Some 3x3x3x3 Questions that I hope this group can help me answer.

[carl.n.hoff]

Back in 2016, I described a puzzle you could call the Sphere 3x3x3x3. https://twistypuzzles.com/forum/viewtopic.php?p=352940&hilit=3x3x3x3#p352940

This puzzle adds additional stickers to each piece type in a manor that is similar to the Circle 3x3x3.

http://wwwmwww.com/3x3x3x3/CircleCube.gif

In the 3x3x3x3 there are 5 piece types:

(1) There is the core. Which is unstickered in the standard picture but has 8 stickers in the sphere 3x3x3x3.


(2) There is the cell center. It has 1 sticker in the standard picture and 6 stickers in the sphere 3x3x3x3.


(3) There is the cell face. It has 2 stickers in the standard picture and 6 stickers in the sphere 3x3x3x3.


(4) There is the cell edge. It has 3 stickers in the standard picture and 5 stickers in the sphere 3x3x3x3.


(5) There is the cell corner. It has 4 stickers in the standard picture and 4 stickers in the sphere 3x3x3x3.


What I want to determine is which piece types have a unique position and orientation in the solved state for the Standard 3x3x3x3 versus the Sphere 3x3x3x3.

Assume you can just rotate the 8 cells and the core hypercubie never moves.

(1) Looking at the yellow cell center hypercube. how many orientations can it reach and still appear solved? It is either 24 or 48. Since it is only rotated by a single cell, I believe it can only reach 24 orientations. I don't believe any mirror states of this hypercubie are reachable, correct?
(2) Looking at the yellow-red cell face hypercube, how many orientations can it reach and still appear solved? It is 4 or 8 and I'm not certain which is correct. It is 8 if mirror states are reachable, looking at the sphere 3x3x3x3 is it possible to swap the green and blue stickers while leaving the cyan, magenta, yellow, and red stickers apparently unchanged?
(3) Looking at the yellow-blue-red cell edge hypercubie, how many orientations can it reach and still appear solved? If mirror states are reachable, I believe the answer is 2. Looking at the sphere 3x3x3, can the small cyan and magenta stickers be swapped while leaving the yellow, blue, and red stickers apparently unchanged?

I know the cell corners can reach mirror image states but with 4 stickers only one of these states appears solved.

I'm pretty sure I'm correct about (1) above but not sure about (2) and (3). If the cell face and cell edge hypercubies can reach mirror image states. Do you know short move sequences that would take an individual cell face or cell edge hypercubie and return it to its initial position but in the mirrored state, without concern for its effect on any other hypercubie? I may want to animate these move sequences on the Sphere 3x3x3x3 to demonstrate how these mirrored states are reachable... assuming they are reachable.

Thanks for any help you can provide,
Carl