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 Post subject: Corner parity on 4x4x4 fisher cube
PostPosted: Tue Jul 01, 2014 4:34 pm 
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I recently bought the recently mass produced 4x4x4 fisher cube.
Reducing it to a 3x3x3 fisher cube is not big job : 4 centers are "super centers" (with orientation to be managed). Rebuilding the edges from dedges (pieces that are actually on the corners of the cube and in the "center" of the faces) is no problem too.

I then solve the reduced cube as a 3x3x3 fisher cube. The trouble is during the last layer resolution. I can place and oriented edges (those "edges that are actually in the corners") but placing/orienting corners (which are actually in the middle of the edges of the cube) brings out a special situation : 2 corners are OK, the 2 others are inverted. This does not look like a fisher cube parity, I believe this is a parity specific to the 4x4x4 fisher cube.

Has anyone met the same parity issue ?

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 Post subject: Re: Corner parity on 4x4x4 fisher cube
PostPosted: Tue Jul 01, 2014 4:54 pm 
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What you describe is not unique to the 4x4x4 Fisher Cube shape mod. For example:
Attachment:
4x4x4_corner_swap_parity.png
4x4x4_corner_swap_parity.png [ 11.47 KiB | Viewed 227 times ]


This parity occurs because the parity of 3x3x3 corners is connected to the parity of 3x3x3 edges and on the 4x4x4 you can reduce the edge groups into an odd permutation in the 3x3x3 group. You need to perform a 2-2 swap of 4x4x4 edge-wings which effectively swaps two 3x3x3 edges. This will match the parity of the edges up with the parity of the corners again and you'll be able to solve the reduced 3x3x3.

There is another way to interpret the parity involving 4x4x4 centers but that's more complicated.

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 Post subject: Re: Corner parity on 4x4x4 fisher cube
PostPosted: Wed Jul 02, 2014 1:52 am 
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I think you are right, this is a 4x4x4 parity case, not something specific to 4x4x4 fisher cube.
A problem remains : the PLL parity algorithms that I use mess up two opposite supercenters (like F and B) of the 4x4x4 fisher cube.
What I will try is turning the 4x4x4 fisher cube so that the white/yellow faces (those with no supercenters) become F and B and then solve again.

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 Post subject: Re: Corner parity on 4x4x4 fisher cube
PostPosted: Wed Jul 02, 2014 9:25 am 
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Here is a double swap of 4x4x4 edges (= seemingly a swap of two 3x3x3 edges) that does not change the orientation of centres on a Super Cube:

Gelatinbrain notation:
B'&2, R, U', B, R', U, B&2, F&2, U', R, B', U, R', F'&2
WCA notation:
b' R U' B R' U b f U' R B' U R' f
Result on a solved cube:
Image

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 Post subject: Re: Corner parity on 4x4x4 fisher cube
PostPosted: Wed Jul 02, 2014 11:53 am 
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Duffman wrote:
A problem remains : the PLL parity algorithms that I use mess up two opposite supercenters (like F and B) of the 4x4x4 fisher cube.

The problem is thinking of this as a special case or using a pre-made sequence to resolve it. The logical approach would be to decompose the 2-2 swap into two 3-cycles of the edge-wings. If you did that you'd never have to worry about the centers moving.

The centers only move with the sequence you have because somebody took as many shortcuts as possible in that sequence to keep the movecounts low. Those shortcuts don't have any effect on a nearly solved 4x4x4 but obviously they have effects on other variations.

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 Post subject: Re: Corner parity on 4x4x4 fisher cube
PostPosted: Wed Jul 02, 2014 3:18 pm 
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Konrad wrote:
Here is a double swap of 4x4x4 edges (= seemingly a swap of two 3x3x3 edges) that does not change the orientation of centres on a Super Cube:

Gelatinbrain notation:
B'&2, R, U', B, R', U, B&2, F&2, U', R, B', U, R', F'&2


Many thanks Konrad. This algorithm was exactly what I was needing (and it does not have the side effect of the r2 U2 r2 u2 r2 u2 U2 algorithm that I was using).

This 4x4x4 fisher cube convinced me that 4x4x4 supercube is probably a great puzzle. I will buy the sticker to build one. :)

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