be difficult to get use to.
 It is an unproven method so any using it will be taking a risk
Potential Improvements
 It is possible that at least some of the CMLL algorithms could be less efficient than is possible as there is a slightly altered configuration of piece though so far this has not been pursued
 TC (Tyrannical Caterpillar): similar to tyrannical caterpillar for Roux, it is possible that the FR edge could be inserted during the algorithm for solving the last 5 corners though this may provide a smaller benefit when compared to Roux and it is possible that it would not give any advantage at all
 EG: similar to PEG, it is possible that the pairs in the blocks could be solved in different slots and these could be solved during L5C using EG algorithms with wide moves so that the pairs and blocks preserved and solved. This, like PEG itself, has also not been pursued.
 NMLL: similar to the Roux technique, this allows the cuber to solve any of the 4 possible SBsquares. while this could increase efficiency, it suffers from the same problem that Roux does: the L5C recognition can become much more difficult (though this is not such a big problem when compared to Roux as all cases would be only the R2 separate block cases would be used).
 F/B L5C: instead of performing an R/R'/L/L' move before L5C, a solver could do an F or B move to reduce the L5C cases though this would require an adapted form of L7E and the extra F or B move can interfere with the ergonomics of L7E.
Advanced Techniques
 FR SBsquare: this is not necessarily an advanced technique and should be a logical step forward where the FR SBsquare is built and the oriented LL corner is placed in UFR before the solver performs an R' move before the conjugated CMLL algorithm is performed. Another extension in the same vein is to solve the right block completely and solve a square on the left of the cube rather than the right though as this may not provide much benefit as most algorithms and cubers are right handed so the lefty algorithms needed may be slower.
 TCMLL: it is possible to use TCMLL algorithms so that the corner in BR slot after the R move does not have to be oriented though this requires more (128) algorithms.
 Nonmatching centres: The first two blocks can be built around incorrect centres. This allows for more efficiency and allows rouxers to take advantage of prebuilt blocks. The centres are corrected directly before or after conjugated CMLL with either u M' u or u' M' u.
 IDL (Influencing During L7E): a set of semiintuitive algorithms similar to EOLR which are used to influence addition edges during the first or second step of L7E
 Nonlinear Blocks: an extension which may become more common as the method grows, this would involve combining the first and second steps so that both more of the cube is solved more easily in inspection as there is much less restriction and less solving when compared to Roux first step and second step where the technique is already quite well established.
 F/F',B/B' L7E: this is where the solver performs an F or B move prior to or during L7E in order to give better cases. This would be most useful when used in conjunction with IDL. However, this means that there will be additional F/B moves in L7E which may interfere with the ergonomics of the step.
 LPEC (L5C (with) Partial Edge Control): it is possible to learn multiple algorithms for each L5C case in order to force better L7E cases.
As a 2x2x2 method
The L5C/conjugated CMLL step can be applied to the 2x2x2 method and can be viewed as an extension to VOP essentially turning the method into a 2step one. However, in this variation it is likely that CLL rather than CMLL algorithms would be used. By using additional algorithms, it is possible that the "V" does not need to "correctly" solved. In a similar way to how EG solves only a face, only the one colour of the "V" may need to be solved. It is possible to use other more advanced Briggs3 techniques such as NMLL. With the techniques listed previously it is possible that the first step may frequently become a "skipped" step have only a 1 or 2 move solution when combined with colour neutrality thereby giving much easier 1looking. However, it is possible that all the techniques may make the recognition much harder. Comparisons in the technique can also be drawn to TCLL or Varasano though alternatively it can also be viewed as an extension.
See Also
External Links
 Original forum post/method proposal
 Video announcement/propsal/description
 Conjugated CMLL case list
 Playlist of extensions, improvements and analyses of the method
 Tutorial