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!