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:
www.jaapsch.net/puzzles/puzst...
www.therubikzone.com/number-o...
hlavolam.maweb.eu/number-of-c...
Music: Icelandic Arpeggios by DivKid
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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
Nah only the 4th Dimension and not even fully at that. From what it looks like, it's just several ordinary cubes contorted to create a mega cube when combined
@@n833u3 No, What you cant see in the pictures is that you can turn the 3x3x3x3 for example in the Fourth dimension too The only reason it looks so weird, is because the only way to display the higher dimension cuben is beacause you have to make one side invisible, because otherwise you wouldnt be able to see the other ones, and the invisible side changes with every turn in the higher dimensions
@@endevomgelende8634 I assumed that this cube was based on an irl toy.
You can create a Rubik's Cube in any number of dimensions using math. And there are some computer programs that let us render and play with them. So while they don't exist in real life, there's still enough of a thing to be interesting, and get a permutations result from
@@RowanFortier There are actual 3d puzzles with states analagous to 4d puzzles
its crazy how some 3d ones have more permutations than higher dimensions
Yeah, it was interesting when I was researching it
Like the 33x33x33 and the 19x19x19
3x3 in 7th dimension: "I WIN!" 150x150 in 3d, "No, son.."
@@A1exP1ays true lol
150x150x150 has 7.2 x 10^86707 and the 3x3x3x3x3x3x3 has 3.3 x 10^8935
I like how the 0x0 exists. Everyone has it but it’s invisible
🧠🧠
Everyone solved it in -9 months because since you were a little cell you already solved it
You have to solve air💀💀💀
You can only hear it
The video is also wrong about it, the empty puzzle has 1 permutation, not 0. There is exactly one way to arrange nothing.
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
🤓
@@Icedonot he’s right tho
@@rax1899 but funny nerd emoji
@@Icedonot 🗿
@@Icedonot 😎
I love how the high ones on this list have more combinations than the amount of atoms in the observable universe
the estimation of Atoms in the observable universe is 10^82~10^87
The 150x150 almost has the amount of atoms in the universe ^10,000. That's like nesting a universe in every atom of the universe 10,000 times.
@@Soviet_Cat1729 isn't it 10^78 - 10^82
@@cythism8106 That's not how you use conjunction operator.
No way I can’t believe that the (insert puzzle name here) had (n) different permutations!
no way!!
Realq
Does rotating a cube count as a permutation? Surely all the pieces can be put in a different location that way, or is permutation the wrong word
@@meraldlag4336 No, because thats just turning the entire cube. Like on the 1x1x1, if rotating the cube was a different permutation, then it wouldn’t be just 1 permutation
@@numbered-as-a-hashtag so how is permutation defined in the video then
I knew gear cube was restricted, but it's honestly insane that it has fewer scrambles than a 2x2...
Good comment, "Ryan", who lives at [[REDACTED]] 😳
@@RowanFortier what
Yo what 💀
@RowanFot
@@RowanFortier bro doxxed the dude 💀
Bro I love this guy he researches so well from what I’ve seen in the comments, and the video was interesting too
Glad you enjoyed!
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 😭
@@cewla3348 0^n=0
🤓🤓🤓
@@Bumpus. the funny
@@Bumpus. 😐😐😐
@@Bumpus. you are being unkind.
Isn't 0x0x0 1? Only one case, which is nothing? Many combinatorics problems (especially recurrence relation problems) has the same logic.
it doesnt literally exist, you can make nothing(0) with it.
@@quantdev 0!=1
i was thinking the same, it should be one as many combinatorics problems also have similar answers
@@quantdev That is still something
Well if its 0x0x0 then it doesnt exist, meaning that its 0, not 1, if it was 1 it would exist edit: nvm
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?
iirc dino has around 20 million, rex has 400 sextillion, and curvy copter has 1.5 sextillion without jumbling, and 15 dectillion with jumbling. also yeah clocks permutation is literally just 12¹⁴
@@PuyoTetris2Fan I thought clock was 12^15?
yeah I love the 4th dimensions cubes, they are hard since you don't see some faces you have to guess where they are 😂
@A Random Gamer oh ok, i'm not really good at understanding all this 4th dimension thing so I just told what passed through my mind
Nah... 3x3x3x3x3x3x3 is harder... You need to see the small piece and also need 1m+ turning face to complete it
@@luparty..gwmatycoysampaiba8701 that's a bit too hard fo me to imagine it
You're how I got into hypercubing :) Love your videos
What about the atlasminx and minx of madness? Also coren's 13 layer pyraminx would be interesting to see too
Respect to the guy who tried out all these combinations 🙏
1:57 This can branch up to infinite x infinite x infinite even in the 11th dimension which is the last one according to string theory
That last one takes "I'm 4 dimensions ahead of you" to a whole other level
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.
🤯🤯🤯
and also its googol^867 x 72000000
despite the 1 permutation, 1x1x1 is still the hardest rubik's cube my fellow cubers know
But 2x2x2 😮 has 3M combinations
I have spent 5 years on that cube that was passed down from my grandpa
@@888_kaiwalyarangle6 you are not worthy
@@888_kaiwalyarangle6 inbicel, if you cant scramble it you cant solve it
back on track be like
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.
Oh yikes - that is really embarrassing. Thanks for pointing that out!
It's 4.3×10^19, not 4.3×10^16, because 3×3×3 has 3 more digits.
Lol i was going to say this
Where is the minx of madness 😡
Thanks to the man who tried all of these combinations!
o7
Could you make a video specifically for cuboids like 1x2x3 or 2x3x4?
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.
0:07:easy cubes 0:43 :medium cubes 1:24 :challenging cubes 1:34 :extremely challenging cubes
omg!1!1(1(1!1(1 this really proves that the 1x1x1 is the hardest puzzle made by man!1!1!1!1!1
CRAZY!!!
What about the maple leaf skewb? We have that one and it's easy enough to solve without any knowledge beforehand and doesn't have that many permutations but I'm wondering where it's at on this list.
very cool !
Awesome video! How did you animate this?
I first made the images for each puzzle section, and then I made pictures of 4 of those at a time. Then just in my editing software, I made them all move to the left.
@@RowanFortier Nice it all looks super clean
@@RowanFortier but why?
Great video
Thanks!
NI LI SITELEN TAWA PONA A! GREAT VIDEO! mi kama sona mute tan sitelen tawa ni! It was very educational!
Toki Riley!
@@RowanFortier toki a, jan Rowan o!
How does the 0x0x0 have zero permutations? It actually has one, and that one permutation is where the “cube” isn’t in existence.
Bug brain
someone said the 0x0x0 exists lol
100x100x100 Rubiks cube:2 x 10^38415 permutations!!!
mans going to the 18th dimension for this
How to be a pro at the rubiks cube 1. Scramble properly 2. Swipe fast 3. Solve it Great job! Now find every combination.
Amazing :)
Thanks! 😄
Props to the person who counted how many permutations each of these puzzles have
Do you are have jokes
Respect that guy who actually counted all this
it made me proud, that there are still a dozen rubik's cube even I can solve, anyways all the best to the guy who manages to solve the 150×150×150 one, I mean he has to knock off 8×7 googol possibilities
Pyraminx has 933120 but only if you dont count the tips. If you count the tips, multiply it by 81
So 75,582,720
The 3^9 has 9,1556069*10^118409 permutations. I did the calculations for the 3^8 and 3^10 too, but i cannot find the numbers nor the calculation rn. But if i remember correctly. the 3^8 had something like 10^35000 permutations and the 3^10 something like 10^500000.
🤯🤯🤯
im no expert, but like, *thats a big number*
sick
Nice, why does the 3 3 4 have less permutations than 3 3 3 tho i dont understand
the 334 has an axis that's restricted to 180 degree turns. So basically the corners and edges are always oriented, unlike a 333 which has edge orientation and corner orientation. Even though it has an extra layer, the piece orientations make it much smaller number of permutations
its crazy how some of them have more permutations than there are atoms in the observable universe
1:45 _casually makes a cube in the 7th dimension_
yeah it's crazy how many permutations a 150x150x150 has, but we can go deeper
can i buy one of the 0x0x0 one? I think its pretty rare that i never saw one before
how does the 3x3x4 have less than the 3x3x3 (if the vertical rotations dont work it would make more sense but it just would be the dodo cube but with 4 instead of 2 layers)
Where can I buy a 4D rubiks cube?
Can you make a tutorial on a 4d 4x4?
damn a hecatonicosahedroid cube exists (120 cell)
Respect to the man who discovered this info.
i declare you put the minx of madness.
All the puzzle permutations: 0x0x0 - 0 1x1x1 - 1 1x1x2 - 4 1x2x2 - 6 1x1x3 - 16 3x3 - 24 1x2x3 - 48 1x3x3 - 192 1x2x5 - 1,152 Gear Cube - 41,472 2x2x3 - 241,920 Pyraminx - 933,120 Skewb - 3,129,280 2x2x2 - 3,674,160 2x3x4 - 418,037,760 3x3x2 - 3,251,404,800 Clock - 1,283,918,464,548,864 (1.2 x 10^15) Corner-Turning Octahedron - 2,009,078,326,888,000 (2 x 10^15) 3x3x4 - 41,295,442,083,840,000 (4.1 x 10^16) 3x3x3 - 43,252,003,274,489,856,000 (4.3 x 10^16) Square-1 - 8,617,338,912,961,658,880,000 (8.6 x 10^21) Face-Turning Octahedron - 31,408,133,379,194,880,000,000 (3.1 x 10^22) 3x4x5 - 41,102,509,778,424,299,529,000 (4.1 x 10^22) Square-2 - 1,240,896,803,466,478,878,720,000 (1.2 x 10^24) 2x2x2x2 - 3,357,894,533,384,932,272,635,904,000 (3.3 x 10^27) 4x4x5 8,881,841,338,276,800,000,000 (8.8 x 10^30) 4x4x4 - 7,401,196,842,564,901,869,874,093,974,498,574,336,000,000,000 (7.4 x 10^45) Pyraminx Crystal - 1,667,826,942,558,772,452,041,933,871,894,091,752,811,468,606,850,329,477,120,000,000,000 (1.6 x 10^66) Megaminx - 100,668,616,553,347,122,516,032,313,645,505,168,688,166,411,019,768,627,200,000,000,000 (1 x 10^68) 5x5x5 - 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 (help me) (2.8 x 10^74) 2x2x2x2x2 - 54,535,655,175,308,197,058,625,263:389,197,058,635,263,389,110,963,764,726,777,446,400,000,000,000,000,000,000,000,000,000,000,000,000 (5.4 x 10^88) 3x3x3x3 - 1,756,772,880,709,135,843,168,526,079,081,025,059,614,484,630,149,556,651,477,156,021,733,236,798,970,168,550,600,274,887,650,082,534,207,129,600,000,000,000,000 (1.7 x 10^120) 4x4x4x4 - 1.3 x 10^344 (yay no more chaos) 3x3x3x3x3 - 7 x 10^560 5x5x5x5 - 1.2 x 10^701 19x19x19 - 6.3 x 10^1,326 Yottaminx - 2.8 x 10^2,950 33x33x33 - 1.8 x 10^4,099 120-cell - 2.3 x 10^8,126 3x3x3x3x3x3x3 (7D) - 3.3 x 10^8,935 150x150x150 - 7.2 x 10^86,707
THE PAIN IT HURTS!
link to the 7th dimension cube?
After the Yottaminx, here comes the Xennaminx!
The 1x3x3 has 192 combinations without shape shifting, but with shape shifting it probably has 10,000-1 million combinations.
i went to another tab for a few seconds and the numbers went from millions to trillions!
I guess I’m not quite understanding the 3x3. I know it’s a 2 dimensional puzzle, but I can’t figure out why it’s 24, I can only see 12 permutations I imagine it’s something incredibly simple, but yeah 😬
For the 2D square puzzles, just imagine mirroring each side. So only the corners can move around. So then it's just 4! factorial, which is 4x3x2x1 = 24
@@RowanFortier hmmm…I’m still not quite getting it, I’m sure if I saw it in action it would be obvious, but I’m just kinda bad at picturing these things. Definitely appreciate the reply though 👍🏻
the pyramnix number is without the tips, so its the tetraminx
the best I can do with a cube is pyramid. I solved it 2 times in 2 days usually took me 1-3 hours to solve one
0:30 not every 0mm smidge you can turn it and it would have different shape?
i can solve the 2x2, the skweb, 3x3x3, 4x4x4, pyramid and the megaminx
If you're interested in the detailed maths behind n*n*n puzzle permutations, I did a video on that (in french)
How do you make these videos?
This went from 0 to a [high number] real fast
Fun fact: the “pyraminx crystal” is basically just a megaminx without the centers
the 150x150 looks so cursed without the long edges
Aight we saw how big the 150x150x150 is, now someone’s gotta go straight for actually building the 150x150x150x150
Its crazy that the 3⁷ has more permutations than a 19x19
1:36 I bet you feel dumb now Tingman
im wondering does a 4d square1 exist perhaps a cube 1?
the last one if you search it on google it just says "undefined and in google translate it just says "Infinity"
The 1x1x1x1: I still have 1 possible scrambled 😢
I was expecting to see something like the 65536x65536x65536 lol
New cube unlocked!! It is as hard as a 15x15x15!! *0x0x0*
what formulas are used to calculate these amounts of permutations?
Some crazy smart math guy found a formula for any nxnxn puzzle
actually, if you think about it, the number of combinations on a 0x0x0 is 1, because there is only 1 way to arrange none.
I understand everyone's arguments that 0^0 is 1, and 0! is one, and that there's 1 way that nothing can be in. The way that I thought of it originally was if you don't have anything in the first place, than what are you arranging? You can't arrange objects that you don't have any of. It's like the question doesn't even make sense, like how anything/0 is undefined. I actually think a 0x0x0 should have Undefined permutations
@@RowanFortier I think it actually makes some sense for there to be undefined permutations, though 1 permutation still makes more sense to me. Ig it depends on the persons perception on a 0x0x0.
I kept thinking about it, and I think I've come to the conclusion that the 0x0x0 has an indeterminate number of permutations. Counting the "permutations" of twisty puzzles is just a tiny bit misleading, because it technically includes both the permutations and the orientations of the pieces For example, a 2x2x2 has 7! permutations of its pieces (assuming one is stationary), multiplied by 3^6 orientations (still assuming one stationary piece, and dividing out the orientation of the last corner which is forced by the others) equaling 3674160 total permutations. If we apply this logic to the 0x0x0, which has 0 pieces, each with 0 possible positions, and 0 possible orientations, we find that the total permutations would equal the number of permutations of the pieces (0!) multiplied by the number of orientations (0^0) This yields the result of 0!*0^0 = 0^0, which is indeterminate. I guess it kinda makes sense for the number of permutations for a puzzle that doesn't even exist ¯\_(ツ)_/¯
0 factorial is 1, and 0^0 is also defined as 1 (I think), which would actually make it have 1 permutation somehow
The one permutation would be nothing?
@@RowanFortier 0^0 is indeterminate because 0^n=0 and n^0 = 1. 0^0 falls into both of these, so it’s not possible to determine an answer
i dont think it makes sense to extend that formula exactly as it is to the 0x0x0. if you look at it from a more practical viewpoint, it seems like it should be 1 for the same reason 0! = 1
@@brcktn But 0^0 is often defined as =1 especially in these cases where it comes to counting permutations
The 150x150x150 Cube Puzzle has 72OVgMnUNi (72 Octovigintimilli-unnongentillion) different Permutations.
Please tell me how this notation works, I’d really like to know how to count past centillion.
@@scrambledmc3772 This notation system is called "Bub's Notation". It goes like this Thousands: K Type-1 ones (before Decillion): M, B, T, Qa, Qi, Sx, Sp, Oc, No Type-1 ones (after Decillon): U, D, T, Qa, Qi, Sx, Sp, O, N Type-1 tens: De, Vg, Tg, Qd, Qt, Se, Sg, Og, Ng Type-1 hundreds: Cn, Du, Tc, Qr, Qn, Sc, St, Oi, Ni Type-2 ones: Mn, Mc, Nn, Pc, Fm, At, Zp, Yt, Xn
@@Baburun-Sama Sorry I didn’t mean the abbreviations, I meant the actual naming system for the numbers.
There's a wikipedia page for big number names I think
@@Baburun-Sama i knew this notation a year ago amd i never knew the name lol
it's crazy that the 33x33x33 is the first cube that has more than a millinillion or millillion Permutations
“I TOLD YOU I FID NOT LEARN ALL PERMUTATIONS” And non-cubers think we lie
For the 1x2x3 just memorize 48 different algorithms. 24 excluding mirrors
Me and the 9th dimensional boys solving a 7 dimensional cube:
Well,technically 0x0x0 should have one permutation cuz “nothing”is a state of the puzzle 😂
what the hell can u arrange if u have nothing in the first place
0:55 wouldn't 3x3x3 be (4.3 x 10^19)?
How does the 4x4x4 have less scrambles than the megaminx (I know megaminx has 12 sides but I just wondered and I can solve megaminx but not a 4x4x4)
Megaminx just has way more pieces I guess
@@RowanFortier o now I know thx
Ah yes, I love my 0*0*0 cube, I highly recommend it
well i already have it
Imagine counting this
Respect for the people who made those Rubik's took like 3 years or higher
1:15 Wow… that’s teh exact same megaminx I have
I'm Japanese, but I'm glad I was able to understand this video.
150x150x150 may sound impossible, but a general person who can solve a 4x4x4 can theoretically solve it
in 4 days
when will we get a Monster (Group theory) puzzle?
0:34 i have the pyraminx and it has 933120 possible permutations!
YESS
I have the gearcube and I can solve it in less then 10 seconds. But I never knew it had 41,472 possible combinations!
Why are there more combinations in a 3x3x3 then a 4x3x3?
1:22 man just passed the fourth dimension