Comparison: Number of Puzzle Permutations

How many different scrambles does each Rubik's Cube puzzle have? This is a comparison video of how many different permutations that different types of twisty puzzles have...

Some sources I used:

Music: Icelandic Arpeggios by DivKid

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[Nitoxym]

Someone might have said it already, but if not I will. Technically, the 0x0x0 has 1 permutation, as nothingness in multiplication is characterised by 1 (compared to 0 in additions). An example is x^0 = (arguably even for 0^0), another is 0! = 1. Nothingness is always in the solved state

[sungkarson]

I love how the cubes casually ascend into the 7th dimension
Edit: came back and 300 LIKES?! That’s the most that I’ve had

[RyanKennelly]

I knew gear cube was restricted, but it's honestly insane that it has fewer scrambles than a 2x2...

[temmie]

No way I can’t believe that the (insert puzzle name here) had (n) different permutations!

[phonetyx]

its crazy how some 3d ones have more permutations than higher dimensions

[JustAPersonWhoComments]

I like how the 0x0 exists. Everyone has it but it’s invisible

[maartenvandermeulen]

I love how the high ones on this list have more combinations than the amount of atoms in the observable universe

[RowanFortier]

Some thoughts about the comments:
1. "Actually 0! = 1, so a 0x0x0 has 1 permutation"
I get the arguments for why a 0x0x0 would have 1 permutation. Because 0! or 0^0 = 1, because there's only 1 way to arrange 0 things. But if you use the mathematical formula to find the number of permutations for any nxnxn, you get a divide by 0 error. So really It should actually maybe be *Undefined*?
2. "How does the 3x3x3 have more permutations than a 3x3x4?"
The 3x3x4 has 4 sides that are restricted to ONLY 180 degree turns. This means that all the edges and corners are ALWAYS oriented, which reduces the amount of permutations by a lot.
3. Also yes, I did mess up the scientific notation for 3x3x3, it was a copy paste error from the previous puzzle, I am sorry 😭

[kangkongfan]

Bro I love this guy he researches so well from what I’ve seen in the comments, and the video was interesting too

[SmartWorkingSmartWorker]

Isn't 0x0x0 1? Only one case, which is nothing? Many combinatorics problems (especially recurrence relation problems) has the same logic.

[o--formerlycalvinlucien]

just think about how many permutations that last cube has
The universe has 10^80 atoms.
If each of these was it's own universe, with it's own 10^80 atoms, it would still only have a googolth of a googolth the atoms.
it would have to have about 1100 nested universes to get the amount of permutations that that monstrosity has.

[Helio_Asou]

It's really interesting to see that the Skyoob and 222 have a really similar number of permutations, same thing with FTO and 345.
I'm curious to know how many states the Dino, Rex and Curvycopter puzzles have.
Also, is it easy to calculate the number of states the Clock has?

[IAmCrit.]

I kinda love how it starts with low numbers and then casually spikes up to 282, 870, 942, 277, 741, 856 536, 180, 333, 107, 150, 328, 293, 127, 731, 985, 672, 134, 721, 536, 000, 000, 000, 000, 000

[-tsvk-]

0:55 The scientific notation for the 3x3x3 seems to be wrong, it should probably be 4.3x10^16 instead of 4.1x10^16, seems to be copy-paste error from the 3x3x4.

[ZachCalin_]

What about the atlasminx and minx of madness? Also coren's 13 layer pyraminx would be interesting to see too

[jan-pi-ala-suli]

despite the 1 permutation, 1x1x1 is still the hardest rubik's cube




my fellow cubers know

[ysuri]

yeah I love the 4th dimensions cubes, they are hard since you don't see some faces you have to guess where they are 😂

[ciragoettig]

I really love this overview and links you provided, thank you -- but I think you may have a couple of mistakes here. the few I noticed are that you took a domino cube *with pictures* number from one of the sources, but that's higher than a regular domino, that you illustrated the entry with, as center orientations are relevant -- its like on a supercube. From one of your sources:

"There are 8 corners and 8 edges, giving a maximum of 8!·8! positions. This limit is not reached because the orientation of the puzzle does not matter. There are 4 equivalent ways to orient the puzzle with a white centre on top, so this leaves 8!·8!/4 = 406, 425, 600 distinct positions.

If the centre orientation is visible, then there seem to be 4·4 possible orientations of the two centres. There is a parity constraint however, as the parity of the number of quarter turns of the centres must be equal to the parity of the corner permutation. This means that the centre orientations only increase the number of positions by a factor of 8, giving 8!·8!·8/4 = 3, 251, 404, 800 distinct positions."

The second one that seems half-wrong is the 3x3x3 . The full number is right, but you must have accidentally copy-pasted the previous entry, 3x3x4 for that number in scientific notation; it says just 4.1 x 10^16 yet the number above is clearly the correct and much greater value of 4.3 x 10^19. Apologies for repeating this, I've seen others have notified you of this one -- after already writing this.

Also I also can't find the 8, 617, 338, 912, 961, 658, 880, 000 for square 1 in your stated sources. It describes a couple of ways of counting, but as far as I can gather even the largest number it gives is the much smaller 62, 768, 369, 664, 000 (and quotes even smaller ones in the table, not that -- so I guess it doesn't think that's the right count either). soo at best just around 1e13 to 1e14, and not on the order of almost 1e22 as stated.

[jakerussell]

Pyraminx has 933120 but only if you dont count the tips. If you count the tips, multiply it by 81

[gilthenrill]

How does the 0x0x0 have zero permutations? It actually has one, and that one permutation is where the “cube” isn’t in existence.