I almost touched on it in the video. I said there were three ways to orient a corner cube, two ways to orient an edge cube, and there are two ways to swap two pieces (either you swap two pieces or you don't). Each of these will put you in a different orbit. So there are 3*2*2=12 orbits.
Hey, Dr. Grime! I wanted to say that I was given the chance to write a paper on a 7x7 Rubik's Cube (that I happened to have with me at college) for a math project of mine. I just finished it about a minute ago. I wish a video could be done about bigger Rubik's Cubes on Numberphile's channel!
+AnimXpert - Everything Animation That's not right, there _is_ a number for it! Of course, it's a massive number, but it's definitely not infinite. The amount of possible 17-rubix cubes and the number 1 are both exactly equidistant from infinity - because it's infinitely distant. To call a finite quanitity infinite just because it's really huge is a terrible generalization!
Good question. Let's say I have two cubes. For the first one I put all the pieces in the correct places. For the second one I put in all the pieces as if I had rotated the cube a half turn. These are not the same because the centre pieces do not move. On the first cube all the orange pieces would match the orange centre piece. On the second cube, the orange pieces are on the opposite side of the orange centre piece. Otherwise you would have been correct.
Ugh, I guess I shouldn't have looked at the comments. Its just like every other comment section on rubiks cube videos from popular youtubers. Someone saying they average 6 seconds, Simone saying to peel the stickers off, people arguing about parity errors (???), and then actual cubers coming in and correcting them.
When Rubik's Cube came out, my brother and I each got one. I gave up, almost immediately, but my brother worked on it, for a few days. Then, he showed us that he finally solved this crazy, new puzzle craze. We were all _very_ impressed. But I was the most impressed when he admitted, only to me, how he managed to solve it - he simply removed, then replaced, the stickers! I thought that was an ingenious solution! How clever!
I did the same thing when i got the first rubik's cube of my life Till now i have changed 2 rubik's cube and finally solved it my the cross method by learning it from youtube
I remember watching a talk show called "Unscrewed with Martin Sargent" and he had a champion Rubik's Cube solver on the show. They gave the guy like 30 seconds to study the cube before attempting it. Then he went at it and solved it in like 20 seconds. There was huge applause, and Martin was floored. After the applause died down, Martin was like "You know, you weren't supposed to be able to solve that! We peeled all the stickers off and stuck them back on randomly!" :-D
@@solaimon3164 The chances are much lower than that as peeling off the stickers enables you to make the puzzle unsolvable in all 12 "universes" of it's current configuration (e.g. putting 3 of the same color on one corner).
BRCuber you can just do R U R' U R' U2 R' until you get the fish case. This is why I learned 2-look OLL. Much more easier than full OLL. Full OLL is a killer, and probably not worth learning, but 2-Look will definately save you your time. Hope that helped.
TeaMMatE11 - Another Music Guy Well full oll is faster I'm already know full oll and I average at around 11 seconds . the thing your saying is just easier but not faster
In case english is your second language: "It annoys me when people *don't* do finger ticks" Because "people" is plural, you should use" don't instead of "doesn't". Or "do" instead of "does"
Fun fact: The amount of ways to arrange a Rubik's Cube doesn't even come close to the amount of ways you can arrange a standard deck of cards of 52. That number would be 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. That means, that if you do a thorough riffle shuffle (at least 7 times), chances are very very very very very likely that array has never happened before, ever in history. Or by comparison: The universe is only about 432,339,120,000,000,000 seconds old. That means that if there were 10 billion people shuffling decks of cards every second for the entire age of the universe, you still wouldn't even come close to that number. (4,323,391,200,000,000,000,000,000,000).
a riffle shuffle 7 times won't have those millions of octodecillions of possible outcomes because the original couple of top cards will remain on the top half for a long time, because of how the mechanics of the shuffle work, so that number is greatly reduced because it would be very difficult to bring the top card all the way to the far bottom positions in only 7 shuffles
of course, the reduced number would still unimaginably huge. I just find it hard to believe that a shuffle that keeps the top cards in the top slots, and the bottom cards on the bottom slots, done only 7 times, can have the same number of possible outcomes as that Original number
Awesome couple of videos! Very enjoyable to watch. What you might not know tho is that 43,252,003,274,489,856,000 - 1 = 43,252,003,274,489,855,999 which is a PRIME number!
0:47 I can't help myself from picturing that guy when he was like 6. Dad: "So what do you want to do later in life?" Kid : "Study squares! :-D" Then some years later... Mate honestly you're my new hero. :-)
This is the exact video i watched and the reason why i started to learn on how to solve a rubik's cube. Thank you numberphile. I average at 22 secs with PB of 16.37 secs and that's without memorizing and learning all the advanced techniques. I recently got back into solving and started memorizing all the algorithms. Hopefully by a few months i'll be sub 20.:)
***** 2x2: 3,674,160 3x3: 43,252,003,274,489,856,000 4x4: approx. 7.4e45 5x5: approx. 2.83e74 6x6: 1.57e116 7x7: approx. 1.95e160 nxn: Kid, the formula isn't so simple. You use factorials for the permutations of each set of pieces, (That's two on a 3x3, three on a 4x4, five on a 5x5, so-on-), then you duplicate the sets for the possible orientations of each piece, then you separate the group of permutations of the system that you can arrive at with a specific set of allowed translations from the initial permutation (solved state), that being slice moves, by identifying changes to the system that can only be done by making an impossible move and eliminating the set that requires that (That's the part where he divides it by twelve, by the way-). Basically, the formula is different for each cube, since it's a different matrix. It's not pre-algebra/algebra, it's early college level probability and carries over into advanced theoretical fields of mathematics.
KingHalbatorix The actual work is simple, it's how you get to the work that's complicated. Figuring out which formula to use and working it out is early high school, but a lot of the theory behind it is from later mathematics.
There was an explanation in Rubik's Cubic Compendium (multiple authors, 1987). Basically, you can swap two corners and two edges and still solve the puzzle because that is an even number. You can swap just two edges or just two corners and not solve it because that is an odd number. You can model a cycle of three pieces as two swaps. To get from ABCDE to ADBCE is a three-cycle or two swaps: ABCDE → ADCBE ADCBE → ADBCE I hope this helped.
I know this is an old comment that's probably been long forgotten, but the reason is this: You can't have just two swapped edge pieces on a cube, that's another universe. So that's two universes, swap or no swap. You also can't have a single flipped edge piece, where the piece is in the correct physical spot but with the colors the wrong way round. Those are two more universes, flip or no flip, and 2*2=4. So there are 4 universes so far. Then the corners, you can't have a single twisted corner, which is the same as a flipped edge but with a corner instead, and that there are 3 orientations for a corner, not 2. So that's 3 more universes of the cube. 4*3 is 12, and as such, there are 12 different universes for the cube to be in. This also means that were you to disassemble a cube and reassemble it completely randomly, then there becomes only a 1 in 12 chance of the cube being solvable, because there are 12 universes for the cube, with exactly one of them being solvable.
"So I know if i do that, that, that, that, that, that, that and then double twist, that will have swapped those two pieces." Flawless explanation, i totally understand. 10/10
PB: 17.79 and I've been cubing since December 2014. I've only gotten 3 sub 20's but I've been very close to beating my PB recently. Like this post if you're a cuber
@Numberphile Can anyone of you help me by writing with simple and understandable words why we divide by 12?plz cuz i need an answer immediately... please...
@@matinatheo8312 the whole explanation was based on taking the cube apart and putting it back together in all sorts of ways...but if you were to solve a cube (without taking it apart) you'd realise that there's no single (or a set ) of moves that can change the position or the orientation of one sub cube only...in simple terms it means that when u shuffle a cube, the sub cubes shuffle relative to each other...the number calculated was for a cube taken apart...so it had to be divided by 12 (the no. Of different "universes") to get the actual no. Of permutations on a cube...also the no. Was calculated for a standard 3x3 cube...it's different for cubes with pictures or numbers on them, larger cubes, etc....play around with a rubiks cube for a while...you'll understand how it works way better than any explanation on the internet :)
I wrote a program to calculate the number of patterns on an n-layer cube given only n as an input. It's interesting how quickly the exponent in the scientific notation goes up as n increases. Going from 2 to 7 the exponents are 6, 19, 45, 74, 116 and 160. If you don't like the sound of the word quintillion, you definitely won't like the (questionably valid) names for the higher-order cube permutations.
He gave the basic details (@4:08) before stating the number (12), without quite connecting them. First, a corner can be twisted +/- 1/3 turn, to get a different orbit (not stated: and there are ways of converting a twist of one corner to a twist of any other, so it doesn't matter which corner), so that gives 3 distinct orbits. Second, a side piece can be flipped, so there are 2 options. And a pair of side pieces can be swapped. So there are 3X2X2=12 orbits.
The total corner twists can add up to an integer + 1 clockwise, 1 counterclockwise or zero. (3 options) The total edge flips can add up to an integer + 1 or zero. (2 options) The total number of two-piece swaps can be odd or even. (2 options) The 12 universes are merely the product of these three possibilities.
The number of universes depends only on whether the cube has an odd or even number of layers. If you have an even-numbered cube you only need to worry about corner twists. There are three universes for that. If you have an odd-numbered cube there are the same 12 universes as with the 3x3x3. The extra pieces that show up from 4x4x4 and higher fall into only one universe unless the extra center pieces are made distinguishable from each other in some way.
Let's look at the 2x2x2. It's possible to swap two corners on it because you could consider a similar odd swap of the center pieces. Since the center pieces are hidden from view, odd swaps of visible pieces become possible. Odd swaps are also possible on the Void Cube (3x3x3 without centers) for the same reason. A way you could explain why even-numbered cubes can have odd swaps is because there are odd swaps of invisible pieces to make the total even.
I know this is the accepted number but it doesn't seem like the right one to me. For example, when you place the first corner piece, there is a giant set of cubes that can be constructed from there. But if you place that piece in any other corner, it generates an entire extra set of cubes that is identical to the first set and never divided out. You get the same thing with the edge pieces and even a similar color symmetry that you can argue is ignored.
The center pieces are never multiplied in because they're static. That functions as the redundant orientations you would otherwise divide out. th-cam.com/video/QV9k6dRQQe4/w-d-xo.html
+Adam Olsen I think I understand now, since the center pieces don't move every different corner position is distinguishable from the other ones. Thanks!
+Melinda Green yes, but if you start with the red/green/white corner in a different position you don't get the same cube configurations because it is in a different place respect to the centers
I don't want to be the bearer of bad news, but. The way you're going about this, is in someway wrong, yes. there is 43,252,003,274,489,856,000 possible combinations. But, not all are solvable If you were to take the rubiks cube apart, and put it back together again, but just flip 1 corner, the cube will become unsolvable, as it is impossible to flip just 1 corner. So neciserally, the rubiks cube, does not have 43,252,003,274,489,856,000 combinations, if you where to mix it by hand.
***** If you disassemble the pieces, you can assemble it again in 5,2*10^20 different ways. And there are 11 unsolvable stiation. So you want to find out the only solvable stiation. Then divide it by 12. (5,2*10^20)/12=43252003274489856 The solution is correct. Please watch the video again more carefully. And sorry for bad English.
***** I know I am a bit late to this, but just as the person before me said, he originally got a number 12 times 43 quintillion. He took into account the number of parities ("orbits"), which is 12, and divided his number by 12 to get 43 quintillion.
The 12 orbits are obtained by these 3 things: 1) flip any edge (2 orbits) 2) rotate any corner (3 orbits) 3) switch any 2 pieces (2 orbits) Each of these 3 are independent, giving 2*3*2 = 12 orbits. This means, somewhat counter-intuitively that if you take a solvable position, switch any two edge pieces AND switch any two corners, you get another solvable position (modulo flips and rotations, of course).
not really. this happens all the time in combinatorics. anytime a 5 and a 2 are multiplied together, the number will end in a 0. in fact, by counting the number of Zeros at the end if a number, you can know how many pairs of the numbers 2 and 5 appear in their factorization. in this case, the zeros come from the 8! (1 zero) and 12! (2 zeros).
+Adam Billman I had a homework assignment to find the number of zeroes that come at the end in 100! That's the method used to find it quick, just count the number of 5's in the prime factorization since there are fewer 5's than 2's. Iirc it's 24: one from each multiple of 5 and another from each multiple of 25
When calculating this, did they take into account that some combinations were unable to be achieved, for example a combination where only one piece is not permuted?
Learning and practicing to solve it in under 10 seconds is very impressive. But why not put all that time into learning a musical instrument or something?
I dont see what's so impressive about speed cubers; they're just following pre-written algorithms. Plain and simple. The ones deserving of praise are the dudes who originally came up with the solution.
MiningChr1s Perhaps he has. I've always considered the whole point of the puzzle (like any puzzle) to be to figure it out yourself. No, no credit to memorisers of other people's algorithms.
Totally McMylastname recognizing, and executing algorithms fast is one part. But for a cfop method, algorithms are less than half of your solve. Intuitice cross, and f2l
Cube Tuber It seems to me that Totally McMylastname's point still applies. You write of a "cfop method", "Intuitice cross" and "f2l". Whatever these are, the praise would go to those who came up with them rather than those who just learn them. I don't mean to call speed-cubing worthless. I'm sure it's a fulfilling hobby. However, I agree that it's a little unimpressive. Personally, I don't see the point in learning somebody else's solution to a puzzle; that just ruins the puzzle. You could even call it cheating.
Sorry intuitive cross and f2l means that there are no algorithms, the cuber figures it out themselves and every situation is different. So we're not just following pre written algorithms. But i guess it is true that it could be called cheating. However, i don't think it is unimpressive especially if you are extremely fast, which is very difficult. That is why very few people are able to average under 10 seconds.
The author had explained the no. of universes (or orbits) is 12. To get to other universes, three ways were mentioned: 1. flip any edge piece once will get you to another universe. But remember that you have normal moves to flip any two edge pieces together. Therefore flipping two edge pieces will leave you in the same universe. So you get two distinct universes by flipping an edge piece. 2. rotate any corner piece. Three orientations of the corner piece means three distinct universes. If you rotate more than two corners, you will be getting into one of these three universes because there are normal moves to rotate two corners together such that the total rotation of the corners are 360 degress. So you can always reduce the rotation of more than two corners into a single rotation of a single corner piece. 3. Swapping two edge pieces / corner pieces. It happens that using normal moves one can turn a swap of edge pieces into a swap of two corner pieces. Therefore the two kinds of swapping are essentially equivalent. Also, doing two swappings of edge pieces will again bring you back to the same universe as there are normal moves to do it. Therefore you get two distinct universes by swapping edge pieces or corner pieces. One should also be aware that these changes are independent of each others. e.g. you cannot undo the effect of a flip of an edge piece by swapping the positions of two pieces, or rotating a corner piece. So the no. of distinct universes is 2 x 3 x 2 = 12.
You said that you can swap the two edges and the two corners, and so the two universes are the same, but you cannot do the two individually. It isn't possible to swap just 2 corners or just two edges, so you are wrong.
UberCuber is right. That is also what I meant. It isn't possible to swap just 2 corners or just two edges by normal moves. If you attempt to swap just 2 corners by normal moves, you will find that you either end up swapping 2 more other corners or swapping 2 more other edges. So the result is a combination of corner swap AND edge swap; or a combination of corner swap AND another corner swap. The same happens to edge swaps. That's why I said if you disassemble the cube and swap two edges AND swap two corners, then you can solve the cube into its original state, i.e. in the same orbit as the state before the disassemble and swap.
Hello Numberphile! I've solved a number of Rubik's cube myself. For cubes with labeled centers, like those giveaways by some companies, you will notice that a solved state can have rotated centers. You don't notice this in a standard cube because the center is just a single color with no indication for orientation. But, if you account for this I believe you should get an extra factor of (4^6) combinations as there are 6 centers and 4 rotations possible for each center.
There are 2 to the power of n squared possible combinations for a minesweeper grid with n grids on each size, or 2 to the power of l * w, where l and w are the length and width respectively.
The video mentions that there are twelve orbits, incidentally the same amount of edges on the cube. Taking off a corner piece, rotating it and placing it on the cube again does not form a new orbit. Only edge pieces will form a new orbit when you flip them since the corner pieces get flipped anyway.
Personal Best 314,159,265,358,979,323,846,264,338,327,950,288,419,716,939,937,510,582,097,494,459,230,781,640,628,620,899,862,803,482,534,211,706,798,214,808,651,328,230,664 Years NO breaks
A = 8 corner positions = 8! B = 3 orientations per corner = 3^8 C =12 edge positions = 12! D = 2 orientations per edge = 2^12 E = can't have single corner twisted = 1/3 F = can't have single edge flipped = 1/2 G = can't have two edges swapped = 1/2 A*B*C*D*E*F*G = 43252003274489856000
Also, The center pieces can be pulled off of most cube to reveal a screw that adjusts the tension, however, it is typically only one center, and since rotation doesn't matter, you won't get anywhere
If you could have as many of any color as you want (i.e. 42 green, 12 red), you would have an astounding 1,047,532,535,594,334,222,593,508,922,191,671,036,215,296 combinations. Crazy, huh?
Another way i like to put the number 43,252,003,274,489,856,000 in to perspective is this. If you were to go through each combination one at a time, one per second, and with the final turn being the solution. I would take about 100 life times (the birth of the universe till now) of the universe to finish. That is (((43,252,003,274,489,856,000/3600)/24)/365.25)/13,700,000,000. Assuming the universe is roughly 13.7 billion years old as of now.
Because those are the number of "orbits" that the cube can be arranged in. (Combinations that are not possible on a traditional cube so you must divide the total combinations by the number of "universes" that aren't possible.
Thank you. It does help. I'll take a look at that book next time I go to the library (they have it, I checked). I was hoping there would be a more general theorem, which could be extended to any puzzle though. For example why is it possible to swap just two edges on a 4x4 and not on a 3x3 (or any odd/even sizes) and remain in the same universe?
I almost touched on it in the video. I said there were three ways to orient a corner cube, two ways to orient an edge cube, and there are two ways to swap two pieces (either you swap two pieces or you don't). Each of these will put you in a different orbit. So there are 3*2*2=12 orbits.
Imagine being the main speaker in a video but your comment in that video's comment section goes 7 years with no likes and comments.
Hey, Dr. Grime! I wanted to say that I was given the chance to write a paper on a 7x7 Rubik's Cube (that I happened to have with me at college) for a math project of mine. I just finished it about a minute ago. I wish a video could be done about bigger Rubik's Cubes on Numberphile's channel!
Why divide by 12 instead of multiplying then???
👍👌
It’s like subtracting the one off of 6561 Using 6560 and 375 and making a run for 5 or 3.
thank you... these took a while so glad to hear some positive feedback! ;)
There exist a 17x17x17 rubiks cube. How many combinations does that have?
+CopCat There are 6.69 x 10^1054 possible combinations on that cube...
+Chrnan6710 Wow, that's a "very" round number!
there is so many combinations there is not a number for it. just consider it infinite
A whole lot more than the original
+AnimXpert - Everything Animation That's not right, there _is_ a number for it! Of course, it's a massive number, but it's definitely not infinite. The amount of possible 17-rubix cubes and the number 1 are both exactly equidistant from infinity - because it's infinitely distant. To call a finite quanitity infinite just because it's really huge is a terrible generalization!
personal best:
4 years with breaks
Padster I once got one side of it right.
Collin Kappa I solve it in under a minute 30
Hanchman Nice
Padster i solved it in 17 secs
Padster i solved it in 17 secs
Permutations, not combinations.
I didn't even notice that.
+Tony Fisher combinations, not permutations. permuting simply means to move around.....
+Chase Brower No one mentioned permuting. I wouldn't have made my statement if it wasn't correct, would I? Look it up, read, learn.
+Tony Fisher lol
+Tony Fisher Isn't combinations a fixed set of elements while not being fixed on permutations?
Good question. Let's say I have two cubes. For the first one I put all the pieces in the correct places. For the second one I put in all the pieces as if I had rotated the cube a half turn. These are not the same because the centre pieces do not move. On the first cube all the orange pieces would match the orange centre piece. On the second cube, the orange pieces are on the opposite side of the orange centre piece. Otherwise you would have been correct.
Ugh, I guess I shouldn't have looked at the comments. Its just like every other comment section on rubiks cube videos from popular youtubers. Someone saying they average 6 seconds, Simone saying to peel the stickers off, people arguing about parity errors (???), and then actual cubers coming in and correcting them.
I completely agree.....
Woah you are here
Ikr
You forgot people saying it takes them x hours or x days or x years etc. to solve it.
legoboyz3! non-cubers
When Rubik's Cube came out, my brother and I each got one. I gave up, almost immediately, but my brother worked on it, for a few days. Then, he showed us that he finally solved this crazy, new puzzle craze. We were all _very_ impressed. But I was the most impressed when he admitted, only to me, how he managed to solve it - he simply removed, then replaced, the stickers! I thought that was an ingenious solution! How clever!
I did the same thing when i got the first rubik's cube of my life
Till now i have changed 2 rubik's cube and finally solved it my the cross method by learning it from youtube
I remember watching a talk show called "Unscrewed with Martin Sargent" and he had a champion Rubik's Cube solver on the show. They gave the guy like 30 seconds to study the cube before attempting it. Then he went at it and solved it in like 20 seconds. There was huge applause, and Martin was floored. After the applause died down, Martin was like "You know, you weren't supposed to be able to solve that! We peeled all the stickers off and stuck them back on randomly!" :-D
The chances of that happening are very slim. Amazing if true bro.
@@alfiedemmon4132 actually, the chance are 1/12 as singingbanana mentinonned. But I don't know where this 12 comes from
@@solaimon3164 The chances are much lower than that as peeling off the stickers enables you to make the puzzle unsolvable in all 12 "universes" of it's current configuration (e.g. putting 3 of the same color on one corner).
I doubt he was a champion speedcuber. Maybe he was the best in his town but thats not a champion. in 2015 the wr was around 4 seconds.
@@venkinta3343 If he won a competition, he is a champion.
0:08 First I thought that Feliks was talking after his solve!:D
They have very similar voices
Lucy Hunt true!
5:23 was anyone else screaming OLL JUST DO OLL!
OLL is pretty hard, there are more than 50 different algorithms you have to memorise.
Well he had an easy "fish case" is what I call it where all edges are flipped and 2 corners ar flipped.
BRCuber you can just do R U R' U R' U2 R' until you get the fish case. This is why I learned 2-look OLL. Much more easier than full OLL.
Full OLL is a killer, and probably not worth learning, but 2-Look will definately save you your time.
Hope that helped.
That wasn't even part of full OLL that was 2-look I'm pretty sure (Haven't watched this video in a couple of months.)
TeaMMatE11 - Another Music Guy
Well full oll is faster
I'm already know full oll and I average at around 11 seconds .
the thing your saying is just easier but not faster
Brady, you have done an exceptional job with this video. I love how many different videos are spliced into this one
TH-cam Recommendations:
2012: No.
2013: No.
2014: No.
2015: No.
2016: No.
2017: No.
2018: No.
2019: Yes.
2012 No.
2013 No.
2014 No.
2015 No.
2016 No.
2017 No.
2018 No.
2019 No.
2020 on the Wii; Yes.
@@david_ga8490 2021: 😢
@@sourabhsaha6865 the "2020 on the Wii" is a reference to Nathaniel Bandy's video "Just Dance 2020 on the Wii"
@@david_ga8490 : O
It annoys me when people doesn't do finger tricks
+GammersFaze It annoys me when people use wrong grammar.
Tejas Kalyan
What do you mean?
+Tejas Kalyan He did as well. lol
Same
In case english is your second language:
"It annoys me when people *don't* do finger ticks"
Because "people" is plural, you should use" don't instead of "doesn't". Or "do" instead of "does"
He's using the beginners method you could see he's using sune over and over again for oll xD
he's a noob
+SuperZecton hes also using a basic rubiks brand cube. nest thing you know hes gonna be speed solving the Cube4you gigaminx
+SuperZecton The guy doing it in conversation was using a DaYan cube, not sure which one, probably a GuHong
Aston Culf looks like a gu hong to me maybe a zhanchi?
youre one of those guys
Fun fact: The amount of ways to arrange a Rubik's Cube doesn't even come close to the amount of ways you can arrange a standard deck of cards of 52. That number would be 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000.
That means, that if you do a thorough riffle shuffle (at least 7 times), chances are very very very very very likely that array has never happened before, ever in history.
Or by comparison: The universe is only about 432,339,120,000,000,000 seconds old. That means that if there were 10 billion people shuffling decks of cards every second for the entire age of the universe, you still wouldn't even come close to that number. (4,323,391,200,000,000,000,000,000,000).
Elroy Kerstens vsause?
a riffle shuffle 7 times won't have those millions of octodecillions of possible outcomes because the original couple of top cards will remain on the top half for a long time, because of how the mechanics of the shuffle work, so that number is greatly reduced because it would be very difficult to bring the top card all the way to the far bottom positions in only 7 shuffles
of course, the reduced number would still unimaginably huge. I just find it hard to believe that a shuffle that keeps the top cards in the top slots, and the bottom cards on the bottom slots, done only 7 times, can have the same number of possible outcomes as that Original number
Awesome couple of videos! Very enjoyable to watch.
What you might not know tho is that
43,252,003,274,489,856,000 - 1 = 43,252,003,274,489,855,999
which is a PRIME number!
I'm calling these the Rubik Primes
@@incription 4x4 - 1 =
@@kymiram7865
4 * 4 -1
= 16 - 1
= 15
= 3(5)
Not a prime lol
0:47 I can't help myself from picturing that guy when he was like 6.
Dad: "So what do you want to do later in life?"
Kid : "Study squares! :-D"
Then some years later...
Mate honestly you're my new hero. :-)
0:47
Now go check out the variations of a deck of cards, or 52 factorial!! Mind blowing
it's 51 factorial actually. One card has an option of 51 other cards
This is the exact video i watched and the reason why i started to learn on how to solve a rubik's cube. Thank you numberphile. I average at 22 secs with PB of 16.37 secs and that's without memorizing and learning all the advanced techniques. I recently got back into solving and started memorizing all the algorithms. Hopefully by a few months i'll be sub 20.:)
I was about to comment "Matt memorizes numbers in his free time."
kind of off topic, but I'd love it if somebody made a ridiculous 'cube' that was something like 1x1x25
or in other words, a big long twisty stick >.
ramiel555 someone made a 2x2x14
Tristan Dueck yeah, I've seen that, it's pretty awesome, but......NEED MOAR! :P
ramiel555 there is a cube made by Oskar's puzzles, it is a 1 x 1 x 27. Cost is close to 1,300 dollars
Zachary Fagan where?
Zachary Fagan that's a 3x3x27 O.o
how many possible combinations are there for a 2x2x2 cube?4x4x4 cube? 5x5x5? 6x6x6? etc
Don't know, but redkb has a video on how many permutations there are for a 7x7
Cubeagami101 And 17x17!
*****
2x2: 3,674,160
3x3: 43,252,003,274,489,856,000
4x4: approx. 7.4e45
5x5: approx. 2.83e74
6x6: 1.57e116
7x7: approx. 1.95e160
nxn: Kid, the formula isn't so simple. You use factorials for the permutations of each set of pieces, (That's two on a 3x3, three on a 4x4, five on a 5x5, so-on-), then you duplicate the sets for the possible orientations of each piece, then you separate the group of permutations of the system that you can arrive at with a specific set of allowed translations from the initial permutation (solved state), that being slice moves, by identifying changes to the system that can only be done by making an impossible move and eliminating the set that requires that (That's the part where he divides it by twelve, by the way-).
Basically, the formula is different for each cube, since it's a different matrix. It's not pre-algebra/algebra, it's early college level probability and carries over into advanced theoretical fields of mathematics.
KingHalbatorix
The actual work is simple, it's how you get to the work that's complicated.
Figuring out which formula to use and working it out is early high school, but a lot of the theory behind it is from later mathematics.
***** so 6^(n^^3)?
I remember it being marketed as having 43 billion combinations. An understatement
There was an explanation in Rubik's Cubic Compendium (multiple authors, 1987). Basically, you can swap two corners and two edges and still solve the puzzle because that is an even number. You can swap just two edges or just two corners and not solve it because that is an odd number.
You can model a cycle of three pieces as two swaps. To get from ABCDE to ADBCE is a three-cycle or two swaps:
ABCDE → ADCBE
ADCBE → ADBCE
I hope this helped.
he had a dayan...
numberphile
2 minute solver
AND HE HAS A FRIGGEN DAYAN ZUCINI
Zhanchi*
+JSK01 - Agario it's a joke tho
Man am I glad he didn't pull a Parker cube.
interesting! I was still wondering how you came to the amount of 12 "universes"?
I know this is an old comment that's probably been long forgotten, but the reason is this:
You can't have just two swapped edge pieces on a cube, that's another universe. So that's two universes, swap or no swap.
You also can't have a single flipped edge piece, where the piece is in the correct physical spot but with the colors the wrong way round. Those are two more universes, flip or no flip, and 2*2=4. So there are 4 universes so far.
Then the corners, you can't have a single twisted corner, which is the same as a flipped edge but with a corner instead, and that there are 3 orientations for a corner, not 2. So that's 3 more universes of the cube.
4*3 is 12, and as such, there are 12 different universes for the cube to be in. This also means that were you to disassemble a cube and reassemble it completely randomly, then there becomes only a 1 in 12 chance of the cube being solvable, because there are 12 universes for the cube, with exactly one of them being solvable.
"So I know if i do that, that, that, that, that, that, that and then double twist, that will have swapped those two pieces."
Flawless explanation, i totally understand. 10/10
Can anybody tell me why there are 12 universes? Its not explained in the video.
REALLY ??????????????????????? AMAZING !!!
I am a speed cubed as well, 16.82 seconds is my best time
Do you still cube?
PB 22.49. Best average of 5: I forgot
@@nanamacapagal8342 Pb - 11.28 , Pb ao5 - 14.58 , Pb mo3 - 12.21
Feliks Zemdegs went to my primary school in Melbourne Australia
I LOVE FELIX
Who
Popular Now world record speedcuber
Great vid
*_I LOVE NUMBERPHILE!!! GREAT JOB!! KEEP IT UP!!!_*
"So, I'm jut gonna give the Rubik's Cube to Matt..."
Who else was thinking of Mats Valk lol
PB: 17.79 and I've been cubing since December 2014. I've only gotten 3 sub 20's but I've been very close to beating my PB recently.
Like this post if you're a cuber
I average 20 and pb is 7.10
+Picklepower26 how did u get your pb,
super luck
+RECuber I average about 21-23 seconds. I have recently been getting into BLD, though.
Sethamajig
-Esperanto
-Cuber
matt is my numberfu
my favorite video thus far numberphile
The sound of how smooth those cubes are ughhhh
What I'm interested is how you get the fact that there are 12 orbits.
Me too..
2 (number of orientations for one edge) * 3 (number of orientations for one corner) * 2 (Because an odd number of swaps isn't possible) = 12
@Numberphile Can anyone of you help me by writing
with simple and understandable words why we divide by 12?plz cuz i need an answer immediately... please...
@@matinatheo8312 the whole explanation was based on taking the cube apart and putting it back together in all sorts of ways...but if you were to solve a cube (without taking it apart) you'd realise that there's no single (or a set ) of moves that can change the position or the orientation of one sub cube only...in simple terms it means that when u shuffle a cube, the sub cubes shuffle relative to each other...the number calculated was for a cube taken apart...so it had to be divided by 12 (the no. Of different "universes") to get the actual no. Of permutations on a cube...also the no. Was calculated for a standard 3x3 cube...it's different for cubes with pictures or numbers on them, larger cubes, etc....play around with a rubiks cube for a while...you'll understand how it works way better than any explanation on the internet :)
The "subcubes" are called cubies. Trust me, I'm a speedcube solver.
ok boomer
@Blockcamp ok boomer
@@MeiZhang-q5k nobody asked
@iBrow nobody asked you either
This video is a goldmine for overused noncuber comments...
Math and the Rubik's Cube is a perfect combination!! I love both!
I wrote a program to calculate the number of patterns on an n-layer cube given only n as an input. It's interesting how quickly the exponent in the scientific notation goes up as n increases.
Going from 2 to 7 the exponents are 6, 19, 45, 74, 116 and 160. If you don't like the sound of the word quintillion, you definitely won't like the (questionably valid) names for the higher-order cube permutations.
43,252,003,274,489,856,000 wtf is this ?? Is there such a number?
Yah 43 quintillion
what is quintillion ? ı thought There are just millions but wtf
FullTimeSlacker no man ım just kidding
+videogamefan004 g(g(64))
fourty-three quintillion, two-hundred fifty-two quadrillion, three trillion, two-hundred seventy-four billion, four-hundred eighty-nine million, eight-hundred fifty-six thousand
2:51 I believe those are called unit Cubes.
cubies!
Marcus Hill actually its called pieces
Marcus Hill cubies
He gave the basic details (@4:08) before stating the number (12), without quite connecting them. First, a corner can be twisted +/- 1/3 turn, to get a different orbit (not stated: and there are ways of converting a twist of one corner to a twist of any other, so it doesn't matter which corner), so that gives 3 distinct orbits. Second, a side piece can be flipped, so there are 2 options. And a pair of side pieces can be swapped. So there are 3X2X2=12 orbits.
4:08
The total corner twists can add up to an integer + 1 clockwise, 1 counterclockwise or zero. (3 options)
The total edge flips can add up to an integer + 1 or zero. (2 options)
The total number of two-piece swaps can be odd or even. (2 options)
The 12 universes are merely the product of these three possibilities.
Wrong the last corner only has one possible orientation.
The last corner still has 3 orientations, the orientation is simply which way the colours are facing
It is impossible to rotate one corner, you always have to rotate at least 2 corners
Siem De Wit He's not talking about a legit move, he said when you take all the peices off and put it back together.
Then you are right.
If you got 43,252,003,274,489,856,000 views on this video.....
takes me 50 seconds - 1 minute to solve it :)
Blaidan what about now? As Brucey always said... Higher or Lower?
Wow, these people are amazing....👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼👏🏼
Combinations: Permutations
Sub-Cubes: Pieces
Quarter Turn: Outer turn
Arrangement: Orientation
Twist: Move
Did anyone catch
For the second corner ill have se-- six left
+Tessa the gymnast he said seh-- seven not six
4D Puzzle game by 2080? eh?
I thought those were the amount of Genders.
Think more in the neighborhood of Graham's number
Ashkar Ibne Awal Not funny.
It's hilarious XD
*nO tHeRE iS OnLy tWo GenDeRs*
The number of universes depends only on whether the cube has an odd or even number of layers.
If you have an even-numbered cube you only need to worry about corner twists. There are three universes for that.
If you have an odd-numbered cube there are the same 12 universes as with the 3x3x3.
The extra pieces that show up from 4x4x4 and higher fall into only one universe unless the extra center pieces are made distinguishable from each other in some way.
Let's look at the 2x2x2. It's possible to swap two corners on it because you could consider a similar odd swap of the center pieces. Since the center pieces are hidden from view, odd swaps of visible pieces become possible. Odd swaps are also possible on the Void Cube (3x3x3 without centers) for the same reason.
A way you could explain why even-numbered cubes can have odd swaps is because there are odd swaps of invisible pieces to make the total even.
I know this is the accepted number but it doesn't seem like the right one to me. For example, when you place the first corner piece, there is a giant set of cubes that can be constructed from there. But if you place that piece in any other corner, it generates an entire extra set of cubes that is identical to the first set and never divided out. You get the same thing with the edge pieces and even a similar color symmetry that you can argue is ignored.
I agree
The center pieces are never multiplied in because they're static. That functions as the redundant orientations you would otherwise divide out.
th-cam.com/video/QV9k6dRQQe4/w-d-xo.html
+Adam Olsen I think I understand now, since the center pieces don't move every different corner position is distinguishable from the other ones. Thanks!
Adam Olsen
I never mentioned the center pieces.
+Melinda Green yes, but if you start with the red/green/white corner in a different position you don't get the same cube configurations because it is in a different place respect to the centers
Personal best: 90 million years
Jk I've never used a rubik's cube
Girly Card Then shut up
PB: N/A
+Tebs Productions actually hes typing. even tho ppl mostly talk when they type, so they type the right way
i did it while even typing this
thechrimsonfucker no new episode yet i see
Numberphile. Keeping the brown paper industry alive.
43,252,003,274,489,856,000 or if you
prefer scientific notation its roughly
~ 4.3252 × 10^19 so yeah I rounded
It off but its accurate enough right
I don't want to be the bearer of bad news, but. The way you're going about this, is in someway wrong, yes. there is 43,252,003,274,489,856,000 possible combinations. But, not all are solvable If you were to take the rubiks cube apart, and put it back together again, but just flip 1 corner, the cube will become unsolvable, as it is impossible to flip just 1 corner. So neciserally, the rubiks cube, does not have 43,252,003,274,489,856,000 combinations, if you where to mix it by hand.
***** that is exactly what i thought, the actual combinations are about 5-6 orders smaller
***** If you disassemble the pieces, you can assemble it again in 5,2*10^20 different ways. And there are 11 unsolvable stiation. So you want to find out the only solvable stiation. Then divide it by 12. (5,2*10^20)/12=43252003274489856 The solution is correct. Please watch the video again more carefully. And sorry for bad English.
deniz is correct he did take unsolvable combos into account in the video.the original number was around 12 times as large
***** I know I am a bit late to this, but just as the person before me said, he originally got a number 12 times 43 quintillion. He took into account the number of parities ("orbits"), which is 12, and divided his number by 12 to get 43 quintillion.
The 12 orbits are obtained by these 3 things:
1) flip any edge (2 orbits)
2) rotate any corner (3 orbits)
3) switch any 2 pieces (2 orbits)
Each of these 3 are independent, giving 2*3*2 = 12 orbits.
This means, somewhat counter-intuitively that if you take a solvable position, switch any two edge pieces AND switch any two corners, you get another solvable position (modulo flips and rotations, of course).
pb:14 am I cool yet?
skater Boi yes, my pb is 16 :(
lraC Ae u got this man,we'll be sub 10 in some time!
skater Boi thanks you, i hope so
my pb is 16
My PB is only 22 ;(
Wow 856 thousand flat. Thats convenient.
What if there were 31,415,926,535,897,932,384 combos that would be legendary.
Amazing
not really. this happens all the time in combinatorics. anytime a 5 and a 2 are multiplied together, the number will end in a 0. in fact, by counting the number of Zeros at the end if a number, you can know how many pairs of the numbers 2 and 5 appear in their factorization. in this case, the zeros come from the 8! (1 zero) and 12! (2 zeros).
Adam Billman
Ohhh....duhh
+Adam Billman I had a homework assignment to find the number of zeroes that come at the end in 100! That's the method used to find it quick, just count the number of 5's in the prime factorization since there are fewer 5's than 2's.
Iirc it's 24: one from each multiple of 5 and another from each multiple of 25
Well that's definitely my new favorite number :3
O_O oh my goodness! That 15 piece puzzle is something I had as a kid XD No idea it was that old!
LOL i have both of the cubes shown in the vid i have the classical and dayan zanchi!
:P
My best is 28 seconds
My best is 2 minutes.
My best is 52 seconds
Legomon Jones
my best was like 3 minutes, which is not impressive by any standards.
now i cant even solve it :(
55 seconds.
1:56 :/
the 15 puzzle! :D a staple widget in desktops for ages.... I have a cube somewhere... I've only ever solved it once.
Now there exits a 19x19 how many combinations does that have?
Did Matt have a Zhanchi?
Yes it was
:)
+PurelyAwesomeCuber He needs some lube its pretty springy lol
Agreed.
+WJ50Skillz no
My best is 8 hours. I am extremley tired.
It's actually really easy to solve a Rubik's cube. First you have any color facing up, then you switch around the stickers. Works every time
General JoBob My cube is stickerless. Try and peel that now!
Charlie Lettau smash it
Charlie Lettau dissassemble it and assemble it correctly
MrTURBOJOHN That's cheating though c;
my best is 10.31, as everyone else seems to be saying their bests.
Wow that's awfully slow. But now i feel awfully sad that I can't remember my PBs from 2 years ago at all
mine is 24.81
As well as being a good tangible representation of symmetry as Matt said. It is also a good tangible representation of entropy
When calculating this, did they take into account that some combinations were unable to be achieved, for example a combination where only one piece is not permuted?
yes, they take it into account
*999999th view!*
wooooo
dat calculation t lol your channel name
Anna Walker, bet you'll love my videos😂
thats one in in a million
dat calculation tho
Learning and practicing to solve it in under 10 seconds is very impressive. But why not put all that time into learning a musical instrument or something?
Cuz cubing is life!
What if you had to deactivate a Rubik's cube-shaped bomb with 10 seconds left on the display? :q
Cubing is more impressive
Ben Adams we like cubes. That is why.
I dont see what's so impressive about speed cubers; they're just following pre-written algorithms. Plain and simple. The ones deserving of praise are the dudes who originally came up with the solution.
You do it then
MiningChr1s Perhaps he has. I've always considered the whole point of the puzzle (like any puzzle) to be to figure it out yourself.
No, no credit to memorisers of other people's algorithms.
Totally McMylastname recognizing, and executing algorithms fast is one part. But for a cfop method, algorithms are less than half of your solve. Intuitice cross, and f2l
Cube Tuber It seems to me that Totally McMylastname's point still applies. You write of a "cfop method", "Intuitice cross" and "f2l". Whatever these are, the praise would go to those who came up with them rather than those who just learn them.
I don't mean to call speed-cubing worthless. I'm sure it's a fulfilling hobby. However, I agree that it's a little unimpressive.
Personally, I don't see the point in learning somebody else's solution to a puzzle; that just ruins the puzzle. You could even call it cheating.
Sorry intuitive cross and f2l means that there are no algorithms, the cuber figures it out themselves and every situation is different. So we're not just following pre written algorithms. But i guess it is true that it could be called cheating. However, i don't think it is unimpressive especially if you are extremely fast, which is very difficult. That is why very few people are able to average under 10 seconds.
Cool they did a video on it and it was accurate infromation!
Yeah, I joined the craze when they first came out and still have a couple that have been kicking around for 20 years or more untouched.
i am a speed cuber
The author had explained the no. of universes (or orbits) is 12. To get to other universes, three ways were mentioned:
1. flip any edge piece once will get you to another universe. But remember that you have normal moves to flip any two edge pieces together. Therefore flipping two edge pieces will leave you in the same universe. So you get two distinct universes by flipping an edge piece.
2. rotate any corner piece. Three orientations of the corner piece means three distinct universes. If you rotate more than two corners, you will be getting into one of these three universes because there are normal moves to rotate two corners together such that the total rotation of the corners are 360 degress. So you can always reduce the rotation of more than two corners into a single rotation of a single corner piece.
3. Swapping two edge pieces / corner pieces. It happens that using normal moves one can turn a swap of edge pieces into a swap of two corner pieces. Therefore the two kinds of swapping are essentially equivalent. Also, doing two swappings of edge pieces will again bring you back to the same universe as there are normal moves to do it. Therefore you get two distinct universes by swapping edge pieces or corner pieces.
One should also be aware that these changes are independent of each others. e.g. you cannot undo the effect of a flip of an edge piece by swapping the positions of two pieces, or rotating a corner piece.
So the no. of distinct universes is 2 x 3 x 2 = 12.
You said that you can swap the two edges and the two corners, and so the two universes are the same, but you cannot do the two individually. It isn't possible to swap just 2 corners or just two edges, so you are wrong.
UberCuber You can swap just two corners.
No you can't...
UberCuber is right. That is also what I meant. It isn't possible to swap just 2 corners or just two edges by normal moves. If you attempt to swap just 2 corners by normal moves, you will find that you either end up swapping 2 more other corners or swapping 2 more other edges. So the result is a combination of corner swap AND edge swap; or a combination of corner swap AND another corner swap. The same happens to edge swaps. That's why I said if you disassemble the cube and swap two edges AND swap two corners, then you can solve the cube into its original state, i.e. in the same orbit as the state before the disassemble and swap.
I hate math with the burning passion of quintillion neutron stars going supernova and I still love Numberphile.
I love that it's a physical representation of metatrons cube. Something quite special about the rubiks cube.. aside from how addictive they are !
Hello Numberphile! I've solved a number of Rubik's cube myself. For cubes with labeled centers, like those giveaways by some companies, you will notice that a solved state can have rotated centers. You don't notice this in a standard cube because the center is just a single color with no indication for orientation. But, if you account for this I believe you should get an extra factor of (4^6) combinations as there are 6 centers and 4 rotations possible for each center.
no
How many combinations are there for the new one with the circular dials? My best estimate is approximately a shit-ton.
There are 2 to the power of n squared possible combinations for a minesweeper grid with n grids on each size, or 2 to the power of l * w, where l and w are the length and width respectively.
The video mentions that there are twelve orbits, incidentally the same amount of edges on the cube. Taking off a corner piece, rotating it and placing it on the cube again does not form a new orbit. Only edge pieces will form a new orbit when you flip them since the corner pieces get flipped anyway.
Personal Best
314,159,265,358,979,323,846,264,338,327,950,288,419,716,939,937,510,582,097,494,459,230,781,640,628,620,899,862,803,482,534,211,706,798,214,808,651,328,230,664 Years NO breaks
awesome video.
A = 8 corner positions = 8!
B = 3 orientations per corner = 3^8
C =12 edge positions = 12!
D = 2 orientations per edge = 2^12
E = can't have single corner twisted = 1/3
F = can't have single edge flipped = 1/2
G = can't have two edges swapped = 1/2
A*B*C*D*E*F*G = 43252003274489856000
Also, The center pieces can be pulled off of most cube to reveal a screw that adjusts the tension, however, it is typically only one center, and since rotation doesn't matter, you won't get anywhere
If you could have as many of any color as you want (i.e. 42 green, 12 red), you would have an astounding 1,047,532,535,594,334,222,593,508,922,191,671,036,215,296 combinations.
Crazy, huh?
Another way i like to put the number 43,252,003,274,489,856,000 in to perspective is this. If you were to go through each combination one at a time, one per second, and with the final turn being the solution. I would take about 100 life times (the birth of the universe till now) of the universe to finish. That is (((43,252,003,274,489,856,000/3600)/24)/365.25)/13,700,000,000. Assuming the universe is roughly 13.7 billion years old as of now.
Because those are the number of "orbits" that the cube can be arranged in. (Combinations that are not possible on a traditional cube so you must divide the total combinations by the number of "universes" that aren't possible.
Thanks! I was thinking and wondering the exact same thing. :)
This is pretty amusing.
Thank you. It does help. I'll take a look at that book next time I go to the library (they have it, I checked). I was hoping there would be a more general theorem, which could be extended to any puzzle though. For example why is it possible to swap just two edges on a 4x4 and not on a 3x3 (or any odd/even sizes) and remain in the same universe?