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Blindfold Cubing with M2

Content

1 A Brief History

Invented by Stefan Pochmann in early 2006, the M2/R2 blindfold cubing method is a significant improvement of the original Pochmann method. Unlike the 4-step 3-cycle method (hereafter simply 3OP...see note below), which separates orientation from permutation in both corners and edges, the Pochmann method and M2/R2 combine the two steps and instead use 2-cycles. Details of both methods can be found on Stefan's website: original Pochmann, M2/R2.

(Note: When I wrote my guide for the 4-step 3-cycle method, I called it the 3-cycle method to distinguish from the original Pochmann, the only other method existing at the time Now that there are methods using 3-cycles that combine orientation and permutation, this is no longer an appropriate name. Following what many others have already started, I will call this method 3OP and its corner and edge steps 2-step edges and 2-step corners, respectively, or simply 3OP edges and 3OP corners.CO/CP and EO/EP, respectively.)

It is becoming more and more clear that the invention of M2/R2 marked the beginning of a major change in the blindfold cubing world. Cubers began to explore 3-cycle methods that combine orientation and permutation. There is freestyle 3-cycle, which uses commutators or conjugated U-permutations without restrictions,and several variations (including TuRBo, one could argue) depending on how "free" the freestyle is. M2/R2 can be seen as a 3-cycle method that relies on very specific types of commutators.

[insert Blindfold Cubing Methods Timeline]

While the original Pochmann method never improved on the world record set using 3OP, M2/R2 and these new methods had by 2007 already began to replace 3OP as the primary method among top blindfold cubers. At WC05, the top two blindfold cubers, Leyan Lo and Tyson Mao, used 3OP, with Jean Pons coming in third with Pochmann. At WC07, Rafal Guzewicz placed first with M2, with myself and Tyson at second and third with 3OP. In 2008, the world record was firsts broken without pure 3OP, using M2 with CO/CP. WC07 will likely be the last WC with pure 3OP cubers in the top 3.

The superiority of M2 and other new edge methods over EO/EP has now been clearly been demonstrated. However, as of 2008, it does not seem likely that top blindfold cubers will ever agree on a single edge method among these. As for corners, many still stick to CO/CP over R2 or freestyle, which are more complicated than their counterparts for the edges.

What I have described below is one possible form of M2 that I have adopted over EO/EP. I provide no explanation of why M2 works. If you are interest,d please see Stefan's site.

2 Preparation

In methods combining orientation and permutation, it is natural to consider cycles of stickers rather than of cubies. Although visual memorization of sticker cycles still works, many cubers find it helpful to use a lettering scheme for the 24 edge stickers. Here's mine:

L stickersM stickersR stickers
ULIDFQURA
LUJFDRRUB
FLKUBSBRC
LFLBUTRBD
DLMUFUDRE
LDNFUVRDF
BLODBWFRG
LBPBDXRFH

UR, for example, refers to the U sticker of the UR edge while RU refers to the R sticker of the same edge.

We assign a sequence, as shown below, to each L/R sticker and to UB/BU:

L stickersR stickers
ULLU'L'UURR'URU'
LUBL'B'RUB'RB
FLU'L'UBRUR'U'
LFL'BL'B'RBRB'R'B
DLLU'LUDRR'UR'U'
LDBLB'RDB'R'B
BLU'LUFRURU'
LBL'BLB'RFRB'RB
UB(nothing)BUB'R'BR'URU'

For any sticker s listed here, the corresponding sequence is denoted (s) and its inverse (s)'. For example,

(UR) = R'URU'
(UR)' = UR'U'R

Note that (s) brings s to position UB without disturbing the other M edges. No move is needed for UB.

3 Phase 1: Memorization

First, locate any edge that is flipped in the correct position. We will call these inactive edges and the rest active edges. Memorize the locations of the inactive edges and note whether the number of inactive edges is even or odd.

Starting with the buffer sticker DF, we will now generate sticker cycles. To do this, locate the sticker position where the DF sticker belongs. Memorize this sticker position or the corresponding letter, and repeat the procedure with the sticker in this position. You will eventually reach a sticker that belongs to either DF or FD. Do not memorize this position. This completes your first sticker cycle.

If this cycle involves all active edges, the memorization phase is complete. Otherwise, we need to "break into a new cycle."

Choose any sticker on any active edge that does not already appear in a cycle. If possible, choose a sticker with an easy corresponding sequence, such as UB or FR. Memorize the position of this sticker, and start a new sticker cycle from it, memorizing each sticker location as before. The cycle ends when you return to this sticker you started with or the other sticker on the same edge position. Unlike with the first cycle from DF, you need to memorize this last sticker position. Thus, you should have two stickers of the same edge memorized, corresponding to the beginning and the end of the cycle.

Repeat the process of breaking into a new cycle until all active edges have appeared in some cycle.

Finally, memorize all sticker positions in pairs of two, ever if a cycle ends with the first in a pair. If there is one sticker remaining at the end, memorize it alone. You do not need to memorize where a cycle begins or ends; this is already coded in since you memorize the first edge of each cycle except the first twice.

Note: Unlike in many versions of M2, it is not necessary to switch UF/FU with DB/BD for the second sticker in a pair.

Examples are found at the end of this guide.

4 Phase 2: Resolution

4.1 Pairs

The memorization has split the sticker cycles into 3-cycles. For each memorized pair of stickers starting with the first pair, we perform the 3-cycle involving DF and the two stickers in that pair. Use the following rules to determine the appropriate sequence:

Case 1: Neither sticker in M
For pair s t, where s and t are both side stickers, perform

(s) M2 (s)' (t) M2 (t)'

Case 2: One sticker in M
Use the following table, where s is the side sticker:

GroupPairAlgorithm
IUB sM2 (s) M2 (s)'
s UB(s) M2 (s)' - M2
BU s (BU) M2 (BU)' - (s) M2 (s)'
s BU (s) M2 (s)' - (BU) M2 (BU)'
IIUF s(s) U2M'U2M (s)'
s UF(s) M'U2MU2 (s)'
DB s(s) MU2MU2M2' (s)'
s DB(s) M2U2M'U2M' (s)'
IIIFU sM2' (s) M (s)' M
s FUM' (s) M' (s)' M2
BD sM2 (s) M' (s)' M'
s BDM (s) M (s)' M2

Tips for memorizing these:
- Group I is really the same rule as Case 1.
- Note the position of (s) and (s)' in each group.
- In group III, the sticker in M travels through the UB edge, not the DF edge. The first move is M2 (or M2') when the M sticker is the first one in the pair.

Case 3: Both stickers in M (only some cases)
These are all commutators involving M'/M, which means you don't actually need to memorize them; some people are so comfortable with commutators that they can come up with these solutions on the fly with little thinking. Of course, memorizing all 24 possibilities fits better with the philisophy of direct letter-pair-to-sequence conversion without having to first convert from letter pair to position. Either way, understanding how these work is the first step. For now I've only shown pairs involving UF and UB edges, but the same type of thinking can be used for the other cases.

GroupPairAlgorithm
No flipUF UBU2M'U2M
UB UFM'U2MU2
Flip on UFU UBD[M',UR2U']D'
BU UFD[M,U'R2U]D'
UB FUD[UR2U',M']D'
UF BUD[U'R2U,M]D'
Flip on FBU FUxD'[UR2U',M']Dx'
FU BUxD'[M',UR2U']Dx'

[X, Y] is commutator notation and means X Y X' Y', where X and Y can be any sequence of moves. For example, D[M',UR2U']D would be D-M'-UR2U'-M-UR2U'-D.

There is a way to do this by handling one piece at a time--see Stefan Pochmann's original M2 page--but this approach makes you switch between UF/FU and DB/BD depending on whether or not the sticker is the second in a letter pair. You may wish to adopt this approach until you are comfortable using commutators.

For a list of algorithms to handle all 24 cases, see this post.

After performing the appropriate sequence for a particular pair, you may erase the pair from memory. Move on to the next pair and repeat until you are left with zero or one sticker.

4.2 Endgame

Case a: Zero sticker left
If there were an even number of inactive edges during memorization, orient them just as in the EO step of 3OP edges. If there were an odd number, orient in addition the DF edge.

The edges should now be solved.

Case b: One sticker left
If there were an even number of inactive edges during memorization, orient them just as in the EO step of 3OP edges. If there were an odd number, there are two cases:
(1) If the last sticker is on U, D, or on F/B in the middle layer (recall the UDRLF2B2 EO definition), then orient in addition the edge containing the last sticker.
(2) Otherwise, orient in addition the DF edge.
In either case, this leaves us with with a 2-cycle of this last sticker and DF with correct orientation according to the UDRLF2B2 EO definition. Finish the parity as in 3OP together with two corners while or after you solve the corners.

Although this forces us to use the 3OP edge orientation in the last step, note that it saves us a whole (s)M2(s) or even more if the last sticker lies in the M slice.

That's it!

5 Examples

Example 1

Scramble

B2 D2 F D' U2 R' B2 U L2 R' D2 L' U2 R' U B R2 D L F R2 U F D2 R2 (Generated using JNetCube)

Memorization

No inactive edge.
BL FR
UF BU
BR FL
LD RD
DB UR (DB goes to DF, ending the first cycle. begin the second cycle with UR)
UL RU (the repeated UR/RU ends the second cycle)
No parity.

Resolution

PairSolutionSolution fully written out
BL FR(BL)M2(BL)' (FR)M2(FR)'U'LU-M2-U'L'U-URU'-M2-UR'U'
UF BUD-[U'R2U, M]-D'D-U'R2U-M-U'R2U-M'-D'
BR FL(BR)M2(BR)' (FL)M2(FL)'UR'U'-M2-URU'-U'L'U-M2-U'LU
LD RD(LD)M2(LD)' (RD)M2(RD)'BLB'-M2-BL'B'-B'R'B-M2-B'RB
DB UR(UR)MU2MU2M2'(UR)'R'URU'-MU2MU2M2'-UR'U'R
UL RU(UL)M2(UL)' (RU)M2(RU)'LU'L'U-M2-U'LUL'-B'RB-M2-B'R'B

Example 2

Scramble

D F' D R2 B2 U F L' F' L2 U2 L2 B R2 B' R U' D B R' F' D2 B' L' R2 (Generated using JNetCube)

Memorization

Inactive edge: UB (odd)
FR UR
RB DR (RB goes to DF, ending the first cycle. begin the second cycle with DR)
LB FL
LD UF
LU BD
DR (the repeated DR/RD ends the second cycle)

Resolution

PairSolutionSolution fully written out
FR UR(FR)M2(FR)' (UR)M2(UR)'URU'-M2-UR'U'-R'URU'-M2-UR'U'R
RB DR(RB)M2(RB)' (DR)M2(DR)'RB'R'B-M2-B'RBR'-R'UR'U'-M2-URU'R
LB FL(LB)M2(LB)' (FL)M2(FL)'L'BLB'-M2-BL'B'L-U'L'U-M2-U'LU
LD UF(LD) M'U2MU2 (LD)'BL'B'-M'U2MU2-BLB'
LU BDM(LU)M(LU)'M2'M-BL'B'-M-BLB'-M2'

DR is on D, so we orient DB in addition to the inactive edge, UB: F2-M'UM'UM'U2MUMUMU2-F2. This leaves us with the 2-cycle (DF DR) with correct orientation according to the usual definition.

6 Links

aoeu

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