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