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[00:00:00]I've got a series of things. Literally a series of things in that a series is mathematically defined as a sum of a sequence and there are lots of important things in maths where you have to add up a bunch of numbers that happen in sequence.

[00:00:11]And actually there are three results that most students who get sort of to the end of school still doing maths, they will learn three sums of important series and they're not a surprise to anyone who's doing maths at school. They are the sum of the integers, the sum of the natural numbers.

[00:00:25]Just as a heads up, I'm not going to infinity. Don't start talking about -1/12. These are actually finite sums.

[00:00:31]So the sum of the integers 1 + 2 + 3 + 4 + 5 up to n. And then like by extension the sum of the squares, so 1 squared + 2 squared + 3 squared + 4 squared all the way up to n squared and the sum of the cubes. And those three are like big deals in school. I'm going to write down some of the results. I'm going to leave one as a secret. And I want to tell you the story of all three of them and how you can see the results even if the formulas look like really arcane. Is that okay?

[00:00:55]>> Sounds good.

[00:00:55]>> All right.

[00:00:56]>> First one up, I'm going to use some notation. This is a Greek letter sigma for the sum. It's just the Greek letter S. Get used to it. I'm going to use R, which is like a dummy variable and R runs from 1 to n. So this is a mathematically terse description of add up the numbers R as they go from 1 to n.

[00:01:12]That's what that symbol means. And I hope many numberphile viewers will know the result here. It's a half times n times n + 1. This turns up all over the place. It's actually the formula for the triangle numbers. There's a classic problem about if there are 10 people in a room and they all shake hands, how many handshakes would there be? It turns out that's a triangle number and the formula comes very like this. It's like n n - 1 instead of n n + 1.

[00:01:34]Thing is this is a famous formula uh and I'm going to prove it without using algebra.

[00:01:39]Maybe a lot of you could do that. Next one though would be sigma again, this time R squared. So this is the sum of squared. R runs from 1 to n. And this one, I mean when I first saw this one I was like, really? Why does it have to be this? It's a sixth n n + 1 2n + 1. That that's annoying. It feels less clean, uh particularly when this one gets familiar, this one doesn't feel so easily memorable.

[00:02:01]>> So, if I do 1 squared plus 2 squared plus 3 squared all the way up to say 20, >> you need to put 20 in for n.

[00:02:07]>> 20 is in there, and that will give me the the result.

[00:02:09]>> It'll give you the correct result. And it'll also, despite dividing by six, it'll always give you a whole number, which is a relief. Otherwise, like, how are you not getting whole numbers? So, but the sick thing there is like, there's a fraction, but we're going to get a whole number anyway. It works, and I'm going to prove it works, hopefully.

[00:02:21]The last one, sigma r cubed this time.

[00:02:24]This is the the third in the sort of triumvirate.

[00:02:28]I'm going to leave as a little bit of a surprise.

[00:02:29]>> Okay.

[00:02:30]>> Some people will know it already, but there's a there's a reason to tell all three of these stories together. Let's start with the triangle numbers, the sum of the integers, the sum of the natural numbers up to n.

[00:02:40]Uh so, this is 1 + 2 + 3 + 4 + dot dot dot up to n minus one plus n. You get to n in the end. And we can do this algebraically.

[00:02:50]There's all sorts of nice stories.

[00:02:52]Maybe there's a famous story about Gauss adding up the numbers from 1 to 100. I'm not going to tell the story again. You can do it algebraically, but I don't want to do it algebraically. I want to draw a picture. And the pictures look very simple. Here is one blob. Here is two blobs. And here is three blobs. Can you see first of all why they call them the triangle numbers? I'm sure you can.

[00:03:08]That's a slightly patronizing question.

[00:03:09]It turns out if you draw a triangles of shapes of blobs, you get these triangle numbers. And you can see that's why 1 + 2 + 3 + 4 gives you a triangle.

[00:03:17]Crucially, this does not help me see that formula until I draw another copy.

[00:03:21]So, I'm going to draw another copy of this one. This is the sum of the first four integers. So, it'll be like up to there.

[00:03:27]If I draw another copy and fill them in this time, it's upside down, but it's the same triangle. Agreed?

[00:03:33]And it turns out because triangles, the way I've drawn them, always fit together like this, I will always make a rectangle, which has got, in this case, four along here, and one more than four here, and it's two copies of it.

[00:03:47]The rectangle size, in this case, is 4 * 5, but I better halve it to get just the original and if I generalize it, this could be n, this could be n plus one, and it'll be a half of n times n plus one. And that is that formula. And just in case you weren't sure that this generalizes, here is the seventh triangle number. Uh and I can crank this up, right? Once you Once you've The nice thing about GeoGebra is it lets you generalize without redrawing the thing. And I can always make a second copy, and it always makes that rectangle, which is one longer that way than it is that way. So, it's n times n plus one, and halve it because I've got two copies. This method of making more than one copy of a thing in order to then like halve it to cuz we've doubled it in the first place is quite a useful technique. We're going to use that again for the next one.

[00:04:31]Part two, the sum of the squares. This formula is really much less intuitive to me, and there is a way to algebraically prove it, and I'm not going to do it, but if you notice that this one ends up being a quadratic. It's got an n times an n in there, so it's got an n squared in it.

[00:04:44]It may be it's not a surprise that this one has to be a cubic. And you can use that information to algebraically come up with this, but I'm not going to do that. I want to show you a better way of adding the squares together. Here is a stack of some squares. Takes a little bit of getting used to. In fact, there are There's a square of four on the bottom.

[00:05:01]Um Let's start with one. That's a square, apparently. Uh There's a square with one and a square of two.

[00:05:08]And if I do a square of three on the bottom of that, you can see I'm stacking squares up. So, the number of balls in this diagram is is kind of the answer of my my question here.

[00:05:17]>> So, at the moment I can see that there's a nine on the >> Yeah, there's a nine plus a four plus a one, so we're at 14. The thing is, once I've got it on GeoGebra, I can generalize it. So, here's a stack of squares. Um actually, if this is looking familiar, go and have a fun time with Matt's cannonball video from a few years ago on Numberphile, where we talk about what numbers do this and are square shape. That's not where I'm going here. I want to know how many are in there. And the reason I'm showing you diagram is that I can make copies of this diagram, and let's do that. Let me I'm going to reduce it slightly so we can see it a bit better. But here's a stack of four squares and I'm going to put two more versions on the screen.

[00:05:55]Red and blue, they are the same copy. Do you just So I've I've made three times the original thing, but they fit together really nicely.

[00:06:04]You see the sort of matching together.

[00:06:05]Now, it's not as nice as I'd like. If it had made a nice cuboid, I could get the volume of the cuboid or the number of balls in it in this case by just multiplying the dimensions. But I've got this lip at the top.

[00:06:14]>> Yeah, that bit on top's a bit offset, isn't it?

[00:06:16]>> It is. It's offset by one row, in fact.

[00:06:18]Which means if I get another copy of these three and put them on top, I think it would fill in the gap. So, let's do that.

[00:06:26]There they are.

[00:06:27]And if you know I mean it's kind of hard to see now. Let me just separate them so you can see them.

[00:06:31]Maybe I'll leave this spinning slightly so we can uh get a hang of it. These are six copies of the original now and I've faded out the ones on top. But if I put them all together, they do make a nice cuboid.

[00:06:42]>> Perfect.

[00:06:42]>> In fact, I can tell you the dimensions of the cuboid. It's This side's got four and remember my original one if I just going back to this, it's just the sum of the squares from four up to the to one or one [clears throat] up to four.

[00:06:52]So, this side's got four, but then on this side it's four plus one. It's always plus one because it's the tip of the of the red tetrahedron going on here. And then the other dimension, which is the height of the thing, is actually two copies. So, four plus another four plus one cuz it's the tip of that green one that went in there.

[00:07:11]So, the dimensions of this cuboid are n, n plus one, and two n plus one. And [clears throat] there are six copies.

[00:07:21]So, n, n plus one, two n plus one and divided by six cuz there are six copies. That is the formula for the sum of the squares. And although it's possible to prove it algebraically and once you've got it, you can prove it by induction. All sorts of ways to do it. I really like the visual proof of seeing the six copies and seeing how they fit together. Now, while this might be not a formal mathematical proof, it really deals with your intuition. And I'm a much happier person about this formula now because I can kind of see where those bits are coming from.

[00:07:48]>> Why does that not count as a proof? That looks like a proof to me.

[00:07:51]>> Yeah, and I mean I'm all all for like they call them proofs without words. The diagram does everything for me, but maybe that's only a proof that it works for four.

[00:08:01]But the the fact that I've done it in GeoGebra means I can generalize it as long as I've programmed it right.

[00:08:04]There's five, uh there's six. Yeah, I'm beginning to see the generalization, but it is technically only one specific example. And maybe that's why mathematicians might argue that a a proof needs to cover in general and show why not just works for that one.

[00:08:19]Maybe it's a fluke. It's got to work always. We've got to be a bit careful, but actually this diagram gives me a way of constructing the formula which I could prove by another way. So the diagram has a lot of value whether whether you think it's formal or not.

[00:08:29]Let's do part three.

[00:08:31]Part three I deliberately suppressed cuz some of you know the answer already. Uh let me start by going straight in with the diagram. We're trying to add the cubes together now. I mean I can I can draw it first. Let's let's do some some blocks, right? There's there's one cube and then two cubes I need to add together >> [snorts] >> that and then I could draw the three cubes or we've got a Rubik's Cube or something. I've got to add all those together and drawing them like that doesn't give me an obvious way of like stacking them together in a way that makes me see an answer.

[00:08:59]But drawing them in a different way gives me a really nice visual thing.

[00:09:03]Okay, so on screen is, believe it or not, the sum of the cubes and this takes a tiny bit of thought. So first of all I can change the number of cubes in here.

[00:09:10]They don't look like cubes, so bear with me here, Brady, but that square down here is one.

[00:09:16]I hope you agree it's just one square.

[00:09:17]The next pile is two squares of two.

[00:09:21]So one square of two would be two squared and it's then multiplied by two again. So I think that the number of squares in that pile is two cubed. And similarly, three squares of three is three cubed. Four squares of size four is four squared times four and five squared times five.

[00:09:37]The number of squares in this figure is the sum of the cubes. It still doesn't look like a super intuitive thing, but I'm going to do something which the first time I saw it was like Okay, I get it.

[00:09:46]>> So, you're saying um one five cubed is equal to the area of that thing.

[00:09:54]>> Exactly that, yeah. The number of squares in there.

[00:09:56]>> Yeah.

[00:09:56]>> And I don't think that's controversial.

[00:09:58]It's It's just like what Why are you bothering to draw that diagram? Well, here's why. Let me turn off the the squares for a moment, but it's still the same area. I just made it go red. And I'm going to draw a line. In fact, I'm just going to draw this line. And it looks like an arbitrary line, but actually it's a 45° line that's going straight through the middle of the first square at 45°. So, it's slightly un non-intuitive where it cuts the other piles, but actually it's cutting them in half. Although, that's not really important. And I'm going to go into three dimensions now, slightly unusually, to see what happens. Because you really should do this with a piece of paper, but then you can't generalize it. So, first of all, whatever I just done does generalize.

[00:10:34]Let's leave it at five for example.

[00:10:35]Those green bits I'm going to twist. And actually, if you imagine the pin in the middle of each sort of pile, you'll see how they twist.

[00:10:46]And let's look from the top again.

[00:10:49]That is a square.

[00:10:51]And this always works. I can generalize like, you know, one cubed plus two cubed three cubed, four cubed, five cubed. It always works. I can always twist it back.

[00:11:02]But now I've got a visualization of the area here, because it's a square and the side of the square is 1 + 2 + 3 + 4 + 5 + 6. The side of the square is the sum of the integers.

[00:11:12]And you remember remember from a few minutes ago that we have a formula for the sum of the integers, which means I can now write down the formula for the sum of the cubes.

[00:11:21]It must be the square of the sum of the integers.

[00:11:23]>> Because the sum of the integers runs along the bottom and square.

[00:11:26]>> The first pile was width one, the second pile was width two, and three, and four, and five, and six. So, this total length is the sum of the integers, and then the area must be that squared, which means I can write down the formula. Let's write it down.

[00:11:37]>> [snorts] >> It must be the same as this one squared.

[00:11:41]Which would make it square the half, a quarter, n squared, n plus one squared.

[00:11:46]And that is the result. But actually written on its own, it's forgivable that you don't notice it's the same as that one squared. And I really think this is the sort of the end of the story. You've got three classic series, one of which is nice, triangle numbers famous, one of which is nasty but has a nice cuboid to see it, and one of which is just the same one again squared. And if you'd asked me to predict that the sum of the integers squared is the sum of the cubes, I'm like, have you just got your numbers mixed up in your head because all everything's involved. You've got the power one, you square that to get the cubes of the Anyway, the fact the fact that it's a beautiful algebraic coincidence, and there's this beautiful twisty proof means that I really like these three results as a little sort of trio of series sums.

[00:12:27]>> Presumably, the sums of the powers of four and powers of five and all that sort of other trends start to show up?

[00:12:33]>> Yeah, the It It turns out that the even powers have a sort of different pattern to the odd powers, and you can kind of see that happening with the one and the three.

[00:12:41]But I'll leave that as an exercise.

[00:12:43]It gets messy, and if you want to do these algebraically, which is presumably how they were originally done, you do have to have a bit of guts to push through. There are some really nice methods, too. Classically in school, you have to prove them by induction, and that's a method which needs you to know the answer before you prove it. And although it's powerful, the question still is like, well, how did they come up with the original answer? And almost certainly they did it algebraically by other methods. They're all out there. But maybe they stumble across across a nice diagram like this which gives them the intuition, which means you can write down the formula, and then you can prove it formally if you care about that sort of thing. Check out the links below for more videos with Ben Sparks. Plus, find out more about what he's up to, chances to hear him speak, training, [music] and maybe pre-order or order his new book. Check that out. As I said, links down below in all the usual places.