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  • So this thing is actually the logo of Trinity Maths Society.

  • Trinity College in Cambridge, Maths Society,

  • when they have lectures about mathematics, and they go out socializing to the pub,

  • well their Society has this as their logo.

  • It's a square made of other squares

  • but all the squares are different sizes.

  • So, see this one here with 50?

  • Well that means it's a 50 unit by 50 unit square.

  • This one here with a 35 is 35 by 35.

  • All these squares are different sizes

  • yet they fit together to make one big square

  • which is 112 by 112.

  • So it's called a squared square.

  • Why have they picked this as their logo for their Society?

  • Because they're nerds!

  • Well, yeah. And that.

  • AND, because it was Trinity Maths students that solved this.

  • This one is actually the smallest possible squared square.

  • This is something that feels like it would be done by the ancient Greeks or something.

  • This feels like an old problem.

  • How can you make a square from other squares of different sizes?

  • That feels like a classic problem.

  • This didn't really turn up until the 20th century, which surprises me.

  • They are integer squares. We're not using the same square twice

  • Or if you do, they're called imperfect

  • so you can use that. But I'd rather they're all different sizes.

  • There were four students at the Trinity Maths Society that really solved this problem.

  • They were called Arthur Stone, Cedric Smith, Roland Brooks and Bill Tutte.

  • The first three were maths students.

  • Bull Tutte was actually a chemistry student, but he was in the Maths Society because he liked maths.

  • So you don't have to be a mathematician to be in the Maths Society.

  • I think it was Arthur Stone who brought this problem in, who heard of this problem before.

  • The only thing that had been done before was there were

  • a couple of rectangles that had been found made out of squares.

  • Not quite the same thing.

  • Here they are. Just two. There were just two rectangles that had been found.

  • One of them uses nine squares.

  • The other uses ten squares.

  • Rectangles they are easier than making squares.

  • But only two had been found so far, so the students started to find more.

  • Their methods were a little bit ad hoc, so they were trying to find these rectangles.

  • They were hoping that they would get lucky and find a square made out of squares just by chance.

  • But it wasn't working.

  • They kept trying. They found rectangles, but they just couldn't find a square.

  • They thought, right, we're going to have to get more systematic.

  • So when they started finding these rectangles made out of squares

  • they thought, okay, let's do this.

  • And they started to extend the horizontal lines like this

  • You see this horizontal line? (tootootoot right)

  • Once you've extended the lines, we're gonna turn this rectangle into a network.

  • And what you do is you connect two horizontal lines if the square are touching

  • So this one here, 25, is touching these two horizontal lines.

  • That's the top of 25 and that's the bottom of 25.

  • So I'm just gonna connect them with a line

  • I'm actually gonna draw an arrow here to show top to bottom

  • and mark it, that's the 25 square.

  • So I'm just gonna do the whole thing, now, with a network.

  • So, 16, that's the top of sixteen, that's the bottom of 16

  • Let's connect those with a line.

  • And I'll mark it with 16. That's the 16 line.

  • And then I'll keep going.

  • So they turned it into a network.

  • And when they did that, they recognized it.

  • They went, that seems familiar.

  • And why they recognized it was because it reminded them of an electrical network.

  • If you connect this to a battery, and if these were wires in a network, and each wire has one unit resistance

  • and you connect it to a battery which has the same potential as the height of the rectangle

  • and put all these numbers in, these are the currents that flows through that electrical network.

  • So, suddenly they realized that this was connected to electrical networks

  • and they could use all this maths that was already there, and they could borrow it completely

  • to solve the squared square problem.

  • In particular, these are called Kirchhoff laws.

  • A couple of Kirchhoff laws are, the current flowing in to a point should equal the current flowing out of a point.

  • So let's look at the current flowing in to here.

  • 36 is flowing in, 2 is flowing in, so we've got 38 in total flowing in.

  • And flowing out, we've got 33 and a 5

  • So it, they are equal.

  • Another thing we can learn from Kirchhoff laws

  • if you've got a circuit, like this circuit here,

  • a circuit should make zero - I'll show you what I mean.

  • So if I'm going from here, you've got a 9, we've got a 7, which makes 16

  • and then I go in the opposite direction here, so this is a minus 16

  • So all together, the whole circuit is a zero.

  • Easy!

  • Yeah, so all this maths suddenly you can just borrow

  • and they could solve the squared square problem.

  • A couple of things they solved on the way,

  • they showed that the smallest squared rectangle just uses 9 squares,

  • and there's two of them actually. That's one of them.

  • That's the smallest squared rectangle, meaning it just uses 9 squares

  • And the other one was that one that was done before then.

  • Remember there were two that was done previously, one of those is a small squared rectangle.

  • They showed that there are no cubed cubes. And that's kind of disappointing.

  • You can't make a cube out of cubes of different sizes.

  • So using these networks, they could find what made valid squared rectangles

  • and what are not valid squared, without having to draw out the squared rectangles.

  • So they started to build up a catalog.

  • One of their ideas was, if they could find a squared rectangle

  • and another squared rectangle that had the same dimensions, they could do it like this.

  • A squared rectangle like that

  • and another one which had the same dimensions

  • but using completely different squares inside.

  • So, they're mutually exclusive but they're the same size.

  • And then you could join them together like this.

  • This would be a square up here, this would be a square here,

  • and it would make a squared square automatically.

  • Or, if you can't do that--

  • Wouldn't it have duplicates in it, though?

  • So you want to avoid duplicates.

  • So you want two squared rectangles which have no squares in common.

  • That's what they were looking for.

  • Or maybe they could have one square in common if it was a corner square

  • because you could do this.

  • You could have a rectangle like this,

  • another one that's meant to be exactly the same dimension

  • that just share one corner, and that would make a squared square as well.

  • So that was their plan, and it worked.

  • And they finally found one. The first squared square!

  • It was massive, the first one they found. It used 55 squares.

  • And then they started to improve it. They wanted to find smaller ones.

  • They found one that used 38.

  • Bill Tutte found one that used 26, and these were getting better.

  • And the question was, what's the smallest squared square?

  • Now, that is something they struggled with

  • because it became too hard a problem.

  • And this was a problem that was then left open for another 38 years

  • until the 1970s, until computers came along.

  • And in using computers, they could use these students' methods that they had done,

  • they could use those methods with a computer and began searching for squared squares,

  • and they found the smallest uses 21 squares.

  • It's definitely the smallest. It's unique. There's only one squared square that uses 21.

  • This is it.

  • There are none that use fewer squares.

  • And then you can look at many that uses more than 21 squares.

  • But those students, all of them became mathematicians.

  • Well, one of them became a civil servant but he was still a maths fan.

  • I think that was Roland Brooks.

  • But the other three became mathematicians, including Bill Tutte, who started off as a chemistry student.

  • He swapped over to maths.

  • He then became a Bletchley Park code breaker in World War II.

  • A very important job.

  • And became a proper mathematician after the war.

  • And he was a pioneer in graph theory, which is the maths of networks,

  • and he credits this problem as his training in graph theory.

  • Okay, so, we've seen this one which is the smallest squared square.

  • The dimensions of it is 112 units by 112 units.

  • There is actually smaller squares if we just talked about dimensions

  • but they use more squares, and they're not unique either.

  • There are three known squares that are that size. They are smaller dimensions but they use more squares.

  • One thing I found that was interesting. Theoretically, we can do this.

  • One squared plus two squared plus three squared plus four squared,

  • imagine consecutive squares,

  • going up to ... 24 squared

  • is equal to

  • a big number, I'm not even sure what it is, well, I can tell you what it is,

  • it's 70 squared

  • which means that it is theoretically possible to make a square which is 70 by 70

  • made out of consecutive squares, one, two, three, four

  • but it can't be done.

  • It's so tempting. The maths looks like it can be done, but geometrically they don't fit together.

  • It's SO disappointing. I know, it'd be like a magic squared square perfect super perfect.

  • And it's so disappointing it doesn't exist.

  • Or does it?

  • No it doesn't.

  • Oh.

  • Another favorite squared square is this one.

  • I say it's favorite because this one was actually found before the Trinity students found theirs.

  • They were scooped.

  • Everyone else is like, not a very nice square, is it? Should we tell Matt? He seems so excited about it.

  • You know what I am excited about it!

So this thing is actually the logo of Trinity Maths Society.

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二乗の正方形 - Numberphile (Squared Squares - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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