> 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".
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.
[1] http://b.chrishunt.co/how-many-positions-on-a-rubiks-cube