# How to construct commutators on any puzzle



## trying-to-speedcube... (Jul 2, 2009)

*PLEASE NOTE: This tutorial is not to learn how to construct cycles of 3 particular pieces, but rather how to construct cycles of 1 particular piece, a random piece in its layer and and other random piece on the puzzle.* 

A commutator can be constructed by constructing an algorithm of the form A B A' B', where A is a move which inserts a piece into a slice without destroying any other than the inserted piece. B is then a move which turns that slice, after which A is undone to restore the rest of the cube, except for the commutated piece, and at last B is undone to restore the rest of the slice. This way one can intuitively cycle 3 pieces of the cube with each other.

On a 3x3 cube this is easy, because the insertion can always be done in 3 moves, for both edge and corner. As we move on to more complex puzzles, like the 4x4, still the insertion can always be done in 3 moves. As we move on to other shapes though, like dodecahedra, things get different. An insertion of three moves on edges of a megaminx is impossible. Here we would have to do something like L' R F L R' as the A-move.

As we move on to even more complex puzzles, the insertion is very hard to find out. Therefore, we will have to break down the A-part of the commutator into a conjugate of the form A C A'.

If we see A as a move which isolates the piece that is wished to commute into a random layer and C as a move of that random layer, you will see that after A C A' the original slice B is unharmed, except for the piece that is wished to commute. 

You can find the A-part by simply picking the piece out of its slice, and further isolating it from all pieces from its slice, until you have isolated it so far, that it is in a slice without any other pieces from its layer. 

This method makes it easier to construct the A-part of the commutator.

Example 1:

http://users.skynet.be/gelatinbrain/Applets/Magic Polyhedra/dodeca_f9.htm

Try to insert a piece in the UF edge center piece.
(Put White on top, and lightblue on front, to refer to colors when notation isn't clear)

As the insertion is hard to find out, we will do r' to isolate the piece from 18 of the other white pieces. Now it has only 11 pieces next to it. Now do (ld) to isolate the piece from 9 more white pieces. It has 2 pieces left in its current layer. Do d' to isolate it from the last 2 pieces. As you can see the pink layer has only one white piece in it; the desired piece. So now you have completed the A part of the conjugate for the insertion. The C-part will just be (LD), to move a random piece into its position. Now undo A; d (ld)' r. You will see that a random edge center piece has been inserted into the desired position. Now do the B-part of the commutator; U. Now undo the whole A-part of the commutator; r' (ld) d' (LD)' d (ld)' r. And undo the B-part of the commutator; U'. Seemingly, you now have swapped two pieces, but you have done a 3-cycle. Because 2 of the 3 pieces are identical, this is impossible.

The full commutator is: *r' (ld) d' (LD) d (ld)' r U r' (ld) d' (LD)' d (ld)' r U';* 16 moves.

Example 2:

http://users.skynet.be/gelatinbrain/Applets/Magic Polyhedra/dodeca_f9.htm

Now try the same with a corner center piece.

This one is a bit harder to isolate. We will start with an r', then a (rd)', so we have 2 pieces left around the desired piece. With l' we move one of the two edge center pieces away, and with d we move the other one away. Now we can do (LD) as the C-part of the conjugate, and undo A; d' l (rd) r. The B-part is U. Now we undo the A-part; r' (rd)' l' d (LD)' d' l (rd) r, and finally undo B: U'.

The full commutator is: *r' (rd)' l' d (LD) d' l (rd) r U r' (rd)' l' d (LD)' d' l (rd) r U';* 20 moves.

Exercise 1:

Commute a wing edge on http://users.skynet.be/gelatinbrain/Applets/Magic Polyhedra/dodeca_f10.htm

Exercise 2:

Commute a wing edge on http://users.skynet.be/gelatinbrain/Applets/Magic Polyhedra/icosa_v14.htm

Ultimate exercise:

Commute an edge center on http://users.skynet.be/gelatinbrain/Applets/Magic Polyhedra/dodeca_f10.htm


----------



## Mike Hughey (Jul 2, 2009)

Nice explanation! Hopefully it will help a lot of people.

But now I'd like to see an explanation for constructing commutators on the kind I have the most trouble with - the ones where the only possible move turns half of the puzzle. So how do you do this on a skewb or a UFO, for instance?

I've always had to just play with those puzzles until I find something that will work. Is there a good way to construct commutators on these types of puzzles? After all, your title says "any puzzle".


----------



## ChrisBird (Jul 2, 2009)

Mike Hughey said:


> Nice explanation! Hopefully it will help a lot of people.
> 
> But now I'd like to see an explanation for constructing commutators on the kind I have the most trouble with - the ones where the only possible move turns half of the puzzle. So how do you do this on a skewb or a UFO, for instance?
> 
> I've always had to just play with those puzzles until I find something that will work. Is there a good way to construct commutators on these types of puzzles? After all, your title says "any puzzle".



I'm pretty sure this isn't the answer you are looking for, but both the Skewb and UFO can be solved with one algorithm each.

The Parity alg on he UFO: R U R U'2 BR U R
R is the slice perpendicular to you when you are solving it.
BR is the slice starting on the lower right, and finishing upper left.
Just build one full side, then use the parity algorithm to fix the other side.

As for Skewb just create the first side intuitively, then align the centers om the top and 4 sides with R' L R L', this algorithm moves the center on top to the front, and front to top, And switches left and right side centers. 

Once you have the centers solved, check your corners, if none are facing up, but the two top stickers facing opposite on your right, and perform that same algorithm twice. If you have two corners oriented correctly, put them in the Front left and back right (rotate the skewb) then perform that algorithm twice.

I am proud to say I came up with the Skewb method (the one I talk about) myself, someone else probably already knew it before me, but I did mine with no help =]

~Monkeydude1313


----------



## trying-to-speedcube... (Jul 2, 2009)

Yeah, "Any puzzle" might be really broad, but on the other hand, I don't own a Skewb or a UFO, and I have no idea how they turn, or how I should seperate pieces from their slice. The problem probably is that on all examples and exercises I gave, all these puzzles have an enormous amount of space where you can just totally screw up the puzzle to seperate that one piece you want to commute. That's why I think that small puzzles are harder than big ones; just because on big puzzles you have a lot more space to store crap in


----------



## Mike Hughey (Jul 2, 2009)

trying-to-speedcube... said:


> Yeah, "Any puzzle" might be really broad, but on the other hand, *I don't own a Skewb or a UFO*, and I have no idea how they turn, or how I should seperate pieces from their slice.


You mean you *do* own the puzzles from your examples? I want to see videos.   

And in case you didn't notice, gelatinbrain has Skewb. So you should be able to play with it to your heart's content.

And thanks, MonkeyDude1313. I've been wanting to find a really easy-to-learn Skewb method - yours looks nice. It's obviously not commutators, but it's nice. Last month in the computer competition, I invented a solution, but it was too hard to remember. Yours looks a little easier to remember.

I'd still love to learn how to do commutators on Skewb - it would make BLD easier.


----------



## trying-to-speedcube... (Jul 2, 2009)

Bad implication interpretation. 

Also, I have solved a Skewb on GelatinBrain, and it was by luck. Wrong computer here, so I can't do GelatinBrain right now, but I'll have a look into it first thing tomorrow.


----------



## Mike Hughey (Jul 2, 2009)

Mike Hughey said:


> And thanks, MonkeyDude1313. I've been wanting to find a really easy-to-learn Skewb method - yours looks nice. It's obviously not commutators, but it's nice. Last month in the computer competition, I invented a solution, but it was too hard to remember. Yours looks a little easier to remember.


I tried it out, and I've learned it. Talk about easy to remember - this is a quality beginner's method! I can average around a minute with it on gelatinbrain, which is pretty fast considering how bad I am at gelatinbrain controls. 



trying-to-speedcube... said:


> Bad implication interpretation.


I figured that.  I must admit I'm a little disappointed, though - I'd love to see those puzzles in real-life!


----------



## ChrisBird (Jul 2, 2009)

Mike Hughey said:


> Mike Hughey said:
> 
> 
> > And thanks, MonkeyDude1313. I've been wanting to find a really easy-to-learn Skewb method - yours looks nice. It's obviously not commutators, but it's nice. Last month in the computer competition, I invented a solution, but it was too hard to remember. Yours looks a little easier to remember.
> ...



Very easy. You can easily average around 30 seconds on a real skewb with it.

*Also note:
With that same algorithm, you can solve the skewb 3D, Skewb Ultimate, Polymorphinx, TFs Golden cube and much more!

Just for center turning in place, you turn the top, front, right, and left centers, by doing the algorithm six (6) times.


----------



## trying-to-speedcube... (Jul 3, 2009)

I have tried this method of constructing commutators on the GB skewb, but there are so many pieces, and there is so little room to mess with, that I think it's almost impossible. I haven't found anything yet, so I'm giving up.

As for UFO, there is none on GB, and I don't have one, but I figure you could pretty easily get one of the pieces out of the D-layer with /(-1,0)/(-1,0)/(1,0)/, as A-part, where /(-1,0)/ is the A-part that seperates a piece, and (-1,0) the C-part that moves the piece around.

This was all done in my head, so forgive me if it doesn't work


----------



## byu (Jul 3, 2009)

I used to have a Rubik's UFO, I'm looking for it now...


----------



## trying-to-speedcube... (Jul 4, 2009)

I just thought about that UFO commutator, and I imagined that if you make the "/"-slice the slice for C, instead of the U-layer, you can make the A-part of the commutator faster;

[/(2,0)/(-2,0)/, (0,1)]


----------

