Odd Rubik's Cubes (extra footage) - Numberphile

0:02 So, let's Let's talk about how to modify

0:06 this this calculation to do to deal with odd-sided number cubes.

0:10 And of course, odd-sided cubes are

0:11 particularly interesting because the first one, well,

0:15 the first one is our old friend, the 1 by 1,

0:17 but the second one is the is the classic.

0:19 The reason I started with even cubes is because there

0:22 are a couple of things which make odd cubes slightly trickier,

0:26 I'm just a little bit.

0:27 And it's really to do with the edges in the 3 by 3 cube.

0:32 Or in fact, for larger odd larger odd-sided cubes as well,

0:35 we'll have a central edge.

0:37 We didn't have a central edge for the even-sided cubes.

0:40 We've got a central edge on the 3 by 3 and any larger odd-sided cube.

0:44 And they behave a little bit differently because

0:47 you can flip them over in in place.

0:51 So, if you just look at the the blue and the the yellow cube,

0:54 don't worry about the other things moving around,

0:56 you can see that I can I can flip it over in the same position,

1:01 which I couldn't do on the 4 by 4 by 4 cube.

1:05 So, that gives us a new a new thing to think about,

1:08 um which I'm going to call M for the middle cube.

1:12 So, the question is, how many arrangements are there of the middle edges?

1:17 The middle edges.

1:19 And well, there's 12 edges, okay?

1:22 And each one can go into any of 12 slots.

1:25 So, we start off with as previously by 12* 11 and so on, so 12 factorial.

1:31 Then we can flip each one.

1:32 So, you might think what we need to do is time multiply by 2

1:37 to the power 12 because each of the 12 edges could be one of two ways up.

1:41 But that's slightly wrong.

1:44 Once you've fixed 11 of them, you can't flip the 12th.

1:51 So, actually we only need to multiply by 2 to the 11.

1:54 And also, there was something slightly wrong in the previous

1:57 argument that gave us 12 factorial as well,

2:00 which is that Can I solve the cube and then [laughter]

2:04 You're filming you're filming me solving it.

2:05 Oh.

2:06 You never know.

2:07 It's almost as bad as doing it in front of Rubik's.

2:09 There's a problem with the original 12 factorial in terms

2:13 of how to position the different edges into the different slots.

2:18 Uh and the the problem is this, which is you can see here I've

2:21 got I've got a cube where I've I've sort of swapped three edges into around,

2:28 but amongst themselves.

2:31 There and there.

2:32 Everything else is solved, right?

2:34 Yeah.

2:35 Um and you can do that, but what you

2:37 can't do is the equivalent thing with just two, right?

2:41 So, once you've got 10 of your edges in particular slots,

2:48 there's actually no choice for the final two.

2:51 That means this 12 factorial* 2 to the 12,

2:55 we've got to divide by two twice, right?

2:57 Once to deal with that, the edges being swapped around,

3:01 and once again to deal with the fact that once 11 are in position,

3:07 there's no choice which way up the 12th goes.

3:09 So, we need to divide by 2 squared, okay?

3:11 So, this M is is 12 factorial* 2 to the 10, right?

3:17 That's 2 to the 12/ 2 squared.

3:19 That number is the number of possible arrangements

3:22 of the middle edge pieces on your classic 3 cube,

3:28 around about 5* 10 to the 11 or 500 billion.

3:32 Just like you had middle edges, you've also got middle centers, right?

3:37 In fact, I mean with with the 3 cube, you've only got middle centers.

3:40 With the a bigger one like a 5 cube,

3:46 so that that red cubelet in the middle, that's a a middle center, okay?

3:51 The middle centers on any odd cube really you can think

3:55 of as just being a like a 1 by 1 cube embedded in it.

4:00 So, you could Now, I could do something like that.

4:03 So, I'm moving It looks like I'm moving the middle centers around,

4:06 and I am moving them around.

4:08 Yeah.

4:08 But I'm not moving them around any more than I'm just,

4:10 you know, moving that around.

4:13 Um so, the way it it makes sense to think

4:16 of those middle centers as being just stuck in one place.

4:19 So, what should I call the this number for the middle centers?

4:22 I don't know.

4:23 Give me a give me a number.

4:24 All right.

4:25 Uh Um so, that's 24 cuz there's just 24

4:28 possible arrangements of the uh the middle centers, okay?

4:33 So, um now we can do the calculation for the for the 3 cube.

4:38 Okay, so we can do the uh we can do the calculation for the 3 cube.

4:41 So, it's just It's really simple.

4:43 It's the number of corners, C, the number of arrangements of the corners,

4:47 times M, that was the middle edges,

4:49 and then Okay, and then we can we can times I, and then we divide by 24.

4:54 And I won't write that again because you can see I is 24,

4:57 so I divided by 24 is just going to be one.

4:59 So, those go.

4:59 So, this is just C* M.

5:02 Right.

5:03 Uh and you work that out, and the answer is,

5:05 as you remember, Brady, Uh something quintillion.

5:10 43 quintillion, yeah.

5:11 Yeah.

5:11 Exactly.

5:12 Sorry, I didn't put you on this one.

5:13 That's all right.

5:15 [laughter] So, it's So, there was no K in this one because there's only an I.

5:18 There's no There was no meat in the middle there other than the Exactly.

5:21 There's no There's no K because there's no

5:22 centers apart from the the middle centers, yeah.

5:25 Yeah.

5:25 Um So, with the 5 by 5 that you showed me a second ago, Yeah.

5:30 Where's it going?

5:31 That is, yeah, that's the one.

5:32 Yeah.

5:32 So, to to to finish it off then, how would we do the 5 by 5?

5:35 Yeah.

5:36 Okay, so um actually it might be easier to jump to the 7 by 7, I think.

5:40 Okay.

5:40 You might see it a bit more clearly.

5:42 So, the the 7 by 7 cube,

5:44 it's the biggest cube in like official World Cubing Association competitions.

5:49 So, this is the this is the biggest

5:51 one which is somehow an official world record for.

5:54 So, this is a good one to It's a good one to do.

5:55 So, the question is, how many different kinds of centers are there here?

5:59 So, not including the the middle middle one, the I.

6:03 The I, yeah.

6:04 Um not including that, how many are there?

6:07 And there's um So, it turns out that what

6:12 we need is this little uh rectangle of yellow cubelets.

6:18 Those are the ones that are different.

6:20 And if we do this as a 2n+ 1 cube,

6:24 so because we want we want an odd number, 2n+ 1.

6:26 So, for the 7 cube, that's going to be three, n equals three.

6:30 What we need is C, corners, times M, the middles,

6:35 and I'm not going to do times I cuz it's going to disappear in a minute.

6:39 Okay, let's do the Let's do the edges.

6:41 How many different edges are there?

6:42 Well, of course, there's the middle edges which we already dealt with.

6:45 What about the other edges?

6:46 Well, there's two others here.

6:48 In general, um there's n minus two.

6:53 So, here in this case, n equals three, and we've got two other kinds of edges.

7:00 It's those two there, right?

7:02 Those are the two fundamentally different types of edges.

7:04 So, two for the 7 cube, and in general for the 2n+ 1 cube, n minus one.

7:09 So, we need to do E to the n minus one.

7:12 And then we've got to think about the the the centers.

7:15 The number of fundamentally different ones is here Well, it's six.

7:19 In general, on a 2n+ 1 cube, that is n* n minus one.

7:25 Right?

7:25 So, here n is three, so 2* 3.

7:29 Okay.

7:30 Okay.

7:30 So, it's a rectangle.

7:31 And you can sort of see that if you

7:33 just watch watch that yellow rectangle as it rotates around,

7:37 it's going to cover all the centers except for, of course,

7:42 the the I in the middle, yeah.

7:44 Yeah.

7:44 Yeah.

7:45 Okay.

7:45 So, we need to multiply by K to the n* n minus one.

7:50 So, that's the formula for the 2n+ 1 cube.

7:54 Uh let's do Let's just plug in n equals three for the 7 cube to see what we get.

7:57 So, for the 7 cube, we get uh C* M* E to the n equals three with when uh So,

8:06 E squared* K to the six, okay?

8:10 And that turns out to be uh approximately equal to 2* 10 to the 160.

8:23 It may have been that the actual most

8:25 efficient way to solve that was fewer than 20.

8:28 Um but a human can look at it and actually write down a list of 23,

8:31 which I think is probably more impressive

8:33 than people who can solve it really quickly.

8:34 You might think, well, how do we know?

8:36 How do we know that?

8:37 Uh do we just check all all 43.25 billion billion different ways of solving it?

8:42 And we effectively did that.

8:47 Uh a move, by the way, is a quarter turn.