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  • I want to draw a square.

  • I would like I've got some dotty paper on the brown paper here so I could draw stuff on these dots.

  • I'm gonna draw a square, but I'd like to tell me how big a square.

  • Uh, let's do, like, a four by four.

  • So you called it four by four.

  • How big is that square in terms of area?

  • I know a trick question.

  • I'm just curious.

  • 16 units, 16 units.

  • So it is possible to draw a square of size 16 and I want to say size.

  • I'm gonna talk about area from now on.

  • 16 years.

  • What other squares are possible to draw on this grid on the rule is I gotta use the dots and straight lines between dots.

  • I don't think this is a trick question, but could you suggest another square size?

  • I could draw what you could do, like a two by two.

  • Every four before going to 16 obvious ones.

  • Uh, wanna one window one.

  • Do you see the soul of numbers we're getting here?

  • You always were getting square numbers because you're drawing squares.

  • And first of all, that's no surprise.

  • The naming is a good thing here so I could draw 149 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 it could one drawer square on this grid using the same rules during the dots.

  • That's no square diagnose or something or slightly squares that are allowed as long as we stick to the dots.

  • Let's try one.

  • I'd like you to tell me.

  • I'm gonna start with this dot I won't go along the vertical or horizontal.

  • Oh, is that where you want to go?

  • I can go a certain amount down to send along.

  • Let's go forward down onto you three.

  • So four down here and on Wrangel go across to cross, too.

  • If that's the side of the square, I better do this one sort of four along.

  • So it was definitely a square, and you asked me to go sort of four down too long, which means all of the triangles like that so there's two.

  • There's four, so maybe we call it the 24 square.

  • And once it's area, I don't think this is much less obvious to me.

  • But it's not too hard to work out.

  • There are many ways you could.

  • You could chop it into bits and work out the air of the bits.

  • Or you could look at the area of the big square, which I think is six by six.

  • So 36 and take away the four triangles.

  • So I think he's triangle is two by four.

  • Let's call it half times two times four, which means it's one times four, which means four in here, all together, four times for 16.

  • So I think this is area 20.

  • Get so we've drawn squares on the grid that are not just the square.

  • 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 out?

  • Do one can we do to Turns out to is really easy.

  • You see, these two triangles makes one.

  • So this is area two and you do the same trick with the outside square.

  • Realize that area four and you got to Anyway, can you do three?

  • This is suddenly no obvious, Andi you start drawing stuff and ends up being very hard to draw three, 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 do.

  • Four.

  • Committee five Let's try 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 to down on one along, so the square's gonna look, let's give myself some space.

  • That's sort of diagonal, which means he's like a a two and a one sort of vector s o.

  • The other side was going to that that as long as I do a two and a one on each vector, I'm pretty sure that's five.

  • Then I enjoy the thing around it, which is now nine, because it's three by three and these triangles have calculated Let's do it.

  • This is 21 So the area off one, the whole thing's nine the four triangles give me four, so this is very fine.

  • Okay, so there's the problem.

  • The problem is, even if you allow slanty squares, which was your genius idea at the door to the square numbers some you can d'oh!

  • Indy five, maybe 20 to 10 Be county three.

  • The question I want to investigate in this video, that is, which squares candy drawer?

  • And is there a short cut of finding what the areas that you could get?

  • So 20 took us a little bit of calculation with them.

  • Trying was the five took us a bit, but I did.

  • I used to mother notations There's this to four notation that when we're 20 and the 21 notation number five and if we use the same notation, this could be kind of like 40 notation because for every side you go along for and down zero, I could write down all the list of notation I could do.

  • I could do 11121314 and I could just check all the list and I could find pretty easily just by search of all the numbers and realize there's some missings.

  • So three is missing.

  • Six is missing, seven is missing, 11 is missing and already we're secrets and like and all prime.

  • One of them's perfect like what's going on and I really don't think is obvious before we go any further, though.

  • What would be nice if first of all, we could find a shortcut for instead of home to draw thing and counting trying course.

  • How can I just write down the notation for the sort of slanty nous and get straight to the area?

  • I've seen this active activity done with sort of a nine year old school students, and they're busy drawing squares and calculated, and they were having a lovely time It Suddenly there's this deep glimpse of math so that, like that goes way beyond what they're ready for.

  • But what they already for is is finding a court collectivism area.

  • So let me draw a general square for you.

  • A slanty square's gonna 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.

  • You were describing my challenges.

  • 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.

  • Do you want to suggest how long that is.

  • It's always gonna be a plus B.

  • Yeah, because there's a symmetry.

  • Thes 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 a way we did it before.

  • We can take away the triangles from that, and we'll get the area of this central bit.

  • I think the triangle is relatively obvious is a B.

  • It's a right angle trying because the whole thing's of square, so half of a time to be so if I subtract or four of them four lots of 1/2 a times B s O.

  • That's the area where we're after.

  • We need to do some algebra.

  • So brace yourselves.

  • I'm going to square this bracket A plus B a plus B.

  • Let's just just the long way around to make sure we don't mess up.

  • This is gonna be a squared.

  • We're gonna get on a times B and again a b times a NATO, but to lots of a B, I'm gonna get be squid.

  • This will realize becomes to a B on.

  • Actually.

  • I mean, this is a familiar bit of algebra.

  • Perhaps, according to the calculation, though, if you haven't maybe slant notation the area This square is a squared plus B squared.

  • I suspect something familiar is occurring to must be watching.

  • This video might be recurring to you as well, but let's just check it works.

  • The 24 notation should give us 20 uh, square to you.

  • Get four square for you.

  • Get 16 Adam together.

  • 20 The 21 notation square to you Get four square When you get one together, you get five.

  • It seems to work, and it obviously works on the ones with zeros because 14 16 she's gone.

  • If you notice what we just pretty if I put I see here on Talk about right angled triangle with Sides, A B and C I'm saying This square on See you just proved my figures.

  • It's one of my favorite proofs of Pythagoras.

  • It turns out, in the way of playing with Squares and Dr Grady and approving one of those facts of geometry that everybody knows.

  • There's literally hundreds of Proust ball, but here's my favorite one, and it kind of arises.

  • We can't we have a problem with doing something else anyway.

  • So bonus Along the way we've proved by faggots.

  • We've now got shortcut.

  • If you've got a Navy slanty square Hey, squared plus B squared.

  • Which means the real question we're on now, which is for May.

  • Which squares can you draw on?

  • Which ones can you?

  • Troy becomes a number theory question.

  • Forget geometry, he becomes.

  • Which numbers can you make by adding two squares together, which introduce of some 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 this is a classic piece of number theory.

  • There's out there in degree level undergraduate courses and I'm not gonna sew it for you.

  • But let me write down in the beginning of the list of the squares that you can't troll.

  • He can't draw three because there are no two numbers you can square to get through.

  • And if you try on the grid, you end up drawing all the squares and very quickly, or 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 could be done by exhaustion.

  • You just keep trying.

  • You can't do six.

  • You can't do seven.

  • 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 It can't be done.

  • So I'm gonna boot our 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 their 14.

  • 50 and you get a little run of consecutive there.

  • Then it jumps in 1921.

  • That's enough for now.

  • These numbers you cannot draw squares on Dottie.

  • Do they become more sparse or less passes?

  • You go up.

  • I'm well out of my death thing is.

  • And what what was started as a simple problem like, I quickly get out of my depth.

  • This isn't a number theory question.

  • None of which numbers can be made is the sum of the squares on.

  • I have no intuition about it.

  • What happened was I When I looked it up, I went and looked up.

  • Which numbers can be the summer Two squares and It's a classic number theory problem that you do an undergraduate degree in mathematics and, it turns out, is it.

  • There's a simple test which numbers can be made is the sum of two squares.

  • You wanna know the test audio?

  • Do wanna guess?

  • I definitely want to know.

  • If a number has a prime factor off the Form four K plus three to an odd power comedy, I'll give you an example.

  • The number 11 1st of all this is prime one line prime number, and it only has.

  • One fact is one factor, which is itself in one but factors other than one.

  • This is just itself, and it's a prime off the form four K plus three, by which I mean it's four times something on the three more equivalently is one less than a multiple of four.

  • So you could say four K minus one is kind of equivalent when you going more for just cycles around before, So these are the same thing.

  • It turns out all primes, except to off the four capers, three type or four K plus one, and these ones the problem.

  • So three is another example.

  • It's one less than 19 in fact, seven as well.

  • These are all one less than a multiple off four.

  • Maybe prime numbers are not surprising sequences like this, but the six is what bothers people.

  • It's not prime and still can't be done.

  • And that's because it has a factor of three sixes three times to effect on.

  • Because it's has a factor off that form.

  • It can't be done before you get into it.

  • Nine has a factor of three.

  • These three squids, so it has a factor of that form.

  • But because it's got two of them multiply together.

  • We're fine.

  • So that number, the power has to be on for it to be a killer.

  • Exactly.

  • So another killer one.

  • Let's pick one off the list.

  • Let's pick 12 If we factories it.

  • It's two times two times three, otherwise known as two squared times three.

  • And the problem factor is a prime of that form.

  • There it is, and it's gone on power.

  • So I'm guessing three times three times 3 27 37 is three.

  • Cute could be done, so it should be on the death list of ones that can't be done.

  • There it is.

  • 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.

  • The factory is that it's two times three times three is definitely got a prime, which is three more than multiple for but it's gone.

  • Even power and 18 is not on the list.

  • It can be done, so there's some complicated interactions with the fuel was.

  • If you want to investigate this, it's a nice, little accessible way of number theory to do.

  • Then you can check.

  • If you found one slanted square that works, you can actually prove where it'd be easily that you multiply it by another slanted square.

  • That works and you can build one.

  • So it's to do with them multiply together, which is why the primes get involved, because everything that comes from multiplying is coming from Prime's.

  • But if you want a very elementary starting problem, where can you draw a square on paper?

  • That sort of escalates relatively quickly into Advanced number theory, which has an accessible proof but is 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 on.

  • 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 get some right, you'll get some wrong, but that's okay.

  • 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 then I'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.

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  • There's lots of stuff you can look there for free.