# A Novel ZZ Variant with a 1LLL, 72 cases, and easy recognition

#### 4Chan

1. 20% of the time, LL is just PLL
2. 20% of the time, 2-gen SHORT LL
3. 1/144 chance for LL skip
4. LL looks like a PLL, easy to see headlights and blocks and easier to recognize than ZBLL
5. ONLY 72 CASES FOR LL
6. During LS you don't have to find the F2L edge, you just have to look at orientation of U layer.

ZZ-A is a good method, but it takes a lot of practice. People are lazy, and practice takes a lot of time.
What if we had a method of comparable move count, easier recognition, and way less cases to store in our heads?

Step 1: EOLine
Step 2: Get to last slot
Step 3: Insert the corner and orient the other corners (HLS)
Step 4: Solve everything all at once. (WLL)

For Step 3, most people look for the edge and the corner. With this method, you don't have to look for the edge, you just look at the corner and the orientation of just the U side.

This is basically CLS, but since you don't have to worry about edge permutation, the algs are simpler and nicer. So you get all the easy cases of WV, SV, OLS, VLS, RLS, etc etc etc.

The F2L edge that floats in the LL can be lined up with the side it belongs, and this makes it so that all rotationally symmetric cases will look the same in our heads.
This means that every LL case to solve everything is just 72 cases.
And a lot of those cases are stupid conjugate PLLs and a ton of them are conjugate ZBLLs ONLY from the L set.
So if you knew L-Set ZBLL and PLL, then you already know the algs to do 80%+ of the cases!

For recognition, the last layer is 100% oriented, so there's lots of blocks and very easy to recognize!

You may ask, how can it be 72 algs?
72 is because there are only 6 possibilities for corner permutation:
1. No swap, all correctly permuted.
2. Front swap
3. Back swap
4. Left swap
5. Right swap
6. Diagonal swap

You may wonder, what about the OTHER diagonal swap? Due to rotational symmetry, there's only one set for diagonal swap!

So since there are 5 possibilities for corner swap, there are only 12 possibilities for edge swaps. Because we have the weird edge in the same place everytime, so not only does that remove rotational symmetries, every other edge in reference to that edge can only move in 12 possible ways.
So 6 * 12 = 72. ezpz.

For the 2gen set with no corner swaps, the algs are nice and short and flow decent too!

Like, REALLY BAD. If the case isn't a conjugated PLL or ZBLL, it's a really really bad case.
For example, you know how Summer Variation has some decent algs, but then there's one case that's just absolutively horrible?
The LL for this method is kinda like that.

So in summary, some closing thoughts:

This move count is roughly equal to ZZ+ZBLL, maybe more by a few turns.
At the highest level, the recognition/execution for ZBLL will be better.
This is a good method if you want to do 1LLL for a very low amount of algs.

Here is an example video:

Here are some example solves:

Scramble #1: L F2 U' R2 B' D B2 L' F U' D2 B U2 F2 B' D2 L2 B' L2 B

EOLine: x' D B' F' L U F' D' L2 D'

F2L -1 : R' U' L' U' L U' L' U R2 L U2 L R U2 R U2 R

HLS: U' R U R' U' R U2 R' U R U' R'

WLL: U' R U R' U2 R' U R U' D' R U2 R' D R2 U R U' R' U' R' U (Conjugated L-Set ZBLL)

Scramble #2: D2 F U2 F2 B D2 B U' L' U2 B2 R2 U2 D2 B2 R2 D L2 D2

EOLine + 2 Blocks : x' B' R B L U' L2 D2 U R U' R' D

F2L -1 : R U' R' L' U L

HLS: U R U' R' U' R U' R2 U' R U' R' U2 R

WLL: U' R U2 R U R' F' R U2 R' U2 R' F R U R U2 R' U R' (Conjugated PLL R-Perm)

Scramble #3: L F D' F2 D F' D' L' U B' D2 B R2 U2 F2 U2 D2 L2 U2

EOLine: x' B L2 D' F L D

First Block: U R U2 R' U R'

Left Block: U' L U' L' U2 L2 U' R' U L

HLS: R U' x' U R' D R U' R' D'

WLL: R2 U R' U R' F R' F' R U' F' U F R2 U' R2 (urgh, some of these cases are so bad)

Last edited by a moderator:

#### 4Chan

Thanks!!!

I've finished the 2-gen set, but I gave up after trying to generate the algs for the diagonal swap because I've been frustrated with how bad they are.
I think maybe I'll finish generating sometime when I feel like it, but you're welcome to try (or anyone else!)

#### Logiqx

##### Member
That's really cool!

When you say 60 algs have you forgotten to count the 12 cases without a corner swap? I.e. All corners solved and everything oriented but the last F2L edge floating in the top layer (aligned with centre) with 12 possible edge permutations.

I'd have thought it is more accurate to say 72 algs for "weird edge" scenarios (equivalent of PLL without AUF) plus the standard 21 PLLs when the F2L edge was accidentally solved during LS, making 93 algs in total.

Nevertheless it is very cool and I can see it having a lot of potential for ZZ. Learning the last corner algs (27 x 3) plus WLL (72) + PLL (21) is something most cubers could achieve!

#### Logiqx

##### Member
Thanks!!!

I've finished the 2-gen set, but I gave up after trying to generate the algs for the diagonal swap because I've been frustrated with how bad they are.
I think maybe I'll finish generating sometime when I feel like it, but you're welcome to try (or anyone else!)
Maybe find one fast alg to insert the edge and do the diag swap to leave EPLL?

#### 4Chan

Ohhhh, darn, you're right!!

I forgot to include no swaps in the total!!! My mistake! Yes, you're absolutely right, it's 72 at that point.
Can a mod edit it to 72?

Maybe find one fast alg to insert the edge and do the diag swap to leave EPLL?
WHOA THATS A SMART WAY TO DO IT

#### bobthegiraffemonkey

##### Member
Cool idea. What about having two algs per CLS to avoid diag? Kinda like the idea of learning at least two OLLCP per OLL. I've no idea if that's a reasonable thing to do here though. Either that, or wait and hope people find better algs.

#### Logiqx

##### Member
Ohhhh, darn, you're right!!

I forgot to include no swaps in the total!!! My mistake! Yes, you're absolutely right, it's 72 at that point.
Hehe. Just add an annotation to the video.

WHOA THATS A SMART WAY TO DO IT
On reflection you could do the same trick for all 6 corner cases, giving people a beginner version of WLL:

1) Insert the F2L edge and permute the LL corners - pick the best of the 12 algs in the corner set
2) EPLL

This could be a reasonable intermediate method in its own right but most importantly gives people a stepping stone to full WLL.

#### 4Chan

Annotation added! It makes me happy that people also think this is feasible!

Cool idea. What about having two algs per CLS to avoid diag? Kinda like the idea of learning at least two OLLCP per OLL. I've no idea if that's a reasonable thing to do here though. Either that, or wait and hope people find better algs.
Ohh, forums are cool things. People add in thoughts that I totally overlooked!

I agree, that'd be a good solution too

#### 4Chan

Well darn, I was hoping for some magical nice short fast algorithm.

Thanks for taking the time to gen
Those were better than the algs I tried to gen for diagonal cases LOL

#### Chree

##### Member
Not entirely sure I understand the method, and actually never gen'd algs before tonight... but here are some half fun RUD algs:

R' U' R D' R' D R' U R' D' R2 U' R2 D R2 U
R U R2 D R U R' D' R2 U2 R' D R U R' D'

Nothing great. I like the first one, though.

anyway... this is an awesome idea

Last edited:

##### Member
Really nice method. I've just gotten into using zz, might actually do this. Just one thing , I thought HLS had 216 algs?
I think it's meant to say CLS which is a substep of MGLS. Basically, place the last corner and orient the remaining ones without worrying about EP or CP (though preserve EO). I think this has 27 algorithms. In thins respect it is quite similar to MGLS-Z.

It was probably confused with HLS (which as you say has 216 algs and is a subset of OLS) because both leave what is essentially an OLL skip though HLS was designed for use with CFOP.

#### Logiqx

##### Member
I think it's meant to say CLS which is a substep of MGLS. Basically, place the last corner and orient the remaining ones without worrying about EP or CP (though preserve EO). I think this has 27 algorithms. In thins respect it is quite similar to MGLS-Z.

It was probably confused with HLS (which as you say has 216 algs and is a subset of OLS) because both leave what is essentially an OLL skip though HLS was designed for use with CFOP.
It's 27 algs for each corner orientation so 81 algs in total.

Inserting into the FR slot, I'd say they are akin to a simplified WV (D colour on front), SV (D colour on right) and their relation (D colour facing up). It should be possible to do them all 2-gen as well if you're so inclined (OH friendly).

Last edited:

#### Saransh Grover

##### Member
I think it's meant to say CLS which is a substep of MGLS. Basically, place the last corner and orient the remaining ones without worrying about EP or CP (though preserve EO). I think this has 27 algorithms. In thins respect it is quite similar to MGLS-Z.

It was probably confused with HLS (which as you say has 216 algs and is a subset of OLS) because both leave what is essentially an OLL skip though HLS was designed for use with CFOP.
Right. Thank You

#### Logiqx

##### Member
1. 20% of the time, LL is just PLL
2. 20% of the time, 2-gen SHORT LL
3. 1/144 chance for LL skip
4. LL looks like a PLL, easy to see headlights and blocks and easier to recognize than ZBLL
5. ONLY 72 CASES FOR LL
6. During LS you don't have to find the F2L edge, you just have to look at orientation of U layer.
I think you need to tweak some more numbers if I'm not mistaken:

- 1/6 of the time, 2-gen SHORT LL (i.e. subset with corners solved)
- 1/360 chance for LL skip
- ONLY 93 CASES FOR LL (72 WLL + 21 PLL)... although most people already know PLL it still needs to be counted.

Why 1/360 for a LL skip?

Calculation 1: 1 / (5! [edge permutations] x 6 [corner permutations] / 2 [cube law - parity]) = 1/360
Calculation 2: 1/5 (correct edge in LS) * 1/72 (PLL skip) = 1/360

Last edited: