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A little while ago, cuBerBruce found the following four (34 qtm, 26 ftm) move solutions with k-solve:

Rw U Rw' U Rw' U' Rw2 U Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U Rw' U' Rw U Rw' U2 Rw U2

Rw' U' Rw U' Rw U Rw2 U' Rw2 U2 Rw U2 Rw' U2 Rw U2 Rw U' Rw U Rw' U' Rw U2 Rw' U2

Rw' U' Rw U' Rw U' Rw2 U2 Rw2 U' Rw U2 Rw' U2 Rw U2 Rw U Rw U Rw' U' Rw U2 Rw' U2

Rw U Rw' U Rw' U Rw2 U2 Rw2 U Rw' U2 Rw U2 Rw' U2 Rw' U' Rw' U' Rw U Rw' U2 Rw U2

Looking at a 4x4x4 supercube after each is applied, we can see that two X-center pieces in the top layer are swapped, and the right center is rotated 90 degrees. With the move restriction <U, Rw>, one cannot possibly do a 3-cycle of X-center pieces inside the top center and the right center.

This made me question whether or not a supercube safe algorithm for this case (or any other 2-cycle of wings case) in <U, Rw> is even possible.

I tried another route to make an algorithm like this, but similar to my poor approach in <U, Rw> it's very long. You'll definitely need a supercube applet to see that the following is supercube safe. It has a scary length of (1543 qtm, 1152 ftm) and (1527 qtm, 1141 ftm) if you cancel moves with each of the main pieces. Each of the main pieces are repeated. So you can look at what those do separately if you're interested to see how I made this by hand.

(SiGN Notation) [Link]

r' (r2 U2 r U2 r2 U' r' U2 r' U r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U r U2 r U r2 U2 r' U2 r2 U2 r2 U2 r U2 r2 U' r' U2 r' U' r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U' r U2 r U r2 U2 r' U2 r2 U2 r U2 r2 U2 r U2 r2 U' r' U2 r' U r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U r U2 r U r2 U2 r' U2 r2 U2 r2 U2 r U2 r2 U' r' U2 r' U' r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U' r U2 r U r2 U2 r' U2 r)3 r (r U r U r U' r' U2 r2 U2 r U2 r' U r2 U' r2 U' r U2 r' U' r U2 r' U r U2 r' U' r U2 r' U' r U2 r' U r U2 r' U' r U2 r' U' r U2 r' U r U2 r' U' r U2 r' U' r U2 r' U r U2 r U r2 U' r U2 r2 U2 r2 U2 r' U2 r U2 r' U2 r' U2 r2 U2 r2 U2 r' U2 r2 U r' U' r' U' r' U' r U2 r' U r U2 r' U2 r U r' U r U2 r' U2 r U r' U r U2 r' U2 r U r' U r U2 r' U2 r U' r' U' r U r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U2 r' U' r U r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U2 r')2 (r U r' U r' U r2 U2 r2 U r' U2 r U2 r' U2 r' U' r' U' r U r' U2 r U2)2 r

The big problem is that it also does the move R.

So the one question we have to answer is if the move R' (on applied on the supercube) can be reached in <U, Rw>.

If it isn't possible to do the move R' on a 4x4x4 supercube with <U, Rw>, then it is definitely impossible for a supercube safe 2-cycle of wings to exist. And if so, is there a way someone could find the optimal solution (with k-solve or whatever program is out there) for my alg? (I would be interested to see how long the optimal solution is for a supercube dedge flip + the move R in <U, Rw>).

Rw U Rw' U Rw' U' Rw2 U Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U Rw' U' Rw U Rw' U2 Rw U2

Rw' U' Rw U' Rw U Rw2 U' Rw2 U2 Rw U2 Rw' U2 Rw U2 Rw U' Rw U Rw' U' Rw U2 Rw' U2

Rw' U' Rw U' Rw U' Rw2 U2 Rw2 U' Rw U2 Rw' U2 Rw U2 Rw U Rw U Rw' U' Rw U2 Rw' U2

Rw U Rw' U Rw' U Rw2 U2 Rw2 U Rw' U2 Rw U2 Rw' U2 Rw' U' Rw' U' Rw U Rw' U2 Rw U2

Looking at a 4x4x4 supercube after each is applied, we can see that two X-center pieces in the top layer are swapped, and the right center is rotated 90 degrees. With the move restriction <U, Rw>, one cannot possibly do a 3-cycle of X-center pieces inside the top center and the right center.

This made me question whether or not a supercube safe algorithm for this case (or any other 2-cycle of wings case) in <U, Rw> is even possible.

I tried another route to make an algorithm like this, but similar to my poor approach in <U, Rw> it's very long. You'll definitely need a supercube applet to see that the following is supercube safe. It has a scary length of (1543 qtm, 1152 ftm) and (1527 qtm, 1141 ftm) if you cancel moves with each of the main pieces. Each of the main pieces are repeated. So you can look at what those do separately if you're interested to see how I made this by hand.

(SiGN Notation) [Link]

r' (r2 U2 r U2 r2 U' r' U2 r' U r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U r U2 r U r2 U2 r' U2 r2 U2 r2 U2 r U2 r2 U' r' U2 r' U' r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U' r U2 r U r2 U2 r' U2 r2 U2 r U2 r2 U2 r U2 r2 U' r' U2 r' U r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U r U2 r U r2 U2 r' U2 r2 U2 r2 U2 r U2 r2 U' r' U2 r' U' r' U' r U r' U' r U2 r2 U2 r2 U2 r U r U' r' U r U2 r' U' r U r' U' r' U2 r2 U2 r2 U2 r' U r U' r' U r U' r U2 r U r2 U2 r' U2 r)3 r (r U r U r U' r' U2 r2 U2 r U2 r' U r2 U' r2 U' r U2 r' U' r U2 r' U r U2 r' U' r U2 r' U' r U2 r' U r U2 r' U' r U2 r' U' r U2 r' U r U2 r' U' r U2 r' U' r U2 r' U r U2 r U r2 U' r U2 r2 U2 r2 U2 r' U2 r U2 r' U2 r' U2 r2 U2 r2 U2 r' U2 r2 U r' U' r' U' r' U' r U2 r' U r U2 r' U2 r U r' U r U2 r' U2 r U r' U r U2 r' U2 r U r' U r U2 r' U2 r U' r' U' r U r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U2 r' U' r U r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U' r' U2 r U2 r' U' r U2 r')2 (r U r' U r' U r2 U2 r2 U r' U2 r U2 r' U2 r' U' r' U' r U r' U2 r U2)2 r

The big problem is that it also does the move R.

So the one question we have to answer is if the move R' (on applied on the supercube) can be reached in <U, Rw>.

If it isn't possible to do the move R' on a 4x4x4 supercube with <U, Rw>, then it is definitely impossible for a supercube safe 2-cycle of wings to exist. And if so, is there a way someone could find the optimal solution (with k-solve or whatever program is out there) for my alg? (I would be interested to see how long the optimal solution is for a supercube dedge flip + the move R in <U, Rw>).