# 4x4 methods



## elrog (Mar 12, 2013)

The usuall approach on solving a 4x4 is to reduce to a 3x3, and use a 3x3 method to solve the cube, with the exception of parity cases. This is good because a fast 3x3 time can mean a fast 4x4 time. It could be improved upon by using less moves, but I'll talk about that later. I noticed a page on a Roux method for the 3x3x4 cube on a link on the wiki. It was a link from the 4x4 KBCM page. Then I began to wondaer, why always reduce to a 3x3? I know reducing to a 2x2 is impractical, butYou could reduce doen to a 3x3x4 or a 4x4x3 or something of the sorts. I think it could be interesting doing a 2x3x3. You could also do this with larger cubes such as the 6x6. Reduce the 6x6 down to a 4x4 or a 3x3x6. The possibilities are massive.

Are there already method out there that reduce to something other than a 3x3?

I've notices that direct solve methods for the 4x4 use many 3 cycle commutators to solve edge permutation, and thus solve orientation. This leads me to beleive that when an edge is in its correct place, it is oriented correctly and it flips while it is not in its correct cycle. Thus by orienting the edges, you limit their possible permutations. Knowing this, how practical would a 2 look ELL be doing orientation and then permutation, or is my assumption even correct? I am making a list of the orientation cases currecntly, and there doesn't seem to be too large of an amount. I'm worried that the permutation could take many, so you could reduce this by AUFing and doing L4C rather than CLL and ELL.

As for reducing the cubes state to that of a different cube/cuboid, you can save quite a bit of moves by solving centers and matching edges at the same time, however this takes more time to recognize and preform. My idea is to save mores 1: by solving some center peices, then match some edges, then do more centers, and by 2: placing edge pieces while you change them out for unmatched edges. I also had the idea of doing F2L on the 4x4 just as it is done on the 3x3, then with 1 slot emptly, go back and keyhole the 3rd layer edges. The last slot could then be solved by placing both edges and the corner at once. These are just some ideas I've been toying around with. I'm not sure how they'd actually work in the hands of a more experienced speedcuber.

I've noticed that a corners first method decreases more and more as the number of layers of a cube increases. This however would not be true of an edges first method, because a cube always has 8 corners. This gives me the idea that a edges first method may not be so bad for something like the 7x7. I do not have a 7x7, 6x6, or a 5x5, so I can't try this out. 

I've noticed that Mf8 4x4s can have the circle either on the 4 center pieces, or on the edges. I don't think there is one, but I'd pay just about anything (if I had anything to pay, that is) for a 4x4 with a circle both on the edges and center pieces. That'd be a real challange .


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## qqwref (Mar 12, 2013)

elrog said:


> I know reducing to a 2x2 is impractical, butYou could reduce doen to a 3x3x4 or a 4x4x3 or something of the sorts. I think it could be interesting doing a 2x3x3. You could also do this with larger cubes such as the 6x6. Reduce the 6x6 down to a 4x4 or a 3x3x6. The possibilities are massive.


Those are all impractical too. For the 4x4x4, solving the cuboids is pretty slow (not much faster than a 4x4x4 itself, and certainly not on the same level as a 3x3x3). Reducing to a 3x3x3 lets you freely use quarter turns on all axes to set things up, but reducing to any cuboid means that two of the three axes only let you do half turns. Thus the inevitable commutators (or "condensed" commutators such as uRUR'u on the 4x4x4) will have to involve half turns and thus involve harder recognition and more qtm.



elrog said:


> This leads me to beleive that when an edge is in its correct place, it is oriented correctly and it flips while it is not in its correct cycle. Thus by orienting the edges, you limit their possible permutations. Knowing this, how practical would a 2 look ELL be doing orientation and then permutation, or is my assumption even correct?


You could do it, but there may be a lot of cases, and the algorithms won't be nice at all (half the permutation algs will have OLL parity). There may be as many as ~220 separate permutation cases, counting mirrors/inverses as separate cases. In normal 3- or 4-look 4x4x4 ELL most of the algorithms are fast 8-move commutators, but in this you might have permutations such as a 4-cycle + a 3-cycle, with absolutely no nice short algs - and to make matters worse, AFAIK, there is no optimal 4x4x4 solver out there.



elrog said:


> My idea is to save mores 1: by solving some center peices, then match some edges, then do more centers, and by 2: placing edge pieces while you change them out for unmatched edges.


It's all well and good to just say this, but how do you plan to actually do it? Some methods try stuff like this already (e.g. Yau, K4) but I think any improvement of this type will require the solver to be a lot more careful about exactly what they do, compared to Reduction - solving a specific number of edges, carefully placing each solved edge as you go, or making sure not to turn certain layers. Maybe this won't be a problem, maybe it will.



elrog said:


> This gives me the idea that a edges first method may not be so bad for something like the 7x7. I do not have a 7x7, 6x6, or a 5x5, so I can't try this out.


Cage, centers last, or sandwich (although that does start with two centers). I used to do a full "centers last" method for computer cubes, and it's not all that bad given the ability to turn slices easily, but I think on a real cube it would be much slower. Doing the centers last pretty much guarantees a lot of 8-move slice-heavy commutators, and you're just not going to do those anywhere near as fast as someone intuitively matching up the centers with a couple of moves per piece.



elrog said:


> I've noticed that Mf8 4x4s can have the circle either on the 4 center pieces, or on the edges. I don't think there is one, but I'd pay just about anything (if I had anything to pay, that is) for a 4x4 with a circle both on the edges and center pieces. That'd be a real challange .


Are you talking about the Crazy 4x4s? The small circle adds an "inner 2x2x2" to the puzzle, and the larger circle adds that and also requires the normal 4x4x4 centers to be solved to the correct locations (like a supercube). I can't think of anything both circles would add over just the large circle, except perhaps changing which stickers map to which pieces.


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## elrog (Mar 13, 2013)

Yes, I can see how the reduction to half turns could pose a problem, but just look at HTA. It reduces to only double moves, so surely it must be possible to reduce a single axis to double moves. The only problem for this is making it easy to recognize.

The number of algorithms may be able to be reduced by doing edges before corners as I mentioned. The algs may be harder to preform, but I'm sure they will be less moves than all of those commutators added up. To make up for the slower turn speed, you could have much better recognition time.

I'm not exactly sure how to place the edges while making them, but I don't see it being as hard as you make it sound. I'm sure I can figure something out that works well.

I was not referring to the cage method. I was saying, save the corners for last, not the centers. This doesn't necessarily mean that you solve all 8 of the corners last though. I'm thinking about just solving 3 of them while solving edges which would allow for much eaiser movement for the edges. The last 5 corners can always be solved with 2 3-cycle commutators.

Yes, I was talking about the crazy 4x4. How would adding this circle not add complexity to the puzzle? You would have to match inside centers up with the outside ones. You may also have to preserve them along with everything else unless you save them for last. My idea is to have some sides that when turned, the large circle stays and the small ones moves, and other sides will have the large one move and the small ones not. You may have a side in which both move, or none move. This would add so many more options to the puzzle. If you really wanted, you could just make it like a circle cube where you can turn the circles independently of the side to mix them up. You would just need different kinds of center pieces that you can change in and out to change what kind of puzzle you want it to be.


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## qqwref (Mar 13, 2013)

elrog said:


> Yes, I can see how the reduction to half turns could pose a problem, but just look at HTA. It reduces to only double moves, so surely it must be possible to reduce a single axis to double moves. The only problem for this is making it easy to recognize.


Oh, of course it's *possible*. You can reduce any even cube to any smaller cuboid. But the question is, is it fast to do so? (Probably not.)



elrog said:


> To make up for the slower turn speed, you could have much better recognition time.


I think the poeple who are good at ELL have pretty good recognition time. You can also do it in 3 steps (one edge, second edge, last two edges) which has really nice recognition, and the algs are mostly commutator-based.



elrog said:


> I'm not exactly sure how to place the edges while making them, but I don't see it being as hard as you make it sound. I'm sure I can figure something out that works well.


I'm not saying it's super hard, just that you will have to plan out a specific method. It's not the kind of thing you can do on the fly or just add on to the current method to consistently save moves.



elrog said:


> I was not referring to the cage method. I was saying, save the corners for last, not the centers.


Why? It would just end up slower, like all the 3x3x3 methods that leave most or all of the corners for last.



elrog said:


> Yes, I was talking about the crazy 4x4. How would adding this circle not add complexity to the puzzle? You would have to match inside centers up with the outside ones.


You'll have to play around with some Crazy 4x4s (or read the twistypuzzles topics on them). They don't work the way you think they do.



elrog said:


> My idea is to have some sides that when turned, the large circle stays and the small ones moves, and other sides will have the large one move and the small ones not. You may have a side in which both move, or none move.


Yeah, that would be pretty interesting for sure. Probably a monster to make though, and good luck deciding which types are the most interesting


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## Christopher Mowla (Mar 13, 2013)

elrog said:


> I've notices that direct solve methods for the 4x4 use many 3 cycle commutators to solve edge permutation, and thus solve orientation. This leads me to beleive that when an edge is in its correct place, it is oriented correctly and it flips while it is not in its correct cycle. Thus by orienting the edges, you limit their possible permutations. Knowing this, how practical would a 2 look ELL be doing orientation and then permutation, or is my assumption even correct? I am making a list of the orientation cases currecntly, and there doesn't seem to be too large of an amount. I'm worried that the permutation could take many, so you could reduce this by AUFing and doing L4C rather than CLL and ELL.


I've already created a method very similar to this. See this post. My method has a 3rd ELL step though. The first step is to solve one dedge slot completely (with one 2 2-cycle algorithm) and then do OLL and PLL. I have calculated the number of OLLs and PLLs of wings for the last layer already (and cuBerBruce helped me correct my mistakes). See the attachment in this post. You can see that solving one dedge significantly reduces the number of algorithms to learn. If you see the method PDF, I recently made 3-cycle + 2-cycle algorithms which could be much better to use in practice. I posted them here. I have been creating algorithms for other PLL cases, but I already have algorithms for all OLL cases, although I haven't posted the full set yet...until now:


Spoiler



The number of STM (btm, BHTM) for each algorithm is listed. As many who are familiar with k4 will notice, some of the algorithms are not new (8 move 3-cycles can handle some of the 2 unoriented wing OLL cases, but I used the following commutator to be able to modify it slightly to handle all 4 cases). But for higher number of wing edges unoriented, you guys might notice some new algs.



Spoiler: In 3 or less Composite Edges



*2 Unoriented wings*
Case 1
 [Rw2 U2: [r', U' R2 U]] (11)

Case 2
[Rw2 U2: [r', U R2 U']] (11)

Case 3
[Rw2 U2: [U R2 U', r]] (11)

Case 4
[Rw2 U2: [U R2 U', r']] (11)

*4 Unoriented Wings*

Case 1
[R B Uw2 Rw2 U2: r2] (11)

Case 2
 U2 r U2 r2 x U2 r U2 l U2 l' r' U2 r x' (13)
r U2 l D2 l' U2 M U2 r D2 r' U2 l' (13)
(Or any opp two edge flip 3x3x3 alg).

Case 3a
L r' U L' U' l r U L U' Lw' (11)

Case 3b
R' l U' R U l' r' U' R' U Rw (11)

Case 4a
Lw U L' U' l' r' U L U' r L' (11)

Case 4b
Rw' U' R U l r U' R' U l' R (11)

Case 5
[r2 U2 F' M' U': r2] (11)

Case 6
[r2 U2 F M' U: r2] (11)

6 Wings
Case 1
Rw' U2 r U2 Rw' x' U2 r' U' R' U' Rw' U2 Rw U R U' Rw R U2 x (19)





Spoiler: In 4 Composite Edges



*4 Wings Unoriented*
Case 1
M r2 U' r2 F2 r2 U r2 U' F2 U M' (12)

Case 2
[F' R u2 R' F, U2] (12)

*6 Wings Unoriented*

Case 1
[r': M U' r' U M' Rw' U2 r U2 Rw U2 M U' r U M' U2] (19)
[z U2: R2 U r U R' U' l' r' U R U' l2 U' R U l' U' R] (20)

Case 2
[z' U2: L' U r U' L' U r2 U L' U' l r U L U' l' U' L2] (20)

Case 3
[r': U2 M U' r' U M' U2 Rw' U2 r' U2 Rw M U' r U M'] (19)
[y' z U2: R2 U l' U R' U' l r U R U' r2 U' R U r U' R] (20)

*8 Wings Unoriented*

Case 1
[B2 r2 U2 F M' U: M2] (13)
[B2 r2 U2 F' M' U': M2] (13)
[x U2 l2 U F L U R2 F: M2] (18)
(Or any 4 flip 3x3x3 alg).





Of course, we can solve all wing edges in the last layer of the nxnxn cube in 1-2 algorithms (1-2 very long conjugates of the inner layer l and r moves), as I proved last year.




elrog said:


> As for reducing the cubes state to that of a different cube/cuboid, you can save quite a bit of moves by solving centers and matching edges at the same time, however this takes more time to recognize and preform. My idea is to save mores 1: by solving some center peices, then match some edges, then do more centers, and by 2: placing edge pieces while you change them out for unmatched edges.


See the spoiler in this post. Is that the type of solution you are referring to here? I did another example solve or two with a full explanation a few posts later in that thread.



elrog said:


> I also had the idea of doing F2L on the 4x4 just as it is done on the 3x3, then with 1 slot emptly, go back and keyhole the 3rd layer edges. The last slot could then be solved by placing both edges and the corner at once. These are just some ideas I've been toying around with. I'm not sure how they'd actually work in the hands of a more experienced speedcuber.


I think I pm'd you a link to my F3L documents to solve both wings in an F3L slot simultaneously, but here is a link. (See the linked post and the one immediately after).


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## cuBerBruce (Mar 13, 2013)

elrog said:


> The last 5 corners can always be solved with 2 3-cycle commutators.


No. For example, 5 corners twisted in place can't be solved with 2 3-cycle commutators. Odd parity cases can't be solved at all using only corner commutators.

EDIT: Technically, odd parity cases can't be solved purely with commutators, even if not restricted to ones affecting only corners. Edge commutators can make dedge parity odd to match the corner parity, but an additional quarter turn would still be needed to return corners and dedges to even parity.


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## elrog (Mar 13, 2013)

I think you may be right about reducing to another cuboid qqwref, but it's worth a shot. Meanwhile, I'll be working on a good way to place edges while solving venters.

Thanks for all the links cmowla. I will be reading up on them. I knew somone must have though of doing ELL in 2 steps. From my understanding, your FMC 4x4 method was pairing edges and centers at the same time. I was meaning more of doing something like solve the top and bottom centers, then match the 4 bottom cross edges, then place the 4 edges wihle matching up your last 4 centers.

That is a good point about the parity for corners bruce, although you did make me feel stupid by pointing out the pure flip cases... That is a good reason not to solve edges before corners. It would still be possible to have corner parity even after the edges are solved, meaning that my idea of reducing ELL cases by doing corners after is a bad one.

Thanks for the input guys.


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## TMOY (Mar 16, 2013)

qqwref said:


> Doing the centers last pretty much guarantees a lot of 8-move slice-heavy commutators


No. Using 4-move comms wisely reduces by a lot the number of 8-move ones needed.


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