Commutators and Conjugates - The Ultimate Instructional Video 

Best Tutorial Ever!

This is the best explanation of Commutators and Conjugates really thankultimate very much!

Wow, this was really helpful. I've already applied this to perform 3-corner cycles on cuboids. Now I can solve a 2x2x4 with my own algorithm. This is really cool. I can't wait to make my own solutions for other puzzles.

As I said I haven't seen any single flip parity so I haven't had to 3 cycle center pieces as must be done after breaking up the flipped edge pieces as you've shown in your video. Thanks again for this video. I must say that even though you don't like doing orientation swaps I find them quite useful. Sometimes I like to make designs in my cubes and the orientation swaps make them easy. However I'm also glad you showed the edge conjugate that you used in the 4x4 corner parity video.

brilliant thorough explenation. I went straight from begginner method onto this and it made sense... thanks!

Great video, I guess it's hard to find a better start on commutators. Exactly what I was looking for, thanks!

I bought a Shen Shou 6x6x6 and did a couple of solves using the method outlined in this video and the other I.H. video showing the solution to corner parity. The corner parity is solved similarly to the 4x4x4 except you solve for pairs of cubes. The edge paritys I've encountered are two flipped on the outside edges of the four edge cubes and the other type was four flipped edges. In the first case I simply moved the row of cubes below one of the flipped edges and fixed the center pieces as

are you saying you would rather perform a 3-cycle and then an orientation swap, as opposed to just perform a 3-cycle with a simple conjugate? I think you missed the point of that example. I purposely did the commutator carelessly to demonstrate that you must be careful when performing commutators. and it was also a nice segue into explaining conjugates.

I LOVE THIS VIDEO I MAKES SO MUCH STUFF EASY TYSM!

For those who have trouble knowing if you'll need a conjugate to avoid orientation swaps, just remember that what happens to the last piece will be the inverse of what happened to the first one you swapped in. So for example if yellow goes from the right face to the bottom, whatever color is on the bottom of the piece you're changing with will end up on the right face. If this color doesn't belong on that face then you need a conjugate.

Yeah, I was into speedcubing for a while but it's mainly memorization. After that, I looked into other methods like block building when I started to get bored with 4x4x4 CFOP. So I think doing a basic CFOP but omitting the O/P algorithms for a more intuitive method of commutators will be much more fun and rewarding. I want to be able to just look at something or fiddle with it and solve it, not memorize a hundred cases. I trust this will help me with awkwardly shaped puzzles too.

Absolutely amazing video. Explained perfectly. Thank you!

@kirbyKosK9 that is probably the most difficult concept to understand in this video. go to 16:16 and pay careful attention to the example i do. if you still dont understand, send me a message.

Underrated

of course. commutators and conjugates can directly or indirectly fix anything, IF you are clever enough :)

This video is very, very helpful! Thanks a lot!

haha, no i just made it today after you asked the question. no problem, glad i could help.

Hello, thank you so much for this video. I learned a lot about commutators (I never heard of it until I watched this video). I can't wait to try and practice it. One question I have is; are you able to solve the rubik's cube without performing any OLL and PLL, and purely using F2L & Commutators? Thank you in advance!

glad i could help!

there is a "limitation" in the sense that the puzzle is physically bound to obey certain rules, but it does NOT mean it is limited as in some pieces are "untouchable". even a very simple commutator can be extremely useful with the right conjugate. Whether C&C can be done on ANY twisty puzzle without ANY limitation.... I am not sure. There are often ways to perform C&C on "bizarre" puzzles, but the commutators are so non-intuitive, it may take me months before i find one.

TUTORIAL OTTIMO

glad i could help! in my opinion, unless you are super hardcore into speedcubing, you should drop algorithms altogether and learn commutators. at least you're stimulating your brain rather than mindlessly doing algorithms.

Thank you for making this video. You are a good teacher. This is a heavy concept to grasp, because it is abstract and unrelated to anything learned in the past. As such, there is no place for the knowledge to attach in my head. I had to create new knowledge, which is more challenging than just adding to existing knowledge. There are other great tutorials out there and great RU-vidrs who I really like, such as J Perm, etc but your treatment of this topic was hands down the best. Ultimately, you taught, while the rest simply skimmed over the surface of the topic. Thank you!

Thanks a bunch. I've been looking for a resource on orientation and pair-swap commutators.

@movietk2000 it takes alot of practice because when doing a commutator and conjugate together, when you're at the "middle of it", aka when you've performed (Z X Y), sometimes you can "panic" and forget what to do because the cube looks really scrambled and instead of automatically performing (X' Y' Z'), you kinda try to intuitively figure out what to do next. in order to get really good you need to kinda ignore how scrambled the cube is and just trust that the commutator/conjugate will work.

Thank you for going slow, explaining parameters and not skipping steps.

Thanks!!!! Really like your video. Really explained well the concepts and lots of example. Every second of the 41 minute was worth it, very concise and thorough.

the same general rules apply, although with more complex puzzles it gets much harder to use commutators successfully. depending on the puzzle, it may not be possible at all. what puzzle are you using? typically i've found that with "symmetric" puzzles, performing commutators is still possible (example: megaminx) but with more "strange" puzzles it is not possible, or at least very difficult (example: square-1)

Amazing video, thanks so much! And I thought I understood commutators, LOL.

Very good turorial, i have seen it around 10 times so far and i always learn something new.

Thanks Im. Hum., I used some of this video to solve the 4x4 picture cube.

Finally got idea of comms and conjugates. Thank you!

thank you very very much for taking your time and making this wonderful video :)

for i x j x k cuboid puzzles, commutators can be used to varying degrees. if you have a really weird puzzle like, for example 3x4x5, then i imagine you'd be very limited in the commutators you could perform. for like 3x3x5 for example, you can still perform several commutators. Because of the "shape-shifting" of a cuboid puzzle, it does make it more difficult and often requires some creative thinking in order to find a useful commutator. that is what i have found from personal experience.

great accomplishment! if you really want to torture yourself, put supercube stickers on your 5x5. it the conjugates get kinda confusing when you're doing a 3-cycle to swap 3 center pieces all on the same side.

Thank you so much for this video. I have never enjoyed solving cubes as much as I do right now with commutators and conjugates. I enjoy solving 3x3, 4x4, 5x5 and 6x6 all with this method and I do it a different way every time. So much more fun than memorizing algorithms. Thanks again!

I prefer to use the minus sign for anticlockwise rotation. Let's name each face of the cube as Right, Left, Up, Down, Front and Back. Then define each rotation as a 90-degree clockwise rotation. R: One 90-degree clockwise rotation of the right face of the cube. 2R = R+R: Two 90-degree clockwise rotation which is equal to 180-degree clockwise rotation. -R: One 90-degree anti-clockwise rotation. 4R = 0: Four rotations is equal to zero since it is a 360-degree rotation and the cube doesn't change. R = -3R: One clockwise rotation is equal to 3 anti-clockwise rotations. With this notation it is possible to add and subtract the rotations. The only problem is that the addition is usually not commutative: R + U ≠ U + R But the addition of parallel layers is commutative: R+L=L+R U+D=D+U F+B=B+F

Hi, sorry for the dealy. I just got some time to practice this week, I tried to use this method to solve the whole cube and I almost did it, but I usually ended up with two unmatched pieces(two corners and two edges) and I have the hardest time trying to finish it. What I don't get it is how to use three cycles to solve the last two pieces, any suggestions..thank you

:D Commutators are so much easier than i thought now now i can solve with heise

May be the best video on commutators. Thank you.

31:50 this is where it gets awesome! 34:36 I had to watch it a few times... Ok, now I got it! You used a different layer as the buffer... that's really complicated to see at first. Thanks for the video!

I can't tell you how much I appreciate your work here. I've now used your technique to three cycle corner pieces both on the outer layer and inner layer. Same goes with the edge pieces. I can't say my knowledge of commutators has gelled but I'm slowly beginning to understand how to handle twisty puzzles using C&C. Bmenrighs videos are very good but still a little over my head.

taught me how to do: A perm, Ab perm, Ua perm, Ub perm, Z perm. And how to switch pretty much any set of 3 edges or corners. and I learned that there was other types of commutators than just three-cycles.

you are having trouble solving the last 2 corners and last 2 edges because it is impossible to do with commutators and conjugates alone! turn any single face 90 degrees and then use commutators and conjugates to solve the rest of the cube. the reasoning behind that 90 degree turn is actually really deep, and difficult to explain in a comment. perhaps I'll make another video sometime explaining that.

Could you do a video solving the last layer of a 3x3 and guiding us through your train of thought?

You're welcome, but doing a full solve using only commutators would be like mowing your lawn with a pair of scissors. Commutators are really useful when the majority of the puzzle is solved and only a few pieces are out of place. Although it is possible to solve a completely scrambled cube using only commutators, it would be very tedious and inefficient.

this is a question difficult to answer in a youtube comment, which is why i made a video response to this video. check it out for an answer of how to solve corner parity on a 4x4 cube with commutators and conjugates. in hindsight, i should have included that topic in this video.

Estoy esperando que bajes otros tutoriales de resolución del cubo de rubiks 3x3 en el cual apliques la técnica de no memorización de algoritmos (Ryan Heise)

there might be some mathematical way to determine if a conjugate is needed, but i'm not sure what that is haha. only way i know of is by checking it out for yourself.

before. In the case of the four flipped pieces I moved two rows of center pieces below a pair of side by side flipped cubes, fixed the center pieces and then solved for the remaining edge pieces. I assume there is one other edge parity case where-in two center edge pieces would be flipped. That would be solved by moving a row of cube under one of the flipped edges up and again fixing the centers via commutator and then solving for the remaining edge pieces via C&C.

Here's how to solve that edge parity problem that I encountered with three edge pieces flipped and a single edge piece in another part of the cube also flipped. You can do an orientation swapped to solve the two center pieces and then move a row of cubes in the same row as one of the two remaining flipped cubes. Then fix the centers in a manner similar to the one used to fix the centers of the 4x4 in this video. You'll then end up with four edge pieces in one slice of the cube and one in the

Here's how to do corner conjugates. The bottom sticker of the bottom cube that will be three cycled to the top corner must have the same color as the cubes surrounding the cube that is being: a) Face moved X= F'U'F or X=FUF' b) Side moved X=RUR' or X=L'U'L c) Top moved X=R U2 R' U' RUR' or X=L' U2 LU L' U' L The Z move positions the bottom cube so that it's bottom sticker matches the stickers facing the same direction as the cube sticker being moved to the bottom side of the bottom layer.

Seems like i'm not clever enough!!!!!! Well thanks by the way for ur video that was the one I was looking for a long time seems nobody was able to explain me why commutators weren't able to solve parities. I'm trying to solve parity for 2 corners, i do a 90° layer like the case u mentionned i fix the corners in 2 commutators but i can't find the commutators for the last 4 wings to swap, any clue?

Wow. Thanks you so much. This is very helpful. My cubing experience got so much better because of this. But i have some problems. How do you know what will be the orientation of a piece after a three cycle? And i have problems with edges too. How can you fix two single edges in a 4x4x4?

in another slice. This is similar to the situation found after fixing the centers of the 4x4. And the remaining edge pieces are solved the same way, i.e. conjugates to move edge pieces which are in the same slice to the top of the cube so that they can be three cycled with edge pieces which are in the same slice.

Ok. I missed the point then. But you did say that you couldn't solve that particular commutator without a conjugate. So, I think you can see how I interpreted it the way I did. You were talking about the original state with the three cycle, but I interpreted it that you were talking about the state with the 2 corners out of orientation.

This video is great! One point of clarification though. Is there a way to tell if you'll ned a conjugate for a commutator? Do you just have to look ahead?

At 15:33, you don't have to use a conjugate to solve that. It's just a corner orientation swap, like you showed at 5:46. Red is now your bottom layer, rather than yellow, and instead of Y being a 90 degree rotation, it's a 180 degree rotation. Great video, though.

Oh sorry for the reply, i didn't look carefully at ur other videos therefore i didn't see that u made a special video for corners parity. I'm going to take a look at that one deeply because u're doing it much more faster so i'll hope i can see it through ur finger tricks. Many thanks.

that is incorrect. commutators and conjugates can be done but only to certain pieces, and you have to be very clever how you go about it.

Do you do any blind solves? Im learning 3 style and this video is very helpful for getting the intuitive aspects of it

Okay, on my third try I did get edge flip parity. A weird one. The three white/blue edge pieces are facing the wrong direction and so is the center yellow/blue edge piece. This requires some thinking and I'll let you know if I solve it.

Whilst I really, truly appreciate you making this video, the whole time I was screaming in my head “WHY?!” You’ve covered the what and how, but, for me at least, without understanding the why, it’s useless.

@kirbyKosK9 or better yet, post a video response with the problem you are encountering and i'll see if i can help you out.

Really helpful video but I wish you would not talk as much and just get on with it struggling to follow when you keep stopping

Thank you so much for this video! I know it's old but it really helped me solve the last two steps of the Heise method!

Thanks a lot! Especially about the 4x4 parity. The corner one I managed to solved intuetivelly by some setup moves. But I tried so many ideas on the edge one with no success (thanks about the group theory note). I will play the video multiple times, train on 3x3 and possibly will be able to intuitevelly solve the 4x4.

You can't use C&C on a 3x3x4 cuboid. There's no way you can get the necessary single cube layer overlap necessary to perform a three cycle.

Is it also possible to indirectly fix permutations of corners parity case in a 4X4 with commutators and conjugates?

Thank you! These explanations are incredible! I was able to understand 3 cycles and now I can come up with algorithms by my own! You are one smart guy and I admire that you took the time to make an awesome video.

Tutorial ottimo , nelle mie risoluzioni preferisco utilizzare i commutari e coniugati , secondo me molto interessanti

how do you tell if you are going to hit an orientation swap after you do the 3-cycle without the conjugate?

Thank you for this very well explaining video!

This is a well thought out video in order to better wrap your head around the Rubik's cube, thank you.

This is SOOO Good! I learned 3style for BLD from this! I cannot believe I could learn from this tutorial when you weren't even teaching blind!

which software are you using there?

He probably used a commutator to set up the examples, and than showed you how to fix it.

This is by far the best commutator/conjugate tutorial on RU-vid!

I don’t know if you still see this, but thank you so much for this. When you showed the two flipped edges next to each other I decided to pause and try to figure it out myself, it took me like 10 minutes but I did it. However I reached a different solution. I like yours better but I am still proud of mine: (x: F E F M F2 M’) (y: D) (x’: M F2 M’ F’ E’ F’) (y’: D’)

Watched your video over and over many times and practiced again and again a few months before slowly getting the skill. Don't know how to thank you ! Once moved to the 4x4, I keep ending with 2 corners located opposite / diagonally. I tried rotating 1 layer with the wrong piece a single turn then proceed with solving but I always end up in corner pair in the wrong position again. Does it matter which layer I make the single turn before proceeding in solving ? One of your comment said "i made a video response to this video. check it out for an answer of how to solve corner parity on a 4x4 cube with commutators and conjugates". I cannot find that video. Can you share it here ?

Thank you so much for this extraordinary explanation of commutators and conjugates! It is, without any doubt, the best one and the most useful that I've seen!

Omg I finally realise the derivatives of the PLL algs yay

This was fascinating and well done, thanks! So excited to try now.

Thank you. That at least solve my unsolvable problem he he he. So if I cannot solve it unless I do the one face move. In your video I find the case that you describe with the corners in 14:16. Two things here please: I usually learn repeating things so I tried to figure out your case with an algorithm, I got lost. Then I visited the virtual cube site that you use. Is there any way in that site to change the colours of the stickers so I can get the case you are explaining from the position...

Where would one acquire the virtual simulator you use in your video?

The other option is to take the pieces out of my cube and start with that position, which I think I will do, but just wondering if I can do in that virtual cube, when I use the stickers cube it goes black, how can I assigned colors, if it is possible I mean. The other one is I think if I study that case carefully it will help me to solve the two pieces as you did. So I will let you know. Another video explaining the 90 degrees (group theory?) would be very much appreciated, many thanks....

Thanks a lot. Your VDO is really helpful. Now I get the idea of commutators & conjugates a lot more but still need a lot of practice, though. I always screw up the cube coz I always forget what I did for the conjugate turns...

nothing is difficult, if you practice enough

@jfkx007 thanks!

lets say on the 3x3x4, for example, you're holding the cuboid such that the right and left sides of the cube are the "3x3" part. therefore you can only rotate the U D F B faces 180 degrees. (you can't do quarter turns on those faces.) a possible commutator would be: in the form of (X) (Y) (X') (Y') (L' E2 L) (D2) (L' E2 L) (D2) E2 is a slice "equator" type move, in line with the U and D faces. so you see commutators can be performed but you're limited because of the "bandaging" of the puzzle.

This video is a very good video, it helped me a lot. Commutators and Conjugates allows you to use your own ways to solve a rubik's cube.For example, I can solve all the corners first (after I solve the top four corners, I solve the bottom four corners), and then solve the edges using 3 cycles, and orientation swaps (if needed), and if I solve the first two layers of the rubiks cube, I can solve the final layer either by solving the corners first, then the edges, or vice versa, neither will conflict with the other. And using commutators and conjugates, if I solve all the corners first, and then the edges, this will be much faster than Beginners corners first solution method for Rubiks cube.

thank you sir for the kind words

Great ! A must ! love you

also, i watched Nan Ma's video. its a good video, but it fails to mention something crucial. the theorem "if supp(X) intersection with supp(Y) is one piece, then X Y X' Y' is a 3-cycle" This is true, but the converse is not. For example, the commutator: (R2 F2 R2) (D2) (R2 F2 R2) (D2) This commutator is a 3-cycle, but the intersection of supp(Y) and supp(X) is 3 pieces. this is actually an extremely useful commutator for cuboids, because it does not use any quarter turns in any layer.

this is a well known algorithm among speedcubers and its actually a commutator but its not immediately apparent why that is the case. it turns out that the conjugate and "algorithm x" actually overlap and therefore the commutator is disguised. If we consider the form, (Z) (X) (Y) (X') (Y') (Z') then, (R2) (R F R') (B2) (R F' R') (B2) (R2) is the form of that commutator (i think you added an extra "ui" accidentally). R2 and R combine to make R', so we have: R' F R' B2 R F' R' B2 R2 Boom.

@jfkx007 i explain how to fix parity on larger cubes with commutators starting at 31:25. technically speaking, commutators cannot solve parity alone. that is why you have to do that extra inner-layer 90-degree turn. in general, solving parity with commutators is a long and tedious process, which is why i didn't spend too much time on it in this video. i showed that example simply to show that it is possible to solve parity intuitively, rather than with an extremely long algorithm.

thanks!

@CubicNL glad I could help!

yea those are some tough ones. a commutator is sometimes very difficult to perform if you cannot find the conjugate. perhaps I'll make a video showing some of those in the future.

move conjugates, then the three cycle and then of course undoing the three move conjugate so we're talking about 12 moves to do the three cycle move. Anyway no more cubes for me. The large cubes simply use the same procedure but involve more cublets.. I'm moving on to other twisty puzzles like Pyraminx, megaminx etc. They're cheaper and more challenging.