Name: Squared Squares - Numberphile
Uploaded: 2020-04-05T02:46:37.000Z
Duration: 10 min 13 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Adverbs of manner

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.

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

yet they fit together to make one big square

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

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.

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

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

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.

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.

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

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.

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

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

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

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

And when they did that, they recognized it.

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

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.

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.

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.

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.

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Squared Squares - Numberphile

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