# Request sequence of commutators to exchange two pieces in a 20x20x20 cube



## Zhiguo (Mar 4, 2010)

Does there exist a sequence of commutators to exchange any two pieces on a 20x20x20 cube?

I am not an expert at cube solving, but I have to write a program to solve 20x20x20 cubes. I have searched many algorithms, but most of them are not esay to understand since in each step they try to move more than two pieces.

So I want to ask this question:
Does there exist a sequence of commutators to exchange any two pieces on a 20x20x20 cube?

If there is such a sequence/sequences, it would be easier to give a solution by computer program, although it would not(and it need not to be) an optimal solution.

Thanks in advance! Any help would be appreciated! ^_^


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## qqwref (Mar 4, 2010)

You can't move only two center pieces, but because you can't tell center pieces apart you can do 3-cycles that look like you are only moving two. Let xR and yR be the xth and yth slices on R, then try this:
xR U' yR U xR' U' yR' U'

For edges you can interchange two pieces but there are no commutators to do so, just an algorithm. Letting r be some slice on R and l be the corresponding slice on L:
r U2 r U2 F2 r F2 l' U2 l U2 r2
(This only works if centers are solved, otherwise it might mess them up a bit.)


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## Zhiguo (Mar 4, 2010)

Thanks for your reply!



qqwref said:


> You can't move only two center pieces, but because you can't tell center pieces apart you can do 3-cycles that look like you are only moving two. Let xR and yR be the xth and yth slices on R, then try this:
> xR U' yR U xR' U' yR' U'


I have tried the sequecece on a 6x6x6 cube(from origin state), with x=3 and y=2. But the sequence does not result in cube with only two pieces messed up.

EDIT: Yes, the sequence works as expected. I made a mistake when executing it. SORRY!



qqwref said:


> For edges you can interchange two pieces but there are no commutators to do so, just an algorithm. Letting r be some slice on R and l be the corresponding slice on L:
> r U2 r U2 F2 r F2 l' U2 l U2 r2
> (This only works if centers are solved, otherwise it might mess them up a bit.)


Why isn't it possible to do anything with commutator sequences?
Maybe I mis-understood the meaning of commutators?
By "commutators", I mean the operation to rotate a single column on a cube.
What human can do to cubes are sequence of (maybe the mis-understood)"commutators".


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## qqwref (Mar 4, 2010)

Oh, no, commutator specifically means something like A B A' B' where A and B are moves or sequences of moves. (Move = a turn of any layer.) People talk about commutators a lot because they're so useful - you can do any 3-cycle (of the same type of piece) with a commutator.


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## Zhiguo (Mar 4, 2010)

qqwref said:


> Oh, no, commutator specifically means something like A B A' B' where A and B are moves or sequences of moves. (Move = a turn of any layer.) People talk about commutators a lot because they're so useful - you can do any 3-cycle (of the same type of piece) with a commutator.



Sorry for my mistake!
Now the question is to: find a sequence of basic operations to set piece at a specified position at a specified face to a specified color. 
For example, to "set L(2,3)=YELLOW", that is, to set the (2,3)-indexed piece on L face to color yellow.
According to my observation, given a specified location, for example L(2,3), there are only six pieces can be rotated to that location. Is my observation right?

P.S. By "basic operation", I mean the operation such as xR, yL, U, 1U, 2U...

Thanks again!


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## cmhardw (Mar 4, 2010)

Zhiguo said:


> Sorry for my mistake!
> Now the question is to: find a sequence of basic operations to set piece at a specified position at a specified face to a specified color.



Yes, this can be done and is the same problem we have for solving a larger cube blindfolded. Please read the stickied threads in this forum about blindfold solving the larger cubes for some in depth explanations of *how* to do this!

Hope this helps!
Chris


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## Zhiguo (Mar 8, 2010)

Your first formula is very very helpful! Except that the last operation should be U instead of U'.
With this single formula, I can solve all the central area of a randomly-messed-up 20x20x20 cube.

Now, I need to solve the edges and corners. But I do not understand your second formula. Can you explain a little more bit?

Thanks again!



qqwref said:


> You can't move only two center pieces, but because you can't tell center pieces apart you can do 3-cycles that look like you are only moving two. Let xR and yR be the xth and yth slices on R, then try this:
> xR U' yR U xR' U' yR' U'
> 
> For edges you can interchange two pieces but there are no commutators to do so, just an algorithm. Letting r be some slice on R and l be the corresponding slice on L:
> ...


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