I tried my best. Couldn't come up with a way to do it in 9 quarter turns. Can you tell how.
I don't think it can be done in 9 quarter turns. You can do this using 9 turns in face turn metric, but make sure that you use an odd number of turns in quarter turn metric.
Really interested in knowing how you find out by counting just 12 cycles instead of 24. Can you please explain.
What I do is sort of like memorizing a 3x3x3 cube blindfolded, so it will require counting only 12 "cycle" rather than the full 24. Basically you don't remember how many transpositions it takes to reduce the edges to the solved state, but rather to a state where edge groups are paired together, and oriented correctly in a 3x3x3 blindfolded orient first sense. For example:
Scramble your 4x4x4 cube with white on top and green on front with the following scramble:
Rw B2 D R2 B Uw2 B2 R B U' L' Uw2 Rw D' F2 D R2 Uw' B' Fw2 D' F2 Rw' R2 D2 F2 L' U2 L D B L2 R U R U L' B' F2 D2
If I was using my cycle counting method on this scramble I would do exactly the following.
1) Orient the cube using the cube rotation
x, or turn the entire cube as if you were doing the turn
R. This will place the paired yellow-red piece such that the yellow stickers are on the U layer (which is where yellow should go for me in order for an edge to be "oriented" in a blindfold solving sense).
2) I will look at the piece in UBl and see which piece it is. It is the orange yellow wing edge. I now find the other yellow orange wing, which is at RBd. The yellow is on the B face, which is where yellow should be in my 3x3x3 edge blindfold orientation scheme to be oriented "correctly". So I will imagine that I am sending the piece at UBl to RBu, which will pair it up with the other yellow orange and make an oriented 3x3x3 edge pair. Now I don't memorize this, nor do I count how many cycles there are. All I do is think that I just did one cycle, so I toggle the parity of the permutation I am analyzing. When beginning I have imagined that I have done no cycles at all, and I now imagine that I will swap my UBl edge with the one at RBu, which is one transposition. So now the parity is odd, and I verbally say "odd"
3) The piece that is in RBu is the yellow green edge, and it's pair mate is at DFr with the yellow on the D layer. This is already oriented, so I send RBu next to it to DFl, one transposition which toggles parity. I now say "even" meaning the permutation I am creating has even parity.
4) The piece that was at DFl is green red, and it's mate is at DBl. This piece is "flipped" in a blindfolded sense for 3x3x3 edge orientation because the green is on the B face. So now I will do two transpositions. I imagine that I am sending the piece at DFl to DBl, and the piece at DBl goes to DBr. Since I have done two swaps in a row, I have not toggled parity. So I don't say anything, and the permutation I am creating is still "even" as it was at the end of the last step. Now, when actually doing this, if edges are flipped I simply don't do anything and just move on, but the reason is because I am actually doing two transpositions.
5) The piece at DBr goes to UBl to pair with it's mate (which is where I started). This piece was already sent to UBl when I did the swap earlier. So this tells me that I completed a partial cycle and I must start at a new piece somewhere else. The parity of the permutation is still "even"
6) I now need to start from a new location, because I completed a cycle. I will now start at URb. This piece is white orange, and pairs with the other one at DRb. The DRb piece is flipped in a 3x3x3 sense, so I will do two transpositions, or I will not toggle parity. Therefore I simply don't say or do anything and just skip along to the next piece. My permutation parity is still "even" as it was at the end of the last step.
7) DRf pairs with UFr, which is flipped. Again do nothing and skip to the next piece.
8) UFl goes to LBd. I say "odd"
9) LBd goes to LFu. I say "even"
10) LFu goes to LRu goes to LRd. I say nothing and move on.
11) LRd goes to DLb goes to DLf. I say nothing and move on.
12) DLf is the piece that pairs with URf (the place adjacent to where I started). So I know the parity of the wing edge permutation is the last word I said, so the parity of the wing permutation is even. This means that the wing edges on the cube are in an even permutation, and I must solve the centers using an
even number of quarter turns to restore the parity of the wing edges to even. This will effectively skip the OLL parity case, and I will know that it is impossible to have the OLL parity during the solve, so I save time.
13-optional) You may wonder how I would know that I am done at step 12. Back when I was trying to do this seriously I used a base 5 counting system with my feet to count how many edge pairs I had built during my analysis. The reason I used base 5 was because it allows you to count to 24 on two feet, and this way I could also use it when counting the full 24 cycles (which I did for a while before coming up with the idea I just described).
To verify that the wing edges actually are in an even permutation go ahead and just count the cycles in the usual sense. After the initial cube rotation, here is the wing permutation using my blindfolded lettering scheme.
(QFJBIG)(NPHOUXATCLV)(RK)(DSW) and two wing edges are solved. If you analyze the actual wing edge permutation you will find that it is indeed in an even permutation.
Sorry for my limited understanding of what an orbit is. When you said there are 2 kinds of centers, did you mean
I meant this option that you wrote:
Kind1: Centers that lie on the geometric diagonal of the cube's face
Kind2: Centers that are not of Kind1
Basically every inner slice contains 8 x-centers from the same orbit, and all the rest of the centers are in groups of 4 oblique centers from the same orbits.
And if you don't mind can you tell what an orbit actually refers to.
Thanks a ton.
I wonder if we are using this in the correct sense, as I have heard some people say that cubers misuse this term in a pure mathematical sense. By orbit, I mean all possible locations on the cube that a certain piece may occupy. So for example the corner orbit consists of 8 locations that the corners may occupy. The wing edge orbit on a 4x4x4 consists of the 24 locations that the wing edges may occupy.
Chris
). For 4x4x4 though those swapped rows are equivalent to rotating the center 180 degrees. This way you can still solve dedges before centers.
