Belt method similiar to LCBM

[miotatsu]

Here I have documented the method that I use to solve the 3x3x3 Rubik's Cube.
This method could be described as a "belt method". Many of the ideas and algorithms come from another method called LCBM.
I came upon this method while trying to optimize HTA for speed solving.
I hope that these documents will be useful to others interested in this method.

Overview of method:
1. Edge orientation - place edges in the group <F2, B2, L, R, U, D> (or equivilant)
2. Belt - solve 4 edges of the equator (E slice)
3. Parity - create a situation in which corners can be solved with OCLL cases
4. EPOCBL - orient all corners while preserving equator and edge orientation
5. Seperation - permute corners to their correct layers
6. EPPCBL - permute all corners while preserving equator and edge orientation
7. L8E - solve the remaining eight edges

I am no teacher, so I will simply provide links that can explain each step better than myself.

Step 1: Edge Orientation:
This is the same as EOLine without the line.
http://cube.crider.co.uk/zz.php?p=eoline
http://www.youtube.com/watch?v=a6tkUlkjnOE

Step 2: Belt:
Intuitive. Remember not to use single turns on F and B.

Step 3: Parity:
If you do not have OCLL cases on U/D:

The idea is this: R2 U' R2 basically replaces DRF with ULF.
So what you want to do is replace one corner on D with a differently-oriented one, fixing the corner parity.
If you want to rotate a bottom corner clockwise, for instance, you could replace solved with clockwise,
or clockwise with counterclockwise, or counterclockwise with solved.
Another option (if you're willing to have the belt be either solved or an R2 off, which might be worthwhile for corner separation)
is to simply do an R2, moving two corners to the bottom layer.
I am not sure this can always create a solvable parity case, but it seems to work most of the time,
so it's probably still useful (and hey, one move is better than 3 or 7, eh?).
I guess, yeah, if you're going to separate later, it wouldn't hurt to leave the belt up to an R2 off -
no point wasting moves keeping it that way when you're just going to intuitively mess it up later.

Step 4: EPOCBL:
Memorize the algorithms for each case.

Step 5: Seperation:
Intuitive. Use only moves of the group <F2, B2, L2, R2, U, D>. I recommend only using <L2, R2, U, D>.
Do not forget that you must restore the equator after breaking it.
For example instead of bringing 2 U face corners from positions DLF and DLB to the top face with L2 you must do
something that will not destroy the belt such as:
U2 L2 U2 L2 U2

Step 6: EPPCBL:
Memorize the algorithms for each case.

Step 7: L8E:
I solve these using the exact same system used in Roux for LSE minus the edge orientation
http://grrroux.free.fr/method/Step_4.html (4b and 4c)
http://wafflelikescubes.webs.com/rouxmethod.htm (4b and 4c)
http://rouxtorial.webs.com/lse.htm (4b and 4c)


Example solves:

F2 R2 D2 R2 F2 D R2 U B2 L2 F U' R2 B' R' B F2 L2 U L'
EO: R' D R U2 B
Belt: U2 D' R' D' L
Parity: none
EPOCBL: U2 D2 L' U' L' U2 L U' L
Seperation: D' L2 U D L2
EPPCBL: D x2 R U R' F' R U R' U' R' F R2 U' R' x2
L8E: U2 M U2 M' D' M2 D M2 U M U2 M U M2 U x U2 M2 U2 M2

B' U2 B' L' R D2 F L R2 D2 U2 B' U' B' U' R2 F2 R D B U2 R U2 F2 L'
EO: D F' R2 D F2 D F'
Belt: U' D2 R' B2 L2
Parity: none
EPOCBL: D' L' U' L2 U' L' U2 D' L2 U' L'
Seperation: L2 U' L2
EPPCBL: D x2 F' L U L F D2 F' L' U' L' F U D x2
L8E: M2 U M U2 M' D M2 D' M2 U' M U2 M U' M2 U' M' U2 M'

U F2 D F2 U2 F2 L2 U' F2 U' R U' L2 D R U2 B' R2 B' R2 F'
EO: F D' B' D B
Belt: F2 R' U R L' D' L
Parity: L2 U' L2
EPOCBL: f R U R' U' f' F R U R' U' F'
Seperation: U2 L2 D' U2 L2
EPPCBL: U F' L U L F D2 F' L' U' L' F U D2
L8E: M' U2 M D M2 D' U M2 U2 M2 U' M U2 M' U2 M2

Algs:

Spoiler:

EPOCBL Beginner (2-look) - 7 cases:
================================================== ==
Sune: on U: R U R' U R U2 R'
antiSune: on U: L' U' L U' L' U2 L
H: on U: y F (R U R' U') (R U R' U') (R U R' U') F'
U: on U: R2 D' R U2 R' D R U2 R
T: on U: r U R' U' r' F R F'
L: on U: y' F' r U R' U' r' F R
Pi: on U: f (R U R' U') f' F (R U R' U') F'

EPPCBL Beginner (2-look) - 2 cases:
================================================== ==
Y: F R U' R' U' R U R' F' R U R' U' R' F R F'
J: R U R' F' R U R' U' R' F R2 U' R' U'

EPOCBL Intermediate (1-look) - 25 cases:
================================================== ==
Sune: R U R' U R U2 R'
H: y F (R U R' U') (R U R' U') (R U R' U') F'
U: R2 D' R U2 R' D R U2 R
T: r U R' U' r' F R F'
L: y' F' r U R' U' r' F R
Pi: f (R U R' U') f' F (R U R' U') F'
sune/sune: R' U' R' U R U2 D R' D' R
anti/sune: R' U' R U2 R (U' D) R2 U R'
H/sune: U' R (U' D) M2 U' r2 U R'
U/sune: R' U2 R' U D R2 U2 R' U' R
T/sune: L D' L' U' r2 U2 L' U' L'
L/sune: R D r2 R' U R U2 r2 R'
Pi/sune: L U D L' U D2 L' U2 D L'
H/H: R2 (U' D) L (U D') M2 (U D') L
U/H: R D R' U D2 l2 U2 R' U' R'
T/H: L D L' U' D2 r2 U2 L' U' L'
L/H: L U' L2 R2 U' L' l2 D U2 L
Pi/H: R' U' L2 U R2 (U' D) r2 R' (U' D) R'
L/U: D' L' U' L' U2 L U' L
Pi/U: L' U R U2 R' U L R U R'
L/T: U R U D2 R' U' D2 R'
Pi/T: L' U' L l2 U' l2 U' L' U2 L
L/L: L U' D' L U D L'
Pi/L: L' U' L2 U' L' U2 D' L2 U' L'
Pi/Pi: L U D L' U2 D2 L U D L'

EPPCBL Intermediate (1-look) - 5 cases:
================================================== ==
Y: F R U' R' U' R U R' F' R U R' U' R' F R F'
J: R U R' F' R U R' U' R' F R2 U' R' U'
J/J: F2 U' R2 U D R2 D' F2
J/Y: F' L U L F D2 F' L' U' L' F
Y/Y: F2 U2 R2 U2 R2 U2 F2


EPOCBL Advanced (1-look) - 70 cases
================================================== ==
One case:
Sune: on U: R U R' U R U2 R'
Sune: on B:
antiSune: on U: L' U' L U' L' U2 L
antiSune: on B:
H: on U: y F (R U R' U') (R U R' U') (R U R' U') F'
H: on B:
U: on U: R2 D' R U2 R' D R U2 R
U: on B:
T: on U: r U R' U' r' F R F'
T: on B:
L: on U: y' F' r U R' U' r' F R
L: on B:
Pi: on U: f (R U R' U') f' F (R U R' U') F'
Pi: on B:
Two cases:
Sune: on B:
Sune: R' U' R' U R U2 D R' D' R
antiSune: R' U' R U2 R (U' D) R2 U R'
H/Double Sune: U' R (U' D) r2 R2 U' r2 U R'
U/Headlights: R' U2 R' U D R2 U2 R' U' R
T/Chameleon: L D' L' U' r2 U2 L' U' L'
L/Bowtie: R D r2 R' U R U2 r2 R'
Pi/Wheel: L U D L' U D2 L' U2 D L'

antiSune: on B:
Sune: L U L' U2 L' (U D') L2 U' L
antiSune: L' U L U' L U' D2 L U L'
H/Double Sune: U' L' (U D') l2 L2 U l2 U' L
U/Headlights: U R U' R2' D2 R U2 R U' R'
T/Chameleon: R' (U2 D2) R' D R2 U2 D R
L/Bowtie: L U R U' L U2 L2 U' R'
Pi/Wheel: R U2 L2 U R U D r2' R U R

H: on B:
Sune: D R (U D') L2 R2 U' r2 U R'
antiSune: D' L' (U' D) L2 R2 U l2 U' L
H/Double Sune: R2 (U' D) L (U D') L2 l2 (U D') L
U/Headlights: R D R' U D2 l2 U2 R' U' R'
T/Chameleon: L D L' U' D2 r2 U2 L' U' L'
L/Bowtie: L U' L2 R2 U' L' l2 D U2 L
Pi/Wheel: R' U' L2 U R2 (U' D) r2 R' (U' D) R'
U: on B:
Sune: R U' R2 D L' R2 U R' U' L
antiSune: L' U R U' L l2 U l2 U R'
H/Double Sune: R U R' U2 D R2 U2 R' U' R'
U/Headlights: U2 L' U L' U' r2 U' L U L'
T/Chameleon: R (U' D) R U' R' U2 D' R'
L/Bowtie: D' L' U' L' U2 L U' L
Pi/Wheel: L' U R U2 R' U L R U R'
T: on B:
Sune: L U' L' D' L2 U2 L' U' L'
antiSune: R' (U2 D2) R' U R2 U D2 R
H/Double Sune: L U L' U2 D' L2 U2 L' U' L'
U/Headlights: L2 U2 L' U' L U2 L' U' L'
T/Chameleon: R (U2 D2) R' (U2 D2) R'
L/Bowtie: U R U D2 R' U' D2 R'
Pi/Wheel: L' U' L l2 U' l2 U' L' U2 L
L: on B:
Sune: R U L2 U2 L' U R' U' L'
antiSune: x2 L U R U' L U2 L2 U' R'
L D R D' L D2 L2 D' R'
H/Double Sune: L U L2 U' L U' D2 L U2 L
U/Headlights: U' L' D' L' D2 L D' L
T/Chameleon: D R U2 D R' U2 D' R'
L/Bowtie: L U' D' L U D L'
Pi/Wheel: L' U' L2 U' L' U2 D' L2 U' L'
Pi: on B:
Sune: L U D L' U2 D L' U D2 L'
antiSune: D R U L2 U D R U2 r2 U' R'
H/Double Sune: R' U L2 U' R2 (U D') L2 R (U' D) R'
U/Headlights: R' U' r2 D2 U' R' U r2 U' R
T/Chameleon: L' U' L2 D2 L U2 D' L U' L
L/Bowtie: D L' U' D' L D R2 U L R2
Pi/Wheel: L U D L' U2 D2 L U D L'

EPPCBL Advanced (1-look) - 35 cases
================================================== ==
Case J/:
L: R U R' F' R U R' U' R' F R2 U' R' U'
R: (y) R' U L' U2 R U' R' U2 L R U'
F: R' U L' U2 R U' R' U2 L R U'
B: R U' L U2 R' U R U2 L' R'
Case Y/:
F R U' R' U' R U R' F' R U R' U' R' F R F'
Case /J:
L:
R:
F:
B:
Case /Y:
Case J/J:
L/L: F2 U' R2 U D R2 D' F2
L/R:
L/F:
L/B:
R/L:
R/R:
R/F:
R/B:
F/L:
F/R:
F/F:
F/B:
B/L:
B/R:
B/F:
B/B:
Case Y/Y:
F2 U2 R2 U2 R2 U2 F2
R2 U2 R2 U2 y' R2 U D' R2
R2 U2 R2 U2 F2 dD' R2
R2 U2 R2 U2 F2 E R2
U' R2 U2 R2 D2 F2 E' L2
Case J/Y:
L: F' L U L F D2 F' L' U' L' F
R:
F:
B:
Case Y/J:
L:
R:
F:
B:

[Erzz]

The last thing I expected when I came to this site was to see "LCBM" in a thread title @_@
I'd try this out, but I really can't figure out your organization of algs. You have things like "Sune on B", which will not happen, since you have already solved parity. "H on B" is the same as having T on both, but harder to recognize (imo). You could clean it up a lot by removing those.
The algorithms I listed are the only possible cases (after solving parity). Without solving parity, there is a ridiculous increase in the number of cases, with most of them not able to be referenced as "known OLL on B".
Another thing, you don't really need to permute the corners of both layers, since it is rather easy to solve the corners of the D layer (or even the whole D layer) during separation. You could just use Y and J perms to solve the top corners, then go into L8E. However, if you can solve the full D layer during separation, a simple PLL will solve the cube (and there is no difference between this and normal LCBM :P) I haven't been around for a while though, and am unsure if the moves saved by not solving the D corners during separation would justify the algorithms for EPPCBL. I would need to test some more.

[StachuK1992]

Example solve with more optimized algs, and better L8E;

R2 D2 F2 R2 D L2 F2 D R2 F2 D L' D L F' R' F2 R' D' R2 F2
EO: D R F' x2 (3/3)
Belt: U' D2 R [right] D' U' L [left] (6/9)
CO parity; skip
CO: L' U' D' L' R2 D R2 U L (9/18)
Separate: L2 U2 L2 (3/21)
CP: D R2 D R2 U2 B2 U B2 U (9/27)
Finish L+R: D' L2 D' S2 D L2 D S2 (8/35)
M: U2 M' U2 M (4/39)
Not sure how many the L+R cases would be, I'll compute this today, likely.

[Pyjam]

Belt: U' D2 R [right]

No mistake ?

[StachuK1992]

I can't see what I did wrong. :/
Here's another, hopefully I don't mess this one up.
U2 R2 F2 L' B2 L D2 L B2 L2 R' D' R2 U B2 L R2 D2 B' D
EO: D' B' (2/2)
Belt: U' D L [left] D' R D R' [finish right] (7/9)
CO Parity: U (R2 (R U R' U R U2 R') R2) simplifies to U R' U R' U R U2 R (8/17)
CO: U2 B2 U B L2 D2 L2 D2 B (9/28)
Separate: x2 U' D R2 U R2 (5/35)
CP: D' U2 [this was a /really/ easy force during separation] (2/37)
EP1: M' U2 M U2 [I've decided to just reduce to Roux after 4a here.] (4/41)
Roux 4b: x2 y M2 U M' U2 M' (5/46)
Roux 4c: U' M2 U2 M2 (4/50)

[Pyjam]

I can't see what I did wrong. :/

I expected R U R' (right). U' D2 R does nothing good on my cube.

Here's another, hopefully I don't mess this one up.
U2 R2 F2 L' B2 L D2 L B2 L2 R' D' R2 U B2 L R2 D2 B' D
EO: D B' (2/2)

It's D' B', I think.
Thanks for this second sample.

[miotatsu]

I'd try this out, but I really can't figure out your organization of algs. You have things like "Sune on B", which will not happen, since you have already solved parity. "H on B" is the same as having T on both, but harder to recognize (imo).

Another thing, you don't really need to permute the corners of both layers, since it is rather easy to solve the corners of the D layer (or even the whole D layer) during separation. You could just use Y and J perms to solve the top corners, then go into L8E.

Sorry if my organization of algs is confusing, now that you mention it I see that it is pretty messy heh. Basically for the corner orientation step I use your system, pretty much all same algs although for mirror cases I prefer mirroring a single alg. Personally what I currently do is use the 25 alg set I have listed, half of the time you do a x2 for setup. In regards to the corner permutation I use EPPCBL because it makes the solving style more algorithm driven without requiring much to memorize. In terms of efficiency though I would not be too surprised if intuitively placing the bottom corners first is better.

Here's another, hopefully I don't mess this one up.

thank you for the examples, very nicely done! In this example I would have done either D2 L2 or U2 R2 and then fixed the belt during separation for the parity step, but it seems I get a bit longer of a solution that way