Entirely Ridiculously Big Numbers - Numberphile

0:00 So, this is an entirely ridiculously ridiculously big number.

0:03 You've spoken to me about bigger numbers.

0:04 I have spoken to you about bigger numbers, but I've never held one in my hand.

0:09 I thought we might talk about Rubik's Cubes today,

0:12 but not about this Rubik's Cube.

0:15 Instead, how about that Rubik's Cube?

0:18 Oh, that's got that's got more cubes in it.

0:20 Yeah, that's What do you call the the elements of a So,

0:23 the opinions are divided.

0:25 I think what most people nowadays call it is a cubie,

0:27 but I'm a bit old school and I call it a cubelet.

0:30 Yeah.

0:31 And that's got more cubelets.

0:32 It's got more cubelets.

0:33 You you can see it's six cubelets wide,

0:36 while your your your classic is three cubelets wide, right?

0:41 So, you might call this a six cube.

0:42 So, the standard three cube has got an awful lot of scrambles,

0:46 lots of different possibilities.

0:47 Do you remember how many different scrambles, Brady?

0:50 I believe 10 seconds ago, before we started the filming,

0:53 you told me it was 43 quintillion.

0:56 43 quintillion on your the standard Rubik's Cube.

0:59 That's Surely it's not a rounded all zeros.

1:01 Is there like a more exact number than that?

1:02 more exact, yeah.

1:03 What's the exact number?

1:04 Uh I don't remember, I'm afraid.

1:06 I'll put it on the screen.

1:08 There it is.

1:09 And what does that number refer to?

1:11 So, it's the number of different possible

1:12 scrambles or different possible states it could be,

1:15 different ways it can be mixed up,

1:16 including of course the unscrambled solved state.

1:20 Is this one of those situations where there are like, you know,

1:22 reflections and mirrors and the same ones exist multiple times

1:25 and you have to do that or Yes, that's right.

1:28 So, we want to make sure when we're counting different states,

1:31 we're not overcounting by counting um ones

1:37 which look identical um to each other.

1:39 In fact, this is going to be more important than

1:41 with this, because you can see it's got all these center pieces.

1:44 And if they get jumbled up between themselves, it doesn't matter, right?

1:47 It's it's still going to be the same state.

1:49 So, we won't have to worry about that, yeah.

1:51 So, if you swapped two of those whites with each other, I wouldn't know.

1:54 Exactly.

1:54 So, we can't we should count them as that as being the same the same state.

1:57 But I thought it might be fun to try and work out

2:00 how many different states there are for a big cube like this.

2:04 Okay, so I'm actually having said that, I think

2:06 it's best to start with a slightly smaller cube.

2:09 So, I'm actually going to start with this cube here, two cube.

2:12 And it's going to be actually easier

2:14 to do the calculation for cubes with even width.

2:19 The question is, how many different states are there for this?

2:23 So, this is a 2 by 2 cube.

2:24 Let's scramble it up a little bit.

2:26 You can see there are eight cubelets, right?

2:30 And there are eight positions that a cubelet could be in, right?

2:34 So, you can put any of the cubelets in any of the positions.

2:39 So, let's just pick a slot, maybe this one here, that position,

2:44 then there's eight choices of which cubelet to put it in.

2:47 So, we can write that down, eight.

2:50 And then once we've got that one,

2:53 for the next slot, there's seven choices, right?

2:55 So, it's times seven.

2:57 And then for the next one, there's six and so on.

2:59 So, you can see how it's going to go.

3:01 The number of choices is 8* 7* 6* and so on down to 1,

3:05 which is usually written as 8 factorial with an exclamation mark.

3:09 That's what that exclamation mark means, okay?

3:12 So, that's the start, but that's not the whole answer,

3:15 because once we've decided maybe we put this white,

3:19 green, and red uh cube cubelets into this slot,

3:25 it can actually be one of three ways round.

3:28 So, if you just look at concentrate on that cubelet,

3:31 if I do that, it's in the same position again,

3:35 but it's in a different it's been rotated, right?

3:39 And we can do it again, okay?

3:40 So, actually there's um for each each

3:44 of these cubelets in each of these positions,

3:46 it can be one of three ways round, right?

3:48 We need to multiply by three for each of these.

3:52 So, you think it's going to be times three to the eight, right?

3:55 Because there's eight of them.

3:56 But actually it's not times three to the eight,

3:59 because it turns out that once seven of them have been fixed,

4:04 the the the final one is automatically fixed.

4:06 You can't do anything more with it.

4:08 So, it's actually times three to the seven.

4:10 So, that's the the number of states for the two cube.

4:13 Well, maybe not quite, because this point we have to rewind and think

4:18 about every cube's favorite thing, the one cube.

4:22 And the question is, how many states does the one cube have, right?

4:25 And I think the what feels like the natural answer is it's got one state, yeah?

4:30 But of course, I mean you could argue that it's got 24 states,

4:33 because you think, okay, which 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 the front?

4:43 So, it could be the orange one

4:45 or the green one and so There's four choices there.

4:49 So, you could argue that the one cube has got 24 states, 6* 4, yeah?

4:54 But we probably don't want that.

4:56 We probably want to count them all as being the same,

4:58 which means we need to Likewise for this, a lot

5:01 of these states are going to be the same,

5:02 but just rotations without actually doing any Right,

5:05 so let me call this number we've got so far C, okay?

5:10 I'll explain why it's called C in a moment.

5:12 That's around about 88 million.

5:13 But then to get the number of states for the two cube,

5:16 we need to divide that by 24, okay?

5:18 So, the two cube is C divided by 24.

5:22 The answer is that's about 3.6 million, just for this little cube, okay?

5:26 Wow.

5:27 When we think about uh the bigger cubes,

5:30 so I mean the one I'll do next is the four cube,

5:32 because it's easier to work with um even sides.

5:35 With a two cube, all the cubelets are basically equivalent, right?

5:39 Any can go in the position of any other.

5:41 That's not the case with here.

5:42 Um in fact, there's three different types of cubelets here.

5:46 You've got the corners, and you've got the edges here,

5:51 and then you've got the centers.

5:53 You've got three different types.

5:55 And to work out the number of states for the four cube,

5:59 what we need to do is work out the number

6:01 of possible arrangements of the corners and the number

6:04 of possible arrangements of the edges and the number

6:07 of possible arrangements of the centers and then multiply them together.

6:10 That's that's the plan.

6:11 And I want to start with the two cube,

6:12 because the corners of the four cube in or indeed the corners

6:17 of any cube are basically exactly the same argument as for the two cube,

6:23 because of this is these are corners, right?

6:25 The first thing for the four cube, we need C again, that same number, right?

6:29 So, the next thing to think about is the edges.

6:33 So, for this cube, the sort of overall shape the the the cubic

6:37 shape has got 12 edges and along each edge there's two cubelets, right?

6:42 So, we've got 24 edge pieces.

6:45 So, we've got 24 in total.

6:47 And you can put one in any of the slots.

6:49 So, we've got 24 slots.

6:51 So, for the first slot, we've got 24 choices of cubelets.

6:55 So, 24.

6:56 And then for the next slot, we've got 23 and so on.

6:59 You can see how it's going to go.

7:00 It's going to be 24 factorial.

7:02 Actually, that's it for the edges, because you might think, well,

7:05 we also need to consider the fact that one edge can be either way up.

7:09 But actually it can't in the sense that um once you fit a fix a slot

7:16 to put this uh this edge piece in, it can only go one way up in it.

7:20 The mechanics of the puzzle mean that.

7:22 So, I'm just going to put a dot on this edge.

7:25 And now I can flip those two edges there.

7:29 So, don't worry about the other pieces, just look at those two edge pieces.

7:37 So, I flipped them over, yeah?

7:39 But you can see in flipping them over,

7:41 I've also moved the one that was here to the one that was there.

7:43 The one with the dot was 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 in the slot, that's it.

7:53 So, that is the that is the arrangement for the edges, okay?

7:56 So, the number of edge arrangements is just that, it's

7:59 it's just 24 uh which actually is a pretty big number.

8:04 This is around uh six times 10 to the 23.

8:10 So, that's 620 sextillion.

8:12 The next thing is the centers.

8:15 Again, it sort of 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 pieces.

8:23 There's 24 center pieces all together.

8:25 So, I'm going to call this I'm going to call this number K,

8:28 which is because C was already taken.

8:30 The number of we can just pick a a slot here and then

8:34 the number of choices of which center piece to put in it is 24.

8:39 And then pick another one and the next one is 23 choices.

8:41 So, it's going to be 24 factorial again.

8:44 But this time we do have to consider something else,

8:46 which is that these four cubelets are identical to each other.

8:52 And if they mix around between themselves, we don't really care, right?

8:57 We don't consider it as a different scramble if you know,

9:00 if I was to swap those two or something.

9:02 It's all it's all the same.

9:03 So, that means every possible scramble is going to have basically

9:10 um different versions of it where the orange centers are swapped around.

9:15 So, that's an overcount at the moment.

9:16 So, we have to correct that.

9:18 How many how many um times is each one counted?

9:21 Well, how many arrangements are of these four centers are there?

9:26 Well, there's four choices for which one to put there times 3 times 2 times 1,

9:32 so 4 factorial, which is 24, okay?

9:35 Yeah.

9:36 So, we need to divide by So,

9:39 this to get the number of total arrangements of all the centers in the cube,

9:43 we need to divide by 24.

9:45 Well, that's for the orange centers, okay?

9:47 And then we have to do the same thing

9:49 for the green centers and for the yellow centers.

9:51 So, we have to divide by 24 six times.

9:54 So, we divide by by to the six.

9:56 And that gives us the number of arrangements of the centers.

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

10:05 which is 3 quadrillion if you like those sorts of words, okay?

10:08 Okay, so now we can all to work out

10:10 the number total number of possible states of the 4 cube,

10:12 we just have to multiply those together, okay?

10:15 So, we've got the number of possible arrangements of the corners, C,

10:19 times the number of possible arrangements of the edges, E,

10:22 times the number of possible arrangements of the centers, K.

10:25 And then we have to divide by 24 for the same reason as previously, right?

10:29 Okay, so that's the answer and that gives us a total

10:31 answer here of uh roundabout 7* 10 to the 45.

10:37 So, that's the number of possible states of a 4 cube.

10:41 How does that differ from our 3 cube,

10:43 that that number that I was supposed to remember?

10:46 [laughter] Septillion.

10:47 43 quintillion.

10:48 Quintillion, that's it.

10:49 43 quintillion.

10:50 How does it differ?

10:51 Is it What sort of magnitude is that?

10:53 For the standard cube, it's 4.3* 10 to the 19.

10:57 And now we've jumped up to 7* 10 to the 45.

11:01 So, it's a very substantial increase, okay?

11:03 Really significant increase.

11:05 Yes.

11:06 Shall we have another substantial increase?

11:08 Always.

11:09 Okay, this is now a 6 cube.

11:11 We can get started because the corners are exactly the same as previously.

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

11:16 So, now we think about the edges.

11:19 So, there's two fundamentally different kinds of edges on this cube, right?

11:22 You've got the ones here which are uh blue and yellow

11:25 and you've got the ones which are yellow and green.

11:27 They're like They're in two sort of different types of corridors, aren't they?

11:30 They're in two corridors and you can never switch between the corridors.

11:34 So, once you're in one corridor, you're always in that corridor.

11:36 The central edges are different from sometimes they're called wing edges, right?

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

11:42 you can get to anywhere else in any of the other wing edges.

11:46 And if you're in one of the central edges,

11:47 you can get to anywhere else in any of the central edges, okay?

11:50 What that Can the one on the left Can can you If you're in a central edge,

11:53 like those two, can you change between left and right?

11:56 You can.

11:56 Yes, you can.

11:57 So, you're not stuck in your corridor, you're stuck in your brand of corridor.

12:00 You're You're stuck in your brand of corridor, yeah, exactly.

12:02 That's right.

12:03 That's right.

12:04 Um And what What that means is that if you just look at the central edges,

12:08 the um the argument is exactly the same

12:11 as the edges for the 4 for the 4 cube, right?

12:14 Because the the the edges here are just the central edges, right?

12:18 So, the the number of possibilities for the central

12:20 edge is that number E as we had before, right?

12:24 So, times E.

12:25 But then the number of possibilities for the wing edges is the same again,

12:29 so it's times E again.

12:31 So, we times by E squared.

12:33 Okay, so now we've got to look about the centers.

12:35 They call it centers.

12:37 It means everything [clears throat] which isn't a corner or an edge.

12:39 So, actually quite a lot quite a lot of cubes cubelets in here.

12:42 There's um there's sort of 16.

12:44 What are the What's the sort of corridor structure?

12:46 How can you get Which can you get between and which can't you?

12:49 Yeah, how are they constrained, yeah?

12:50 How How are they constrained?

12:51 Yeah.

12:52 If you look at this white white cubelets, right?

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

12:58 So, it can on this face,

13:02 there's four places it can sort of obviously get to, right?

13:06 Those four.

13:07 So, it can be just above the bottom left corner,

13:10 just to the right of that corner, just underneath that corner, right?

13:14 So, there's four four places it can get to.

13:18 Are there any others?

13:19 And actually the answer is no.

13:20 So, you might think you would be able to get

13:22 it get the white cube into that position there, but actually you can't.

13:27 And likewise, you can't get it down there.

13:28 That's sort of maybe more believable.

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

13:33 Just use a different face.

13:34 So, actually the sort of key thing here is

13:39 if you look at this blue square of four cubelets,

13:42 those are the four fundamentally different kinds of centers.

13:47 So, there's four kinds of centers.

13:49 And then for each kind of center,

13:52 the argument goes exactly as the centers for the 4 cube, okay?

13:57 So, the number of ways of arranging those kinds

14:01 of centers is that number K we had.

14:03 And then the number of ways of arranging those kinds of centers is also K,

14:07 and those kinds of centers are also K, and those kinds of centers are also K.

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

14:12 Then of course, we do divide by 24, which matters less and less as these

14:17 [laughter] numbers get bigger and bigger.

14:18 It's a bit of a formality at this point.

14:19 But when you work that out, we're up to 1.6* 10 to the 116.

14:26 We're like beyond atoms in the universe.

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

14:30 Yeah, there's there's more possible scrambles of this puzzle

14:33 than there are atoms in the universe.

14:36 This is my biggest cube.

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

14:41 Um this is a 10 cube.

14:44 And I haven't actually yet dared properly scramble it cuz I

14:47 dread to think how long it would take me to solve.

14:49 Um at some point I will.

14:51 Um So, but I think it might be fun to work out how many I mean,

14:54 has it gotten to a point now where for even numbered cubes,

14:57 we can create an algorithm here or Yeah, we can.

15:00 We can.

15:01 So, the way to think about it is let's call it a 2n cube, right?

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

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

15:11 We're going to let n be sort of half the size.

15:14 You could let n equal 10, but then the algebra comes out harder.

15:17 You've got C, that's the number of corners, that never changes.

15:19 Okay, now we think about the edges.

15:21 How many different edges have we got?

15:23 Now, let's put some different colors on.

15:32 Okay, but the point is how many different types are there?

15:35 Well, there's the central ones, then the next ones out,

15:38 then the next ones out, and then the wings, okay?

15:40 So, we've got four, right?

15:43 1 2 3 4 fundamentally fundamentally different types of edges.

15:48 Um and in general, so here in this cube, n equals 5.

15:54 So, in general, you're going to have n minus 1.

15:56 Yeah?

15:56 And the reason it's n minus 1 is because n is half the width of the cube,

16:01 and we just don't want the the outermost ones, so so n minus 1, yeah?

16:05 Okay?

16:06 So, that means the number of possible arrangements

16:08 of the edges is E to the n minus 1.

16:11 And now we've got to think about the centers.

16:14 Yeah, just give me a second.

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

16:22 So, now if we think about the centers, this white square of 16 cubelets,

16:28 that's representative of the different corridors.

16:30 So, each of those is different to each of the others.

16:33 You can never get from any of one of those to any of the others.

16:36 And once we've chosen one of those types, so there's 16 types here,

16:43 um or in general, that's n minus 1 squared, half the width,

16:48 so that's n, then we take away the edge,

16:51 so that's that width there is n minus 1, then we square it, okay?

16:56 So, there's n minus 1 kinds of centers.

16:59 That kind, by the way, is my justification for using K.

17:02 Um pretty weak, I think.

17:03 And then the argument for each one is exactly as the previous day,

17:06 that's the number of possible combinations.

17:07 So, we need to then multiply here by K 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 why these numbers

17:20 are getting so big so quickly cuz it's exponential in n,

17:24 but it's not actually just exponential in n, it's exponential in n squared.

17:28 And that's why why it gets so big.

17:29 Of course, we have to go through the formality of dividing by 24.

17:32 Of course.

17:33 Of course, of course.

17:34 Um Okay.

17:35 that does knock one off the power, doesn't it?

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

17:38 It knocks one off the power.

17:39 Um so, for 10, if you plug in um n equals 5 into this, well,

17:44 you get C times E to the 4 times K to the 4 squared, so 16, divided by 24.

17:53 Work it out, you get to 10 to the 349 or thereabouts, okay?

17:57 Which is a which is really ridiculously ridiculously big number.

18:00 I mean, that's the sort of place where, you know,

18:03 if you took every atom in the universe and replaced it with a whole universe,

18:08 and then repeated that four times,

18:13 and then counted the number of atoms in that whole thing,

18:16 this has still got more scrambles than that, right?

18:18 Okay, so it's it's an entirely ridiculously ridiculously big number.

18:22 You've spoken to me about bigger numbers.

18:24 I have spoken to you about bigger numbers, but I've never held one in my hand.

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

18:30 with an even number of pieces along the edges.

18:33 If you want to find out how it gets a bit

18:34 more technical and mathematical with an odd number, like that, [music]

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

18:41 We've got an extension of the video there.

18:43 We go into way more detail, but before you do that, go and order one of these.

18:50 Order a copy of Huge Numbers.

18:52 It's Richard's new book.

18:53 It's available now to order or 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 fantastic read.

19:00 It deals with really big numbers, like you've been seeing just now.

19:05 I'll also put some links below to other things that may interest you,

19:08 previous videos we've done about Rubik's Cubes,

19:10 previous videos we've done about big numbers, and of course,

19:14 the link to that extra video, and the order link for Richard's book.

19:18 Go and check it out, people.

19:19 Thanks for your time, and thanks for watching.

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