Entirely Ridiculously Big Numbers - Numberphile

[00:00] So, this is an entirely ridiculously

[00:02] ridiculously big number. You've spoken

[00:03] to me about bigger numbers. I have

[00:05] spoken to you about bigger numbers, but

[00:06] I've never held one in my hand.

[00:09] I thought we might talk about Rubik's

[00:11] Cubes today, but not about this Rubik's

[00:14] Cube. Instead, how about

[00:17] that Rubik's Cube? Oh, that's got that's

[00:19] got more cubes in it. Yeah, that's What

[00:21] do you call the the elements of a

[00:23] So, the opinions are divided. I think

[00:25] what most people nowadays call it is a

[00:27] cubie, but I'm a bit old school and I

[00:29] call it a cubelet.

[00:30] Yeah. And that's got more cubelets. It's

[00:32] got more cubelets. You you can see it's

[00:34] six cubelets wide, while your your your

[00:37] classic is three cubelets wide, right?

[00:41] So, you might call this a six cube. So,

[00:42] the standard

[00:44] three cube has got an awful lot of

[00:46] scrambles, lots of different

[00:47] possibilities. Do you remember how many

[00:48] different scrambles, Brady? I believe 10

[00:51] seconds ago, before we started the

[00:52] filming, you told me it was 43

[00:55] quintillion. 43 quintillion on your the

[00:58] standard Rubik's Cube.

[00:58] >> That's Surely it's not a rounded all

[01:00] zeros. Is there like a more exact number

[01:02] than that?

[01:02] >> more exact, yeah. What's the exact

[01:04] number? Uh I don't remember, I'm afraid.

[01:06] I'll put it on the screen. There it is.

[01:09] And what does that number refer to? So,

[01:11] it's the number of different possible

[01:12] scrambles or different possible states

[01:14] it could be, different ways it can be

[01:16] mixed up, including of course the

[01:17] unscrambled solved state. Is this one of

[01:20] those situations where there are like,

[01:22] you know, reflections and mirrors and

[01:23] the same ones exist multiple times and

[01:25] you have to do that or Yes,

[01:27] that's right. So, we want to make sure

[01:29] when we're counting different states,

[01:31] we're not overcounting by

[01:34] counting um

[01:37] ones which look identical

[01:39] um to each other. In fact, this is going

[01:40] to be more important than with this,

[01:41] because you can see it's got all these

[01:42] center pieces.

[01:44] And if they get jumbled up between

[01:45] themselves, it doesn't matter, right?

[01:47] It's it's still going to be the same

[01:48] state. So, we won't have to worry about

[01:50] that yeah.

[01:51] >> So, if you swapped two of those whites

[01:52] with each other, I wouldn't know.

[01:54] Exactly. So, we can't we should count

[01:55] them as that as being the same the same

[01:57] state. But I thought it might be fun to

[01:58] try and work out how many different

[02:01] states there are for a big cube like

[02:03] this. Okay, so I'm actually having said

[02:06] that, I think it's best to start with a

[02:08] slightly smaller cube. So, I'm actually

[02:09] going to start with this cube here, two

[02:12] cube. And it's going to be actually

[02:13] easier to do the calculation for cubes

[02:16] with even

[02:19] width. The question is, how many

[02:20] different states are there for this? So,

[02:23] this is a 2 by 2 cube. Let's scramble it

[02:25] up a little bit.

[02:26] You can see there are eight

[02:28] cubelets right?

[02:30] And there are eight positions that a

[02:32] cubelet could be in,

[02:33] right? So,

[02:36] you can put any of the

[02:38] cubelets in any of the positions. So,

[02:39] let's just pick a slot,

[02:41] maybe this one here, that position,

[02:44] then there's eight choices of which

[02:47] cubelet to put it in. So, we can write

[02:49] that down, eight. And then once we've

[02:51] got that one,

[02:53] for the next slot, there's seven

[02:54] choices, right? So, it's times seven.

[02:57] And then for the next one, there's six

[02:58] and so on. So, you can see how it's

[03:00] going to go. The number of choices is 8

[03:03] * 7 * 6 * and so on down to 1, which is

[03:06] usually written as 8 factorial with an

[03:09] exclamation mark. That's what that

[03:10] exclamation mark means, okay? So, that's

[03:12] the start, but that's not the whole

[03:14] answer, because once we've decided maybe

[03:18] we put this white, green, and red

[03:21] uh cube cubelets into this slot,

[03:25] it can actually be one of three ways

[03:26] round. So, if you just look at

[03:29] concentrate on that cubelet,

[03:31] if I do that, it's in the same position

[03:34] again, but it's in a different

[03:37] it's been rotated, right? And we can do

[03:39] it again, okay? So, actually there's

[03:42] um

[03:43] for each

[03:44] each of these cubelets in each of these

[03:46] positions, it can be one of three ways

[03:47] round, right? We need to multiply by

[03:51] three for each of these. So, you think

[03:53] it's going to be times three to the

[03:54] eight, right? Because there's eight of

[03:56] them. But actually it's not times three

[03:58] to the eight, because it turns out that

[04:00] once seven of them have been fixed,

[04:04] the the the final one is automatically

[04:06] fixed. You can't do anything more with

[04:07] it. So, it's actually times three to the

[04:09] seven. So, that's the the number of

[04:11] states for the two cube. Well, maybe not

[04:14] quite, because this point we have to

[04:17] rewind and think about

[04:19] every cube's favorite thing, the one

[04:21] cube. And the question is, how many

[04:22] states does the one cube have, right?

[04:25] And I think the

[04:26] what feels like the natural answer is

[04:28] it's got one state, yeah? But of course,

[04:31] I mean you could argue that it's got 24

[04:33] states, because you think, okay, which

[04:35] which face will I put on the top?

[04:38] I'll put the the white face on the top.

[04:41] Um and then which face should I put on

[04:43] the front? So, it could be the orange

[04:44] one or the green one and so

[04:47] There's four choices there.

[04:49] So, you could argue that the one cube

[04:51] has got 24 states, 6 * 4, yeah? But we

[04:55] probably don't want that. We probably

[04:56] want to count them all as being the

[04:57] same, which means we need to Likewise

[05:00] for this, a lot of these states are

[05:01] going to be the same, but just rotations

[05:04] without actually doing any Right, so let

[05:06] me call this number we've got so far C,

[05:09] okay? I'll explain why it's called C in

[05:11] a moment. That's around about 88

[05:13] million. But then to get the number of

[05:14] states for the two cube, we need to

[05:16] divide that by 24, okay? So, the two

[05:19] cube is C divided by 24. The answer is

[05:23] that's about 3.6 million, just for this

[05:25] little cube, okay? Wow. When we think

[05:27] about uh the bigger cubes, so I mean the

[05:31] one I'll do next is the four cube,

[05:32] because it's easier to work with um even

[05:34] sides. With a two cube, all the cubelets

[05:37] are basically equivalent, right? Any can

[05:39] go in the position of any other. That's

[05:41] not the case with here. Um in fact,

[05:43] there's three different types of

[05:45] cubelets here. You've got the corners,

[05:48] and you've got the edges here,

[05:51] and then you've got the centers. You've

[05:53] got three different types. And to work

[05:56] out the number of states for the four

[05:57] cube, what we need to do is work out the

[06:01] number of

[06:02] possible arrangements of the corners and

[06:04] the number of possible arrangements of

[06:05] the edges and the number of possible

[06:07] arrangements of the centers and then

[06:09] multiply them together. That's that's

[06:10] the plan. And I want to start with the

[06:12] two cube, because

[06:14] the corners of the four cube in or

[06:16] indeed the corners of any cube

[06:18] are

[06:19] basically exactly the same argument as

[06:23] for the two cube, because of this is

[06:24] these are corners, right? The first

[06:25] thing for the four cube, we need C

[06:27] again, that same number, right? So, the

[06:29] next thing to think about is the

[06:33] edges. So, for this cube, the sort of

[06:35] overall shape the the the cubic shape

[06:37] has got 12 edges and along each edge

[06:40] there's two cubelets, right? So, we've

[06:42] got 24 edge pieces. So, we've got 24 in

[06:46] total.

[06:47] And you can put one in any of the slots.

[06:49] So, we've got 24 slots. So, for the

[06:51] first slot,

[06:53] we've got 24 choices of cubelets. So,

[06:55] 24.

[06:56] And then for the next slot, we've got 23

[06:58] and so on. You can see how it's going to

[06:59] go. It's going to be 24 factorial.

[07:02] Actually, that's it for the edges,

[07:03] because you might think, well, we also

[07:05] need to consider the fact that one edge

[07:07] can be either way up. But actually it

[07:10] can't

[07:12] in the sense that

[07:13] um once you fit a fix a slot to put this

[07:17] uh this edge piece in, it can only go

[07:19] one way up in it. The mechanics of the

[07:21] puzzle mean that. So, I'm just going to

[07:22] put a dot on

[07:24] this edge.

[07:25] And now I can flip those two

[07:28] edges there. So, don't worry about the

[07:29] other pieces, just look at those two

[07:31] edge pieces.

[07:37] So, I flipped them over, yeah? But you

[07:40] can see in flipping them over, I've also

[07:41] moved the one that was here to the one

[07:43] that was there. The one with the dot was

[07:44] there before, now it's there.

[07:46] Um and it's always going to be that way.

[07:48] So, once you've put the cube the cubelet

[07:51] in the slot, that's it. So, that is the

[07:54] that is the arrangement for the edges,

[07:56] okay? So, the number of edge

[07:57] arrangements is just that, it's it's

[07:59] just 24 uh

[08:02] which actually is a pretty big number.

[08:04] This is around uh six

[08:07] times 10 to the 23.

[08:10] So, that's 620 sextillion. The next

[08:12] thing is the centers. Again, it sort of

[08:16] starts off as the same as the edges,

[08:18] because we've got six faces of the cube.

[08:21] Inside each face, we've got four center

[08:22] pieces. There's 24 center pieces all

[08:24] together. So, I'm going to call this I'm

[08:26] going to call this number K, which is

[08:28] because C was already taken. The number

[08:31] of we can just pick a a slot here and

[08:34] then the number of choices of which

[08:35] center piece to put in it is 24.

[08:39] And then pick another one and the next

[08:40] one is 23 choices. So, it's going to be

[08:42] 24 factorial again. But this time we do

[08:45] have to consider something else, which

[08:46] is that these four

[08:49] cubelets are identical to each other.

[08:52] And if they mix around between

[08:54] themselves, we don't really care, right?

[08:57] We don't consider it as a different

[08:58] scramble if you know, if I was to swap

[09:01] those two or something. It's all it's

[09:02] all the same. So, that means

[09:05] every possible scramble

[09:07] is going to have basically

[09:10] um different versions of it where the

[09:11] orange centers are swapped around. So,

[09:15] that's an overcount at the moment. So,

[09:16] we have to correct that.

[09:18] How many how many um times is each one

[09:20] counted? Well,

[09:22] how many arrangements are of these

[09:25] four centers are there? Well, there's

[09:27] four choices

[09:29] for which one to put there times 3 times

[09:31] 2

[09:32] times 1, so 4 factorial, which is 24,

[09:35] okay? Yeah.

[09:36] So, we need to divide by So, this to get

[09:40] the number of total arrangements of all

[09:42] the centers in the cube, we need to

[09:43] divide by 24. Well,

[09:45] that's for the orange centers, okay? And

[09:47] then we have to do the same thing for

[09:49] the green centers and for the yellow

[09:51] centers. So, we have to divide by 24 six

[09:53] times. So, we divide by by

[09:55] to the six. And that gives us the number

[09:58] of arrangements of the centers. This

[10:00] comes out to be around 3 * 10 to the 15,

[10:05] which is 3 quadrillion if you like those

[10:07] sorts of words, okay? Okay, so now we

[10:08] can all to work out the number total

[10:11] number of possible states of the 4 cube,

[10:12] we just have to multiply those together,

[10:15] okay? So, we've got the number of

[10:16] possible arrangements of the corners, C,

[10:19] times the number of possible

[10:20] arrangements of the edges, E, times the

[10:22] number of possible arrangements of the

[10:23] centers K.

[10:25] And then we have to divide by 24 for the

[10:27] same reason as previously, right? Okay,

[10:29] so that's the answer and that gives us a

[10:30] total answer here of uh roundabout 7 *

[10:36] 10 to the 45. So, that's the number of

[10:39] possible states of a 4 cube. How does

[10:41] that differ from our 3 cube, that that

[10:44] number that I was supposed to remember?

[10:46] Septillion.

[10:47] 43 quintillion. Quintillion, that's it.

[10:49] 43 quintillion. How does it differ? Is

[10:51] it What sort of magnitude is that? For

[10:53] the standard cube, it's 4.3 * 10 to the

[10:56] 19. And now we've jumped up to 7 * 10 to

[10:59] the 45. So, it's a very substantial

[11:02] increase, okay? Really significant

[11:04] increase. Yes.

[11:06] Shall we have another substantial

[11:07] increase? Always. Okay, this is now a 6

[11:11] cube. We can get started because the

[11:12] corners

[11:13] are exactly the same as previously. So,

[11:15] we start with the same number C. So, now

[11:17] we think about the edges. So, there's

[11:19] two fundamentally different kinds of

[11:21] edges on this cube, right? You've got

[11:22] the ones here which are uh

[11:24] blue and yellow and you've got the ones

[11:26] which are yellow and green. They're like

[11:28] They're in two sort of different types

[11:29] of corridors, aren't they? They're in

[11:30] two corridors and you can never switch

[11:33] between the corridors. So, once you're

[11:34] in one corridor, you're always in that

[11:36] corridor. The central edges are

[11:38] different from sometimes they're called

[11:39] wing edges, right?

[11:41] If you're in one of the wing edges, you

[11:42] can get to anywhere else in any of the

[11:44] other wing edges.

[11:46] And if you're in one of the central

[11:47] edges, you can get to anywhere else in

[11:48] any of the central edges, okay? What

[11:50] that Can the one on the left Can can you

[11:52] If you're in a central edge, like those

[11:54] two, can you change between left and

[11:56] right? You can. Yes, you can.

[11:57] >> So, you're not stuck in your corridor,

[11:59] you're stuck in your brand of corridor.

[12:00] >> You're You're stuck in your brand of

[12:01] corridor, yeah, exactly. That's right.

[12:03] That's right. Um And what What that

[12:05] means is that if you just look at the

[12:07] central edges, the um the argument is

[12:10] exactly the same as the edges for the 4

[12:13] for the 4 cube, right? Because the the

[12:15] the edges here are just the central

[12:17] edges, right? So, the the number of

[12:19] possibilities for the central edge

[12:21] is that number E as we had before,

[12:24] right? So, times E. But then the number

[12:26] of possibilities for the wing edges is

[12:29] the same again, so it's times E again.

[12:31] So, we times by E squared. Okay, so now

[12:33] we've got to look about the centers.

[12:35] They call it centers. It means

[12:37] everything which isn't a corner or an

[12:38] edge. So, actually quite a lot quite a

[12:40] lot of cubes cubelets in here. There's

[12:42] um there's sort of 16. What are the

[12:45] What's the sort of corridor structure?

[12:46] How can you get Which can you get

[12:48] between and which can't you? Yeah, how

[12:49] are they constrained, yeah? How How are

[12:51] they constrained? Yeah. If you look at

[12:52] this white

[12:54] white cubelets, right?

[12:56] The question is where can it get to?

[12:58] So, it can

[13:00] on this face,

[13:02] there's four places it can sort of

[13:03] obviously get to, right?

[13:06] Those four. So, it can be just above the

[13:09] bottom left corner, just to the right of

[13:11] that corner,

[13:12] just underneath that corner, right? So,

[13:14] there's four four places it can get to.

[13:18] Are there any others? And actually the

[13:19] answer is no. So, you might think you

[13:21] would be able to get it get the white

[13:23] cube into that position there, but

[13:26] actually you can't. And likewise, you

[13:27] can't get it down there. That's sort of

[13:29] maybe more believable.

[13:31] And you can't get it there either. Just

[13:33] use a different face. So, actually the

[13:35] sort of key thing here

[13:37] is

[13:39] if you look at this blue square of four

[13:42] cubelets, those are the four

[13:44] fundamentally different kinds of

[13:46] centers. So, there's four kinds of

[13:48] centers.

[13:49] And then for each kind of center,

[13:52] the argument goes exactly as the centers

[13:56] for the 4 cube, okay? So, the number of

[13:58] ways of arranging

[14:00] those kinds of centers is that number K

[14:02] we had. And then the number of ways of

[14:04] arranging those kinds of centers is also

[14:06] K, and those kinds of centers are also

[14:08] K, and those kinds of centers are also

[14:10] K. So, overall we get K to the 4. Then

[14:12] of course, we do divide by 24,

[14:14] which matters less and less as these

[14:16] numbers get bigger and bigger. It's a

[14:18] bit of a formality at this point. But

[14:19] when you work that out, we're up to 1.6

[14:23] * 10 to the 116. We're like beyond atoms

[14:28] in the universe. We're beyond atoms in

[14:29] the universe. Yeah, there's there's more

[14:31] possible scrambles of this puzzle than

[14:33] there are atoms in the universe.

[14:36] This is my biggest cube.

[14:38] Um it's a recent and rash purchase. Um

[14:42] this is a 10 cube.

[14:44] And I haven't actually yet dared

[14:46] properly scramble it cuz I dread to

[14:47] think how long it would take me to

[14:49] solve. Um at some point I will. Um So,

[14:52] but I think it might be fun to work out

[14:53] how many I mean, has it gotten to a

[14:54] point now where for even numbered cubes,

[14:57] we can

[14:58] create an algorithm here or Yeah, we

[15:00] can. We can. So, the way to think about

[15:03] it is let's call it a 2n cube, right?

[15:06] So, um I'll do this as an example. So, a

[15:08] 2 * 5 cube, so that's my 10 cube, right?

[15:11] We're going to let n be sort of half the

[15:13] size. You could let n equal 10, but then

[15:16] the algebra comes out harder. You've got

[15:17] C, that's the number of corners, that

[15:18] never changes. Okay, now we think about

[15:20] the edges. How many different edges have

[15:22] we got? Now, let's put some different

[15:24] colors on.

[15:32] Okay, but the point is how many

[15:33] different types are there? Well, there's

[15:35] the central ones,

[15:37] then the next ones out, then the next

[15:38] ones out, and then the wings, okay? So,

[15:40] we've got four,

[15:42] right?

[15:43] 1 2 3 4 fundamentally fundamentally

[15:46] different types of edges.

[15:48] Um and in general,

[15:51] so here in this cube, n equals 5. So, in

[15:54] general, you're going to have n minus 1.

[15:56] Yeah? And the reason it's n minus 1 is

[15:58] because n is half the width of the cube,

[16:01] and we just don't want the the outermost

[16:03] ones, so so n minus 1, yeah? Okay? So,

[16:07] that means the number of possible

[16:08] arrangements of the edges is E to the n

[16:10] minus 1. And now we've got to think

[16:13] about the centers. Yeah, just give me a

[16:14] second.

[16:16] Sorry, it's just a bit slow, this thing.

[16:22] So, now if we think about the centers,

[16:24] this white square of 16 cubelets, that's

[16:28] representative of the different

[16:29] corridors. So, each of those is

[16:32] different to each of the others. You can

[16:34] never get from any of one of those to

[16:35] any of the others. And once we've chosen

[16:38] one of those types, so there's 16 types

[16:42] here,

[16:43] um or in general, that's n minus 1

[16:46] squared, half the width, so that's n,

[16:49] then we take away the edge, so that's

[16:52] that width there is n minus

[16:54] 1, then we square it, okay? So, there's

[16:56] n minus 1

[16:57] kinds of centers. That kind, by the way,

[17:00] is my justification for using K.

[17:02] Um pretty weak, I think. And then the

[17:04] argument for each one is exactly as the

[17:05] previous day, that's the number of

[17:06] possible combinations. So, we need to

[17:08] then multiply here by K

[17:11] to the n minus 1 squared, right?

[17:15] And K, remember, is 3 * 10 to the 15.

[17:18] And that's why you can sort of see here

[17:20] why these numbers are getting so big so

[17:21] quickly cuz it's exponential in n, but

[17:24] it's not actually just exponential in n,

[17:26] it's exponential in n squared. And

[17:28] that's why why it gets so big. Of

[17:29] course, we have to go through the

[17:31] formality of dividing by 24. Of course.

[17:33] Of course, of course. Um Okay.

[17:35] >> that does knock one off the power,

[17:36] doesn't it? It It knocks one off the

[17:37] power, yeah. It knocks one off the

[17:39] power. Um so, for 10, if you plug in um

[17:42] n equals 5 into this, well, you get C

[17:46] times E to the 4

[17:48] times K to the 4 squared, so 16, divided

[17:52] by 24. Work it out, you get to 10 to the

[17:54] 349 or thereabouts, okay? Which is a

[17:57] which is really ridiculously

[17:59] ridiculously big number. I mean, that's

[18:01] the sort of place where,

[18:03] you know, if you took every atom in the

[18:05] universe and replaced it with a whole

[18:07] universe,

[18:08] and then repeated that

[18:11] four times,

[18:13] and then counted the number of atoms in

[18:14] that whole thing,

[18:16] this has still got more scrambles than

[18:18] that, right? Okay, so it's it's an

[18:19] entirely ridiculously ridiculously big

[18:21] number. You've spoken to me about bigger

[18:23] numbers. I have spoken to you about

[18:24] bigger numbers, but I've never held one

[18:25] in my hand.

[18:28] You may have noticed we've only dealt

[18:29] with cubes with an even number of pieces

[18:32] along the edges. If you want to find out

[18:34] how it gets a bit more technical and

[18:35] mathematical with an odd number,

[18:38] like that,

[18:39] go and have a look on Numberphile 2.

[18:41] We've got an extension of the video

[18:43] there. We go into way more detail, but

[18:45] before you do that,

[18:49] go and order one of these. Order a copy

[18:51] of Huge Numbers. It's Richard's new

[18:53] book. It's available now to order or

[18:55] pre-order, depending on where you are.

[18:57] The cover might look different,

[18:58] depending on where you are, but it's a

[18:59] fantastic read. It deals with really big

[19:02] numbers, like you've been seeing just

[19:03] now.

[19:05] I'll also put some links below to other

[19:07] things that may interest you, previous

[19:09] videos we've done about Rubik's Cubes,

[19:10] previous videos we've done about big

[19:12] numbers, and of course, the link to that

[19:14] extra video,

[19:16] and the order link for Richard's book.

[19:18] Go and check it out, people. Thanks for

[19:19] your time, and thanks for watching.

[19:22] puzzle can be solved in 20 or 15