I want to draw a square. I would like you to - I've got some dotty paper on the brown paper here. So I can draw stuff on these dots. I'm gonna draw a square, but I'd like you to tell me how big a square to draw. (Brady: Uhh... let's do like a 4 by 4?) Ben: So you called it a 4 by 4. How big is that square in terms of area? Not a trick question, just curious. - (Brady: 16 units?)
- Ben: 16 units. So it is possible to draw
a square of size 16, and when I say size and I'm gonna talk about area from now on, 16 units. What other squares are possible to draw on this grid, and the rule is I've got to use the dots and straight lines between dots I don't think this is a trick question, but can you suggest another square size I could draw? - (Brady: Well you could do, like a 2 by 2?)
- Ben: Area 4. So we can do 4, we can do 16, obvious ones?
- (Brady: Uh... a one?) - Ben: A 1! We can do a 1! Do you see the sort of numbers we're getting here?
- (Brady: Always square numbers?) We're getting square numbers because we're drawing squares. And first of all, that's not a surprise, The naming is a good thing here, so I could draw a 1, a 4, I could do a 9. We didn't do that. It's not a surprise that you can draw a square number size squares. Can you draw a square, or could one draw a square on this grid, using the same rules with the dots, that's not a square number?
- (Brady: Like diagonals or something?) Oh, slanty squares are allowed, as long as we stick to the dots. Let's try one. Umm I'd like you to tell me - I'm gonna start at this dot. I won't go along the vertical or horizontal, though so - where'd you want me to go? I can go a certain amount down or a certain amount along. (Brady: Let's go 4 down?)
- Ben: One, two, three... Ah, so 4 down here - and at an angle...?
- Brady: And go across 2? Ben: Across 2. If that's the side of the square I better do this one sort of 4 along. So we have - it was definitely a square and you asked me to go, sort of, 4 down and 2 along, which means all of the triangles are like that. So there's a 2 there, there's a 4, there's a sort of... So maybe we call it the "two-four" square and what's its area? I don't think - this is much less obvious to me now, but it's not too hard to work out There are many ways you could do it You could chop it into bits and work out the area of the bits, or you could look at the area of the big square, which I think is 6 by 6, so 36. And take away the four triangles? So I think each triangle is 2 by 4 Let's call it: 1/2 x 2 x 4, which means it's 1 x 4 which means 4 and here Altogether 4 x 4 is 16. So I think this is area 20. Good! So we've drawn squares on the grid that are not just the square numbers. So, I guess the natural question. It's a pretty elementary starting point. What squares can you draw? We can definitely do square numbers. We can definitely do the number 20 Could we for example work our way up we can do 1 - can we do 2? Turns out 2 is really easy. You can see that each two triangles makes 1. So this is area 2. And you do the same trick with the outside square and realise that's area 4 and you've got two there - anyway, Can you do 3? This is suddenly not obvious, and you start drawing stuff and it ends up being very hard to draw 3 at least. Even if it's possible, it's not easy to do, and I think you can then convince yourself quite quickly that it can't be done. But we know we can do 4 Can we do 5? Let's try it. It can't be a vertical horizontal one so we're gonna have to go a bit of a slanty thing and if I go 2 down and 1 along. So the squares gonna look - let's give myself some space. That sort of diagonal which you can see is like a 2 and a 1 sort of vector. So the other side I was gonna do that and that. As long as I do a 2 and a 1 on each vector; I'm pretty sure that's 5. Then I can, I can draw the thing around it which is now 9, because it's 3 by 3, and these triangles if I calculate them. Let's do it. This is 2 and 1 so the area of 1. The whole thing is 9, the four triangles give me 4, so this is area 5 Ok, so there's the problem. The problem is even if you allow slanty squares, which was your genius idea to dodge the square numbers, some you can do. You can do 5. You can do 20. You can do 10. You can't do 3. The question I want to investigate in this video, then, is which squares can you draw? And - is there a shortcut of finding what're the areas that you could get? So 20 took us a little bit of a calculation with some triangles, the 5 took us a bit, but I did, I used some other notation. There's this (2,4) notation, that went with 20. And the (2,1) notation, number 5, and if we use the same notation this could be kind of like (4,0) notation. Because, for every side you go along 4 and down 0. I could write down all the list of notation I could draw. I could do (1,1), (1,2), (1,3), (1,4), and I could just check all the list, and I could find, pretty easily, just by a search of all the numbers, and realise there are some missings. So 3 is missing. 6 is missing. 7 is missing. 11 is missing. And already we are at a sequence, and I'm like: they're not all prime. One of them is perfect. Like, what's going on? I really don't think it's obvious. Before we go any further though, what would be nice if first of all we can find a shortcut for instead of having to draw the thing and count all the triangles; how can I just write down the notation for the sort of slantiness and get straight to the area? I've seen this activity done with, sort of, nine-year-old school students, and they're busy drawing squares and calculate it and they were having a lovely time, and suddenly there's this deep glimpse of maths that goes way beyond what they're ready for. But what they are ready for is is finding a quicker way to do some area. So let me draw a general square for you. A slanty square is going to fit inside another square. If I say that this distance is a and this one is b - then we've kind of got this - this is like our (a,b) notation that we had, that you were describing earlier. So my challenge is then, how big is the square in the middle? Because that's the bit we can draw and it's not too bad. I can work out the side of the big square. So you want to suggest how long that is? (Brady: It's always gonna be a plus b) - Ben: Yeah, because there's the symmetry, these triangles turn up all the way around. So the whole square is a plus b, squared. Happy with that? What about the triangles - because that was the way we did it before. We can take away the triangles from that and we'll get the area of this central bit. And I think the triangle is relatively obvious, there's just an a and a b And it's a right-angled triangle because the whole thing is a square So half of a x b? So if I subtract the four of them, four lots of 1/2(a x b) So that's the area we're after - we need to do some algebra, so, you know, brace yourselves. I'm going to square this bracket. Uhh, a plus b, a plus b. Let's just do it the long way around to make sure we don't mess it up. This is gonna be a squared. We're gonna get an a x b. And again, a b x a. So I've two lots of ab. And I'm gonna get a b squared This we'll realise becomes 2ab. And actually,
(Brady: Yeah...) Ben: I mean this is a familiar bit of algebra, perhaps. According to the calculation though, if you have an (a,b) slant notation, the area of this square is a squared plus b squared. I suspect something familiar is occurring to lots of people watching this video, might be occuring to you as well, but let's just check it works. The (2,4) notation should give us 20 Square 2 you get 4. Square 4 you get 16. Add 'em together - 20. The (2,1) notation: Square 2 you get 4. Square 1 you get 1. Add 'em together you get 5. It seems to work and it obviously works on the ones with zeros because 4 squared is 16. Have you noticed what we've just proved? If I put a c here, (Brady: Oh, right) and I talk about a right-angled triangle with sides a, b and c. I'm saying the square on c. We just proved Pythagoras. This is, it's one of my favourite proofs of Pythagoras it turns out, in a way of playing with squares on a dotty grid, you end up proving one of those facts of geometry that everybody knows, there's literally hundreds of proofs for, but here's my favourite one and it kind of arises, organically, out of a problem when we're doing something else. Anyway, so bonus along the way we've proved Pythagoras we've now got a shortcut. If you've got an ab slanty square: a squared plus b squared. Which means the real question we're on now, which is, for me, which squares can you draw, and which ones can't you draw it becomes a number theory question. Forget geometry, it becomes "Which numbers can you make by adding two squares together?" Which integers are the sum of two squares? And the nine-year-olds I saw tackling this problem don't make a lot of progress for this, because this is a classic piece of number theory, that's out there in degree level undergraduate courses. And I'm not gonna solve it for you, but let me write down the beginning of the list of the squares that you can't draw. You can't draw three. (Brady: 'cause there are no two numbers you can square to get three.) - Ben: And if you try on the grid, you end up drawing lots of squares, and very quickly you've exhausted all the possibilities and all the answers you're getting after that are bigger than three. So, you know that three is never gonna turn up. So it can be done by exhaustion, you just keep trying. You can't do 6. You can't do 7. You can't do 11. In fact, because it's an exhaustion thing, you might as well do a computer program just to check, sort of manually, which ones can and can't be done. So I'm gonna boot up a computer just to get the list So this list of numbers is weird. I'm gonna just write down a few more of them 12 is in there, 14, 15. You get a little run of consecutive there, and then it jumps to 19, 21. That's enough for now. These numbers you cannot draw squares on dotty paper. (Brady: Do they become more sparse or less sparse as you go up?) I, I'm well out of my depth there The thing is - and what started as a simple problem like I quickly get out of my depth This is a number theory question now: which numbers can be made as the sum two squares, and I have no intuition about it. What happened was that I went and looked it up. I went and looked up which numbers can be the sum of two squares. And it's a classic number theory problem that you do in undergraduate degree in mathematics and it turns out There's a "simple" test which numbers can be made as the sum of two squares. Do you want to know the test or do you, do you wanna guess?
(Brady: I, I definitely want to know) Ben: If a number has a prime factor of the form (4k+3) to an odd power, it can't be done. (Brady: Sorry? Say that again?) Ben: I'll give you an example: The number 11. First of all, this is prime, well known prime number. And it - - (Brady: So it only has one factor?)
- Ben: It only have one factor, which is itself and 1, but the factors other than 1 is just itself and it's a prime of the form (4k+3) by which I mean it's 4 times something and then 3 more. Equivalently it is one less than a multiple of four So you could say (4k-1) It's kind of equivalent when you're going mod(4) it just cycles around every four. So these are the same thing. It turns out all primes, except 2, are of the (4k+3) type or (4k+1) and these ones are the problem. So 3 is another example. It's one less than a multiple - 19. In fact 7 as well These are one less than a multiple of four. Maybe prime numbers are not surprising in sequences like this, but the 6 is what bothers people because it's not prime. It still can't be done. And that's because it has a factor of 3. 6 is 3 x 2 in fact. And because it has a factor of that form, it can't be done. Before you get into it, 9 has a factor of 3. It's 3 squared. So it has a factor of that form, but because it's got two of them multiplied together, we're fine. (Brady: So that number, the power, has to be odd for it to be a killer?) Exactly. So another killer one, let's pick one off the list, let's pick 12. If we factorise it, it's 2 x 2 x 3. Otherwise known as 2 squared times 3 and the problem factor is a prime of that form There it is, and it's got an odd power. (Brady: So I'm guessing, 3 x 3 x 3, so 27 - ) 27 is 3 cubed. Could it be done? (Brady: So it, it should be on the death list of ones that can't be done? There it is!) There it is. So this test is good and if we checked some other ones, so for example 18 might be a good one to check. We factorise that, it's 2 x 3 x 3, it's definitely got a prime which is 3 more than multiple of 4, but it's got even power. And 18 is not on the list. It can be done. So there's some complicated interactions with a few ones if you want to investigate this, which is a nice little, accessible bit of number theory, to do, then you can check: if you found one slanted square that works, you can actually prove relatively easily that you multiply it by another slanted square that works and you can build one. So it's to do with them multiplying together, which is why the primes get involved because everything that comes from multiplying is coming from primes. But if you want a very elementary starting problem: "Where can you draw a square on dotty paper" that sort of escalates relatively quickly into advanced number theory which has an accessible proof, but it's definitely much harder than nine-year-old maths - this is a really lovely example of it. Videos like this one always leave me wanting to get my brain in better shape. And one way to do that is to check out all the online courses at Brilliant. The number theory course is a great place to start. You'll get some right, you'll get some wrong, but that's ok, that's all part of learning. And all the courses on Brilliant have been crafted by people who *really* care. They really think about how you navigate through these things, how you'll learn Everything goes step by step. You'll learn one new thing and that'll help you in the next bit. There's loads of mathematics, science, all sorts of stuff to keep you occupied, and there's more stuff coming all the time. Have a look at these ones in the works. Go and have a look at the Brilliant site. There's lots of stuff you can look there for free. If you sign up for the Premium Membership that unlocks all the stuff, you can get 20% off that by going to brilliant.org/numberphile. We spend so much time exercising and getting our bodies in shape. Brilliant is a great place to go to get your brain in shape. [Preview of next video] The older generation used to get told Pi was pretty much exactly the ratio 22 over 7. It's not quite, but it's unreasonably good, and you can see on this diagram that it's unreasonably good because this is irrational but it's really well approximated by something to do with the number 22
Did anyone solve the 3 dimensional version of this problem?
i.e.
What volumes of cubes are possible to draw on a cubic lattice?
Thought this was a relatively interesting problem popping up out of a simple question, but also a really nice look at this was posted by 3Blue1Brown a while ago - in case anyone's interested at why this is the answer and wants an interesting tangent about a formula for pi.
Edit:Though I should note, rewatching 3Blue1Brown's video - a key step is not included. The basic idea is that something can be represented as the sum of two squares if it is composite in the Gaussian integers. There's a point where it is taken on faith that all primes 4k + 3 are still primes in the Gaussian integers (easy to see - all squares are 0 or 1 mod 4, so 4k + 3 has to not be the sum of squares) but it's not shown that all primes 4k + 1 have to be Gaussian composite.