[0:00] So, this is an entirely ridiculously
[0:02] ridiculously big number. You've spoken
[0:03] to me about bigger numbers. I have
[0:05] spoken to you about bigger numbers, but
[0:06] I've never held one in my hand.
[0:09] I thought we might talk about Rubik's
[0:11] Cubes today, but not about this Rubik's
[0:14] Cube. Instead, how about
[0:17] that Rubik's Cube? Oh, that's got that's
[0:19] got more cubes in it. Yeah, that's What
[0:21] do you call the the elements of a
[0:23] So, the opinions are divided. I think
[0:25] what most people nowadays call it is a
[0:27] cubie, but I'm a bit old school and I
[0:29] call it a cubelet.
[0:30] Yeah. And that's got more cubelets. It's
[0:32] got more cubelets. You you can see it's
[0:34] six cubelets wide, while your your your
[0:37] classic is three cubelets wide, right?
[0:41] So, you might call this a six cube. So,
[0:42] the standard
[0:44] three cube has got an awful lot of
[0:46] scrambles, lots of different
[0:47] possibilities. Do you remember how many
[0:48] different scrambles, Brady? I believe 10
[0:51] seconds ago, before we started the
[0:52] filming, you told me it was 43
[0:55] quintillion. 43 quintillion on your the
[0:58] standard Rubik's Cube.
[0:58] >> That's Surely it's not a rounded all
[1:00] zeros. Is there like a more exact number
[1:02] than that?
[1:02] >> more exact, yeah. What's the exact
[1:04] number? Uh I don't remember, I'm afraid.
[1:06] I'll put it on the screen. There it is.
[1:09] And what does that number refer to? So,
[1:11] it's the number of different possible
[1:12] scrambles or different possible states
[1:14] it could be, different ways it can be
[1:16] mixed up, including of course the
[1:17] unscrambled solved state. Is this one of
[1:20] those situations where there are like,
[1:22] you know, reflections and mirrors and
[1:23] the same ones exist multiple times and
[1:25] you have to do that or Yes,
[1:27] that's right. So, we want to make sure
[1:29] when we're counting different states,
[1:31] we're not overcounting by
[1:34] counting um
[1:37] ones which look identical
[1:39] um to each other. In fact, this is going
[1:40] to be more important than with this,
[1:41] because you can see it's got all these
[1:42] center pieces.
[1:44] And if they get jumbled up between
[1:45] themselves, it doesn't matter, right?
[1:47] It's it's still going to be the same
[1:48] state. So, we won't have to worry about
[1:50] that yeah.
[1:51] >> So, if you swapped two of those whites
[1:52] with each other, I wouldn't know.
[1:54] Exactly. So, we can't we should count
[1:55] them as that as being the same the same
[1:57] state. But I thought it might be fun to
[1:58] try and work out how many different
[2:01] states there are for a big cube like
[2:03] this. Okay, so I'm actually having said
[2:06] that, I think it's best to start with a
[2:08] slightly smaller cube. So, I'm actually
[2:09] going to start with this cube here, two
[2:12] cube. And it's going to be actually
[2:13] easier to do the calculation for cubes
[2:16] with even
[2:19] width. The question is, how many
[2:20] different states are there for this? So,
[2:23] this is a 2 by 2 cube. Let's scramble it
[2:25] up a little bit.
[2:26] You can see there are eight
[2:28] cubelets right?
[2:30] And there are eight positions that a
[2:32] cubelet could be in,
[2:33] right? So,
[2:36] you can put any of the
[2:38] cubelets in any of the positions. So,
[2:39] let's just pick a slot,
[2:41] maybe this one here, that position,
[2:44] then there's eight choices of which
[2:47] cubelet to put it in. So, we can write
[2:49] that down, eight. And then once we've
[2:51] got that one,
[2:53] for the next slot, there's seven
[2:54] choices, right? So, it's times seven.
[2:57] And then for the next one, there's six
[2:58] and so on. So, you can see how it's
[3:00] going to go. The number of choices is 8
[3:03] * 7 * 6 * and so on down to 1, which is
[3:06] usually written as 8 factorial with an
[3:09] exclamation mark. That's what that
[3:10] exclamation mark means, okay? So, that's
[3:12] the start, but that's not the whole
[3:14] answer, because once we've decided maybe
[3:18] we put this white, green, and red
[3:21] uh cube cubelets into this slot,
[3:25] it can actually be one of three ways
[3:26] round. So, if you just look at
[3:29] concentrate on that cubelet,
[3:31] if I do that, it's in the same position
[3:34] again, but it's in a different
[3:37] it's been rotated, right? And we can do
[3:39] it again, okay? So, actually there's
[3:42] um
[3:43] for each
[3:44] each of these cubelets in each of these
[3:46] positions, it can be one of three ways
[3:47] round, right? We need to multiply by
[3:51] three for each of these. So, you think
[3:53] it's going to be times three to the
[3:54] eight, right? Because there's eight of
[3:56] them. But actually it's not times three
[3:58] to the eight, because it turns out that
[4:00] once seven of them have been fixed,
[4:04] the the the final one is automatically
[4:06] fixed. You can't do anything more with
[4:07] it. So, it's actually times three to the
[4:09] seven. So, that's the the number of
[4:11] states for the two cube. Well, maybe not
[4:14] quite, because this point we have to
[4:17] rewind and think about
[4:19] every cube's favorite thing, the one
[4:21] cube. And the question is, how many
[4:22] states does the one cube have, right?
[4:25] And I think the
[4:26] what feels like the natural answer is
[4:28] it's got one state, yeah? But of course,
[4:31] I mean you could argue that it's got 24
[4:33] states, because you think, okay, which
[4:35] which face will I put on the top?
[4:38] I'll put the the white face on the top.
[4:41] Um and then which face should I put on
[4:43] the front? So, it could be the orange
[4:44] one or the green one and so
[4:47] There's four choices there.
[4:49] So, you could argue that the one cube
[4:51] has got 24 states, 6 * 4, yeah? But we
[4:55] probably don't want that. We probably
[4:56] want to count them all as being the
[4:57] same, which means we need to Likewise
[5:00] for this, a lot of these states are
[5:01] going to be the same, but just rotations
[5:04] without actually doing any Right, so let
[5:06] me call this number we've got so far C,
[5:09] okay? I'll explain why it's called C in
[5:11] a moment. That's around about 88
[5:13] million. But then to get the number of
[5:14] states for the two cube, we need to
[5:16] divide that by 24, okay? So, the two
[5:19] cube is C divided by 24. The answer is
[5:23] that's about 3.6 million, just for this
[5:25] little cube, okay? Wow. When we think
[5:27] about uh the bigger cubes, so I mean the
[5:31] one I'll do next is the four cube,
[5:32] because it's easier to work with um even
[5:34] sides. With a two cube, all the cubelets
[5:37] are basically equivalent, right? Any can
[5:39] go in the position of any other. That's
[5:41] not the case with here. Um in fact,
[5:43] there's three different types of
[5:45] cubelets here. You've got the corners,
[5:48] and you've got the edges here,
[5:51] and then you've got the centers. You've
[5:53] got three different types. And to work
[5:56] out the number of states for the four
[5:57] cube, what we need to do is work out the
[6:01] number of
[6:02] possible arrangements of the corners and
[6:04] the number of possible arrangements of
[6:05] the edges and the number of possible
[6:07] arrangements of the centers and then
[6:09] multiply them together. That's that's
[6:10] the plan. And I want to start with the
[6:12] two cube, because
[6:14] the corners of the four cube in or
[6:16] indeed the corners of any cube
[6:18] are
[6:19] basically exactly the same argument as
[6:23] for the two cube, because of this is
[6:24] these are corners, right? The first
[6:25] thing for the four cube, we need C
[6:27] again, that same number, right? So, the
[6:29] next thing to think about is the
[6:33] edges. So, for this cube, the sort of
[6:35] overall shape the the the cubic shape
[6:37] has got 12 edges and along each edge
[6:40] there's two cubelets, right? So, we've
[6:42] got 24 edge pieces. So, we've got 24 in
[6:46] total.
[6:47] And you can put one in any of the slots.
[6:49] So, we've got 24 slots. So, for the
[6:51] first slot,
[6:53] we've got 24 choices of cubelets. So,
[6:55] 24.
[6:56] And then for the next slot, we've got 23
[6:58] and so on. You can see how it's going to
[6:59] go. It's going to be 24 factorial.
[7:02] Actually, that's it for the edges,
[7:03] because you might think, well, we also
[7:05] need to consider the fact that one edge
[7:07] can be either way up. But actually it
[7:10] can't
[7:12] in the sense that
[7:13] um once you fit a fix a slot to put this
[7:17] uh this edge piece in, it can only go
[7:19] one way up in it. The mechanics of the
[7:21] puzzle mean that. So, I'm just going to
[7:22] put a dot on
[7:24] this edge.
[7:25] And now I can flip those two
[7:28] edges there. So, don't worry about the
[7:29] other pieces, just look at those two
[7:31] edge pieces.
[7:37] So, I flipped them over, yeah? But you
[7:40] can see in flipping them over, I've also
[7:41] moved the one that was here to the one
[7:43] that was there. The one with the dot was
[7:44] there before, now it's there.
[7:46] Um and it's always going to be that way.
[7:48] So, once you've put the cube the cubelet
[7:51] in the slot, that's it. So, that is the
[7:54] that is the arrangement for the edges,
[7:56] okay? So, the number of edge
[7:57] arrangements is just that, it's it's
[7:59] just 24 uh
[8:02] which actually is a pretty big number.
[8:04] This is around uh six
[8:07] times 10 to the 23.
[8:10] So, that's 620 sextillion. The next
[8:12] thing is the centers. Again, it sort of
[8:16] starts off as the same as the edges,
[8:18] because we've got six faces of the cube.
[8:21] Inside each face, we've got four center
[8:22] pieces. There's 24 center pieces all
[8:24] together. So, I'm going to call this I'm
[8:26] going to call this number K, which is
[8:28] because C was already taken. The number
[8:31] of we can just pick a a slot here and
[8:34] then the number of choices of which
[8:35] center piece to put in it is 24.
[8:39] And then pick another one and the next
[8:40] one is 23 choices. So, it's going to be
[8:42] 24 factorial again. But this time we do
[8:45] have to consider something else, which
[8:46] is that these four
[8:49] cubelets are identical to each other.
[8:52] And if they mix around between
[8:54] themselves, we don't really care, right?
[8:57] We don't consider it as a different
[8:58] scramble if you know, if I was to swap
[9:01] those two or something. It's all it's
[9:02] all the same. So, that means
[9:05] every possible scramble
[9:07] is going to have basically
[9:10] um different versions of it where the
[9:11] orange centers are swapped around. So,
[9:15] that's an overcount at the moment. So,
[9:16] we have to correct that.
[9:18] How many how many um times is each one
[9:20] counted? Well,
[9:22] how many arrangements are of these
[9:25] four centers are there? Well, there's
[9:27] four choices
[9:29] for which one to put there times 3 times
[9:31] 2
[9:32] times 1, so 4 factorial, which is 24,
[9:35] okay? Yeah.
[9:36] So, we need to divide by So, this to get
[9:40] the number of total arrangements of all
[9:42] the centers in the cube, we need to
[9:43] divide by 24. Well,
[9:45] that's for the orange centers, okay? And
[9:47] then we have to do the same thing for
[9:49] the green centers and for the yellow
[9:51] centers. So, we have to divide by 24 six
[9:53] times. So, we divide by by
[9:55] to the six. And that gives us the number
[9:58] 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