Random Cubing Discussion

How many ways are there to resticker a Rubik's cube given a 6 colors, 9 identical stickers per color? 1 sticker per square obviously. As a useless bound, it's somewhere between 43 quintillion*12 and 54!

And how 'groups' are there in those? A 'group' is a bunch of stickerings which can be turned into one another by turning the Rubik's cube.
 
How many ways are there to resticker a Rubik's cube given a 6 colors, 9 identical stickers per color? 1 sticker per square obviously. As a useless bound, it's somewhere between 43 quintillion*12 and 54!

And how 'groups' are there in those? A 'group' is a bunch of stickerings which can be turned into one another by turning the Rubik's cube.
The first question is easy. It's 54! = (2.31x10^71) if there are no restrictions on combinations of stickers.

The second question is a lot more interesting. FYI, these are called orbits. My first guess as a rough estimate would be 54! / 43 quintillion. This won't be correct but I imagine it's close.
 
Oops. That somehow slipped my mind. You have to divide by the number of ways each group of 9 stickers can be arranged. This means the answer is 54!/((9!)^6) = 1.01x10^38. The relevant formula is permutations of multisets.
 
Oops. That somehow slipped my mind. You have to divide by the number of ways each group of 9 stickers can be arranged. This means the answer is 54!/((9!)^6) = 1.01x10^38. The relevant formula is permutations of multisets.
Ah, I actually know this formula. Didn't think of using it. I think this number should be further divided by 24 to account for double counting different orientations of the same state...?
 
There are states that don't change after being rotated
yup, so its a slightly bigger number than what youd get just dividing by 24
so how would you calculate it then? Would you have to do a case by case analysis to see which states are symmetrical?
 
So apparently the upcoming Cong's Design pyra and skewb will have magnets instead of ball bearings. Seems pretty interesting, but it'd be impossible to change the clickiness without getting new magnets. Thoughts on this?
 
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Tfw. (not PB, but finished the avg100 for today like this)
 
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