What is the superflip on the 4x4x4 Rubik's cube?

Question

The superflip is the state on a rubik's cube that require the most amount of moves to be solved. This is how it looks like:

This state requires 20 moves (where half-turns counts as one move) to be solved.

This got me wondering, what is the superflip on the 4x4x4 rubik's cube? I saw This video that showed how to do the superflip on bigger cubes such as the 4x4x4, but I'm pretty sure that it isn't the "real" superflip, because I could just move the right slice on the 4x4x4, and it requires one more move to be solved and therefore isn't the state that requires the most moves to solve.

If you want to, you can include the superflips on 5x5x5 and up.

EDIT

With superflip I mean the state (or one of the states if there exists more than one) that takes the most amount of moves to be solved.

Answer 1

The 4x4 doesn't have a corresponding superflip state, because the definition of the superflip is ambiguous, and no good definition leads to a known superflip position.

If you define the superflip state as "all edges flipped," then it is possible to treat the edge pairs like 3x3 edges and the center four pieces as 3x3 centers. This will allow you to create a superflip orientation on the 3x3 reduction of the 4x4. This is not, however, a state requiring the maximum moves to solve on the 4x4, because the centers remain solved.

If you define the superflip state as a checkerboard going in from each edge (roughly speaking), then no even numbered cube has one. To see why, remember that the corners stay fixed on a regular superflip to preserve the cube's symmetry - this can't happen for even puzzles.

Imagine the red-blue edge. The first corner has red on top, then the next edge has blue, next edge red, and last corner is blue. The first corner has red atop it, and the second has blue, which means one was necessarily swapped with the corner on the opposite side of the cube. This makes the checkerboarding pattern not possible to complete in any sense without swapping corners.

Finally, if you define the superflip as "the position taking the longest number of moves to solve," it may be some variant of these positions, but it likely wouldn't be recognizable as a regular superflip. This position isn't known, though. The 3x3 superflip position was demonstrated to take the maximum number of moves when it was proved the 3x3 can be solved in 20 moves.

No such analysis has been done on the 4x4, which would be required to find and prove such a state needs the maximum moves.

(If I recall, though I can't seem to find a reference for this, on the 4x4, the estimate for optimum move sequence is 33.)

Answer 2

The algorithm in the video you linked is correct - you perform the exact same algorithm as with a 3x3x3 cube, except the "middle slice" on each face is the middle two layers instead of the middle one. This flips all the edge pieces, which is the definition of superflip.

The superflip is only one example of a "maximum-move scramble" on the 3x3x3 Rubik's Cube. It doesn't necessarily correspond to a maximum-move scramble state on any of the others.

Answer 3

The 20 move upper bound (face turn metric) on the 3x3x3 rubik's cube was only proven in 2010, by extensive computer search (many machine years, run in time donated on the Cloud.) https://en.wikipedia.org/wiki/God%27s_algorithm

Before then it was known that the superflip, along with a number of other highly symmetrical positions, took 20 moves. It had long been conjectured that the maximum moves would be required for a highly symmetrical position. But it was still required to analyse all the unsymmetrical positions.

It's easy to prove that the superflip must be a local maximum. Given that it has the full 48-fold symmetry of the cube, all moves performed on the superflip must lead towards or away from the solved state. As the latter could only be true if it was totally impossible to solve the cube, the former must be true.

The quarter turn metric was only proven to require max 26 moves in 2014. Interestingly, the longest move sequence is not the superflip, but rather is exemplified by a position called superflip composed with fourspot, which has only 16-fold symmetry. http://www.cube20.org/qtm/

Therefore, the longest move sequence on the 4x4x4 rubik's cube by whatever metric you choose is almost certainly unknown, but highly symmetrical. And choice of metric is complicated, because you have to consider whether moves of single internal slices are allowed.