??? Are we talking about a physical cube? I don't think you can have multiple identical permutations; each of the 26 external pieces of a 3x3x3 cube is unique. You would have to somehow have more than one way to get, say, a Yellow-Blue-Red corner, or a White-Green edge for this to be true, no?
> Now it’s up to you if you consider that a separate permutation or not. (I don’t)
This isn't really up to you; it's determined by the method you use to count how many permutations there are.
In this case, I strongly suspect that we count 43,252,003,274,489,856,000 and then may or may not say "but 4,096 of those are equivalent for any given coloration pattern, so divide by 4,096". I would bet we don't count 10,559,571,111,936,000. Unless you have a method that yields that result naturally, you should consider the rotated centers to be separate permutations.
> In this case, I strongly suspect that we count 43,252,003,274,489,856,000 and then may or may not say "but 4,096 of those are equivalent for any given coloration pattern, so divide by 4,096".
I don't think that's true. I went looking for the formula, and found this[1].
They include a term for the permutations of the corner cubes (8!∙3⁸), a term for the edge cubes (12!∙2¹²) and a term for the center cubes (1!∙1¹), then divide that by 12 (2∙3∙2) to eliminate impossible configurations. That term for the center cube permutations seems to agree with my view on it. There's no such thing as "rotation" for a plain colored center cube. It's the same no matter how you look at it.
I think a cube with designs on the center faces will have more permutations than the 43e18 we're discussing here.
> They include a term for the permutations of the corner cubes (8!∙3⁸), a term for the edge cubes (12!∙2¹²) and a term for the center cubes (1!∙1¹), then divide that by 12 (2∙3∙2) to eliminate impossible configurations.
But that matches exactly what I said was being done, modulo the particular number and an error in the formula you provide. (It should be 1·1⁶, not 1!·1¹.) You have 4 orientations for each center cube, and then you divide 4⁶ by 4,096 to get 1.
I agree that the way your link presents it, with a glaring error in the term for the centers, suggests that they didn't put any thought towards the centers beyond "they don't and can't change in any way".
If the brushed lines are uniform, they might have half the state, right? A center cube could be at 0˚ or at 180˚ and the pattern could still line up.
I've never seen one, so I don't know if that's the case or not. I also don't know if there are natural constraints in the topology of the cube that make it possible or not for center faces to be 180˚ out of alignment without others being off by a quarter turn. I do assume that any center face that's flipped would have to be paired with another one in the same orientation. But that's just my intuition.
A brushing pattern, even if perfectly straight, consists of scratches with random thicknesses. It does not have 180 degree symmetry; if the piece is reversed, the scratches do not match, and that is pretty obvious if held up to the light.
I believe all the center face re-orientations are paired, 180 degree rotations: there are no quarter turns.
