On matching a scrambled 4x4x4 Rubik's cube

Question

Has anyone tried to match a scrambled 4x4x4 cube? It's been a fun challenge that requires learning supercube-safe algs (especially for PLL), and I can succeed 99% of the time. Roughly 50% of the time, though, I have to swap two corners on the last face. There are no supercube-safe algs that can do this, however. You can find t- and j-perms that work great, but rotate the top face 90 degrees. Then if you rotate back and fix the corners and edges, you end up with a completely matched cube except for a 90 degree rotation on the top face (see below--trust me, the rest of the faces match exactly). I know it is impossible to rotate a single face 90 degrees without affecting another part of the cube.

I found a parity alg that swaps two corners and only two adjacent center pieces, so if there is a bar on the upper face, this alg can be used to match the given cube. The bottom line is that whenever there is a bar on one of the centers of the initial scrambled cube (which happens almost every time), I'm just sure to set up my match so that this center ends up on the top face at the end. But what if the initial cube has no bars in any of the centers and I'm left with this last-two-corners parity?

Any advice would be appreciated!

Answer 1

You have many options.

• The "rotated" top centre has two blue pieces. Instead of trying to rotate it back (which would be an odd permutation and not possible without affecting corners) fix it by doing two swaps.

• If "rotated" top centre has four distinct colours, use one of the centre-twisting algorithms for the 3x3x3 cube to twist the top centre back while simultaneously twisting one of the other centres - one that does have two of the same colour. Then fix that other centre as per the previous option.

• Before swapping the corners, just put a bar on the top centre with one or two slice set-up moves. Then do the corner swap you have, and undo the set-up moves. The fancy word for using such set-up moves is conjugation.

• Swap the corners, also swapping two centres. Then "swap" those centres again by doing a 3-cycle involving a centre from another face that has the same colour. You can use the 3-cycle algorithm r'd'r U' r'dr U, possibly with some setup moves.

Basically, in every case you will have to stop thinking of the centre as a single piece to be rotated, and think of a centre as four separate pieces, to be moved around, possibly with the help of some centre piece from another face with the same colour.

Answer 2

I don't know of any algorithm to fix the entire 90 degree rotated 2x2 center in one go, but since the center pieces are interchangeable (which is what causes the issue here), you could use the center algorithms from the 4x4x4 Cage solving method to fix just those.

To purely 3-cycle the $\color{orange}{FDL}→\color{purple}{FDR}→\color{blue}{UDR}$ center pieces in the image above (green is the front and yellow is the top), without disturbing any of the already solved corners, edges, nor other center pieces, you can use the algorithm:

r U l' U' r' U l U'

Using some setup moves, you can use this 3-cycle multiple times until all your centers are correct.

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An alternative is to rotate the 2x2 center so it's correctly oriented, and solve the edges/corners around it with 4x4x4 layer-by-layer algorithms (here part 1; part 2; and part 3a of that same layer-by-layer tutorial, although only the linked part 3a is relevant for your current case).