Finger-friendly ZBLL algs: F-<R,U>-F' (with an interesting observation)

[blah]

Bunch of ZBLL algs. I have more, but I'll just post these first

F U R U2 R' U R U R' F' (10)
F R U' R' U' R U2 R' U' F' (10)

F R U R2 U2 R2 U R2 U R F' (11)
F R' U' R2 U' R2 U2 R2 U' R' F' (11)

F U2 R U' R U2 R' U' R U' R2 U2 F' (13)
F U2 R2 U R' U R U2 R' U R' U2 F' (13)
F R U R' U' R U' R' U' R U R' F' (13) - ZZLL
F R U' R2 U2 R U R' U R2 U R' F' (13) *
F R U' R2 U' R U' R' U2 R2 U R' F' (13) *
F R U' R' U R U R' U R U' R' F' (13) - ZZLL
F R2 U' R U2 R U R' U R' U R2 F' (13) *
F R2 U' R U' R U' R' U2 R' U R2 F' (13) *

F U R U2 R' U2 R U R' U R U' R' F' (14) *
F U R U' R' U R U2 R' U' R U R' F' (14) - ZZLL
F U R U' R' U R U' R' U R U' R' F' (14) * - ZZLL
F U R2 U2 R2 U2 R2 U R2 U R2 U' R2 F' (14) *
F U R' U' R2 U' R2 U2 R U2 R U R' F' (14) * - ZZLL
F U2 R U R2 U' R' U' R2 U R U R F' (14)
F U2 R U2 R U R' U R' U' R2 U' R2 F' (14) * - ZZLL
F U2 R2 U R2 U' R2 U R2 U2 R2 U2 R2 F' (14)
F U' R2 U R U2 R' U R2 U2 R' U' R F' (14) *
F U' R2 U R' U R U2 R2 U' R U2 R' F' (14) *
F U' R2 U2 R2 U2 R2 U' R2 U R2 U' R2 F' (14) *
F U' R' U' R' U' R2 U R U R2 U' R' F' (14) *
F R U R2 U' R' U' R2 U R U R U F' (14) *
F R U R' U' R U R' U' R U R' U' F' (14) * - ZZLL
F R U R' U' R U' R' U2 R U2 R' U' F' (14) *
F R U2 R' U R2 U2 R' U' R U' R2 U F' (14) *
F R U' R' U R U2 R' U' R U R' U' F' (14) - ZZLL
F R U' R' U2 R' U2 R2 U R2 U R U' F' (14) * - ZZLL
F R2 U R2 U R U' R U' R' U2 R' U2 F' (14) * - ZZLL
F R2 U R2 U' R2 U R2 U2 R2 U2 R2 U F' (14) *
F R2 U R2 U' R2 U' R2 U2 R2 U2 R2 U' F' (14) *
F R2 U2 R2 U2 R2 U' R2 U R2 U' R2 U2 F' (14)
F R' U R U2 R2 U' R U2 R' U' R2 U F' (14) *
F R' U' R' U' R2 U R U R2 U' R' U2 F' (14)

Couple of things you need to know:

I went through each and every one of these algs to filter out stupid useless ones like F U R' U' R U' R' U2 R U F' (Sune) and F R2 U2 R2 U2 R2 U2 R2 U2 R2 U2 R2 U2 F' (identity). So you can be sure that every single one of those algs up there does something useful.

HTM move count in parentheses.

Trivial: Every alg has its inverse somewhere in there as well, duh.

Excellent for 2H and OH For algs that begin with "F U/U'/U2 ..." you can always do "R U/U'/U2 y ..." (I'd do this for both 2H and OH as well).

And of course, the interesting observation: See the asterisks? Those cases are the ones in which 2 LL corners are diagonally swapped. Considering there's only a 1/6 probability of this occurring, its frequency of occurrence is abnormally high here. I don't yet know why though, anyone? If anything, I think these are worth learning because they kill off the longest and most annoying PLLs: N, N, E, V and Y (not exactly annoying, but still, long ). In case you still don't get what I just said: 2-gen OLL + N/N/E/V/Y = F-<R,U>-F' Sweet.

I have all the algs up till the 17-movers. Just ask me if you want it, but I doubt anyone will be interested I truncated those with 18 or more moves because I think it's an unreasonably high number of moves. But then again, 18 is short considering it's a 1LLL :rolleyes:

Of course, in case you haven't realized, this list is completely useless if you want to learn ZBLL because I haven't sorted the algs according to similar LL orientations/permutations yet, and I don't even have ZBLL diagrams They're just listed in order of move count because that's the raw output I got.

I bet this thread is gonna get less than 3 replies because it's useless to the "general population" - no one really cares about ZB huh?

Edit: I have a theory about those asterisk-ed algs that can be easily proven or disproven simply by performing them, but I'm too tired now, haven't slept for the last 26 hours So if anyone's interested, they can try it out I think they might all be the same alg, and if this is true, I'd feel stupid for starting this thread Because that implies that most of the other algs are probably the same too, and this list is actually just a couple of algs manifesting themselves again and again in different forms. I'm praying it's not. I have no idea what I'm talking about, this is what staying awake for 26 hours does to you.

Edit 2: Been through them once again, added indication for ZZLL algs

 

[byu]

Just for that, I'm going to learn ZBLL.

 

[deco122392]

im interested but i want to finish coll before i begin with zbll. btw how many coll case are usable in zbll?(im assuming all but im not sure)

 

[blah]

im interested but i want to finish coll before i begin with zbll. btw how many coll case are usable in zbll?(im assuming all but im not sure)

Yup, all.

 

[deco122392]

sweet business !!! alrite then im book marking this page =D

 

[blah]

Oops, looks like 14 is a magic number. I just went through a bunch of 15-movers and I couldn't get that anomaly again

F U R U2 R' U2 R U2 R' U' R U R' U' F' (15)
F U R U2 R' U' R' U2 R2 U R2 U R U' F' (15)
F U R U' R U R2 U R2 U2 R' U2 R' U2 F' (15) *
F U R2 U2 R2 U' R2 U R2 U' R2 U' R2 U F' (15) *
F U R' U2 R2 U R U' R U R U R U' F' (15)
F U R' U' R2 U' R2 U2 R U R U2 R' U' F' (15)
F U R' U' R' U R U R' U2 R U2 R U2 F' (15) *
F U R' U' R' U' R' U R' U' R2 U2 R U' F' (15)
F U2 R U R' U' R U' R' U2 R U2 R' U2 F' (15)
F U2 R U2 R U2 R2 U' R2 U' R' U R' U' F' (15) *
F U2 R U2 R' U2 R U R' U R U' R' U2 F' (15)
F U2 R2 U R2 U' R2 U' R2 U2 R2 U2 R2 U2 F' (15)
F U2 R2 U2 R2 U2 R2 U R2 U R2 U' R2 U2 F' (15)
F U2 R2 U2 R2 U' R2 U R2 U' R2 U' R2 U' F' (15)
F U2 R' U R2 U R U' R U R U2 R U2 F' (15)
F U2 R' U2 R' U2 R U' R' U' R U R U' F' (15) *
F U2 R' U2 R' U' R' U R' U' R2 U' R U2 F' (15)
F U' R U2 R' U2 R U' R' U' R U R' U F' (15)
F U' R U' R' U R U R' U2 R U2 R' U F' (15)
F U' R2 U R U' R' U' R U2 R' U2 R2 U F' (15)
F U' R2 U R2 U R2 U' R2 U R2 U2 R2 U' F' (15) *
F U' R2 U2 R U2 R' U R U R' U' R2 U F' (15)
F U' R' U2 R' U2 R U' R' U' R U R U F' (15)
F U' R' U' R' U R U R' U2 R U2 R U F' (15)
F R U R U2 R2 U' R U R U2 R' U' R' F' (15) *
F R U R U2 R' U' R' U R2 U2 R' U' R' F' (15) *
F R U R2 U' R2 U' R2 U2 R U R U R' F' (15)
F R U R' U R U2 R U2 R2 U' R2 U' R2 F' (15) *
F R U' R2 U2 R2 U2 R2 U2 R2 U' R2 U R F' (15)
F R U' R' U' R' U2 R2 U R2 U R2 U' R' F' (15)
F R2 U R2 U R2 U2 R' U2 R' U' R U' R' F' (15) *
F R2 U' R2 U R U R' U R U2 R U R2 F' (15)
F R2 U' R2 U R U2 R U R U R2 U' R F' (15)
F R2 U' R2 U' R U2 R' U' R U' R U R2 F' (15)
F R2 U' R' U R' U R U2 R' U R2 U R2 F' (15)
F R2 U' R' U2 R' U' R U' R' U' R2 U R2 F' (15)
F R' U R2 U' R' U' R' U2 R' U' R2 U R2 F' (15)
F R' U2 R U R2 U R2 U R2 U2 R U2 R F' (15)
F R' U2 R U2 R2 U2 R' U2 R' U2 R2 U2 R F' (15)
F R' U2 R2 U R' U' R2 U R' U' R U2 R F' (15)
F R' U2 R2 U2 R U2 R U2 R2 U2 R' U2 R F' (15)
F R' U2 R' U R U' R2 U R U' R2 U2 R F' (15)
F R' U2 R' U2 R2 U' R2 U' R2 U' R' U2 R F' (15)
F R' U' R2 U R U R' U2 R U R U R F' (15)
F R' U' R2 U R2 U2 R2 U2 R2 U2 R2 U R' F' (15)
F R' U' R' U' R' U2 R U' R' U' R2 U R F' (15)

 

[brunson]

Just for that, I'm going to learn ZBLL.

You let us know how you're getting along. I'm pretty sure there's still a "first person to learn full ZBLL" UWR slot to be filled. Jason Baum is about 150 algs up on you, so get cracking.

 

[masterofthebass]

Just for that, I'm going to learn ZBLL.

You let us know how you're getting along. I'm pretty sure there's still a "first person to learn full ZBLL" UWR slot to be filled. Jason Baum is about 150 algs up on you, so get cracking.

Ville has some more algs on Jason. I think if he utilizes all mirrors/inverses he knows, he's up to about 40% of ZBLL.

 

[mazei]

I thought you have work and still you got enough time to do this. Amazing. In any case, nice. I might learn a few cases. Like the ones where OLL is that symmetrical one.

 

[cuBerBruce]

I've been using GAP to look at the 126 right cosets (for the 2x2x2 cube) that you get when you consider the <U,R> subgroup of the <U,R,F> group. I've noticed that some cosets require 2 or sometimes more F-layer quarter-turns (along with any number of U and R layer turns) to be reached.

Of these 126 right cosets, 6 preserve the position and orientation of the DLF cubie. These 6 right cosets basically correspond to the 6 ways the corners of the top layer can be permuted with respect to another when the D layer is preserved. So I looked at what cosets can be reached when doing an alg of the form F <U,R> F'. What I found is that of the 6 right cosets, only 5 could be reached (with equal distribution among them). The coset that could not reach corresponds to the one that swaps only the UFL and ULB corners. So I believe algs of the form F <U,R> F' that preserve the bottom 2 layers and perform a diagonal swap of the U layer corners will have a probability of 1/5 rather than 1/6 of occurring. (I imagine that filtering out certain "useless" cases would also increase this value somewhat.)

I also looked at algs of the form F' <U,R> F. In this case, I also got only 5 of the 6 right cosets to be reachable. In this case, the unreachable coset corresponded to the diagonal swap case. So I believe that if you use F' <U,R> F instead of F <U,R> F', you won't get any LL algs corresponding to a diagonal swap of corners.

 

[Robert-Y]

"I also looked at algs of the form F' <U,R> F. In this case, I also got only 5 of the 6 right cosets to be reachable. In this case, the unreachable coset corresponded to the diagonal swap case. So I believe that if you use F' <U,R> F instead of F <U,R> F', you won't get any LL algs corresponding to a diagonal swap of corners."

I think that swapping any corners is impossible using F' <U,R> F because doing F' leaves you only 3 corners to play around with. The other 3 corners on the U and R face must remain in the same position after doing the <U,R> part. In order to permute anything using only U and R moves, you need at least 4 corners which are allowed to have a different ending position

Am I right? Correct me if I'm wrong

 

[blah]

"I also looked at algs of the form F' <U,R> F. In this case, I also got only 5 of the 6 right cosets to be reachable. In this case, the unreachable coset corresponded to the diagonal swap case. So I believe that if you use F' <U,R> F instead of F <U,R> F', you won't get any LL algs corresponding to a diagonal swap of corners."

I think that swapping any corners is impossible using F' <U,R> F because doing F' leaves you only 3 corners to play around with. The other 3 corners on the U and R face must remain in the same position after doing the <U,R> part. In order to permute anything using only U and R moves, you need at least 4 corners which are allowed to have a different ending position

Am I right? Correct me if I'm wrong

Yup, you're right. I had the impression that this was pretty trivial so I don't get what Bruce is trying to say

 

[cuBerBruce]

"I also looked at algs of the form F' <U,R> F. In this case, I also got only 5 of the 6 right cosets to be reachable. In this case, the unreachable coset corresponded to the diagonal swap case. So I believe that if you use F' <U,R> F instead of F <U,R> F', you won't get any LL algs corresponding to a diagonal swap of corners."

I think that swapping any corners is impossible using F' <U,R> F because doing F' leaves you only 3 corners to play around with. The other 3 corners on the U and R face must remain in the same position after doing the <U,R> part. In order to permute anything using only U and R moves, you need at least 4 corners which are allowed to have a different ending position

Am I right? Correct me if I'm wrong

Yup, you're right. I had the impression that this was pretty trivial so I don't get what Bruce is trying to say

Yes, Robert is right.

Although 5 of the 6 cosets were reachable, you would need to do additional <U,R> moves to restore the positions of both the DFR and DRB corners. So what I was saying really applies to algs of the form F' <U,R> F <U,R>.

 

[trying-to-speedcube...]

Just for that, I'm going to learn ZBLL.

As if we haven't got enough races on this forum, let's organize a race to full ZB (not ZBLL, full ZB)

Progress: I now know 32 ZBF2L cases and about 50 ZBLL cases.

 

[Lord Voldemort]

What if some of us are trying to use ZBLL with ZZ?

 

[trying-to-speedcube...]

Then don't compete in the race

You're right... Maybe we should do 2 races (because we don't have enough) one for ZBF2L and one for ZBLL.

 

[brunson]

Sounds fun, but how do you verify?

 

[MistArts]

Sounds fun, but how do you verify?

ZBLL attack.

 

[Lofty]

Can anyone make a list of all the algs achievable by F<R,U>F'?
Can you explain your terms cuberbruce? I have taken a group theory class this semester so I do know what cosets are I just don't know how they apply to this specific case.

 

[cuBerBruce]

Can you explain your terms cuberbruce? I have taken a group theory class this semester so I do know what cosets are I just don't know how they apply to this specific case.

When I originally posted my first message in this thread, I used the term "right coset" based upon the definition in Jaap's Puzzle Page (and Jaap mentions that some books define the terms left coset and right coset in the opposite manner). David Joyner (author of Adventures in Group Theory) and GAP itself seem to consider these left cosets. (GAP's built-in functions are for what it considers "right cosets" and I was passing it inverse elements so I was using what GAP would consider to be "left cosets." In any case, I'll just use the term coset in the discussion below.)

The positions of the 2x2x2 cube can be considered to consist of the positions of the <U,R,F> group. That group does not allow the DBL cubie to move (or change orientation), so the DBL cubie becomes a reference cubie with which the positions of the other cubies can be described. Let's call this group G. Also, consider the group H = <U,R>. For the 2x2x2, this group (H) consists of 29160 elements. It does not allow the DLF cubie to move, and the 6 remaining cubies can be permuted 120 different ways, with 243 orientation possibilities. 120*243=29160.

Since H is a subgroup of G, we can define subsets of G, known as cosets, of the form gH where g is an element of G, and H is the subgroup defined above. The coset gH for some element of g in G is the set of all elements gh in G such that h is an element of H. For example, we could pick g to be F. Then the coset FH would be any position reached by doing F followed by doing any element of <U,R>.

Since H has 29160 positions, the coset FH also has 29160 positions. Since G has 3674160 elements, there are 3674160/29160 = 126 separate cosets of the form gH. Since H=<U,R> does not allow the cubie at DLF to move, the cubie at DLF is one characteristic of such a coset. Since 7 different cubies can occupy that position (since the DBL cubie is considered a fixed reference, only the other 7 can occupy DLF), we have a factor of 7. That cubie can have three orientations, giving another factor of 3. Of the other 6 remaining corner cubies, if we know the position of 3 of them, we can determine the positions of the other 3, although this is somewhat complicated. There are 6 total ways the last three can be permuted (if we weren't restricting ourselves to one particular coset), that gives another factor of 6. So 7*3*6 = 126 cosets.

Let me try an example of taking a cube and making some maneuvers. Start with a solved cube. The solved cube is a position in the identity coset H. Then do F. Now the cube position is now a position within the FH coset. Now do as many U or R layer moves as you like. The cube will remain in the FH coset. Then do F'. The cube will no longer be in the FH coset. It might be back in the original coset H, or it might be some other coset. The cube will now remain in that coset as long as we only do U or R layer moves. Once we do another F layer move, the cube will switch to another coset.

We can similarly define <U,R,F> and <U,R> subgroups for the 3x3x3. The concepts are similar but we will now have 2322432 cosets, 73483200 elements per coset, and a total of 170659735142400 positions in <U,R,F>.

We can also define cosets using Hg instead of gH. These generally result in a different set of subsets of G, but the total number of cosets, and the number of positions per coset will be the same either way.

Hope this helps.