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

[Zhiguo]

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! ^_^

 

[qqwref]

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.)

 

[Zhiguo]

Thanks for your reply!

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!

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".

 

[qqwref]

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.

 

[Zhiguo]

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!

 

[cmhardw]

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

 

[Zhiguo]

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!

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.)