Staple off recreational mathematics.

But it's always the hex of flex again that you see one based on the hex against got six edges and so I have some good bye Hearts has made a whole series of videos about these things.

I'm going to go for the forgotten flex again The Tetra flex again one with four edges And to make it, we're gonna use a square of paper.

It's the first step is to fold it in half whichever way you want them both the same.

And one of the reasons why I love the Tetra Flex again is you don't need any gluten.

If you're making any of the hex of flex against, you have to get some glue and stick bits together, whereas this one is just folding and cutting.

So the first half and half has split it into four sections.

Long ways.

I'm not gonna fold it up this way.

And if I make the creases really like good creases because they're going to kind of be used as hinges later, I should end up with.

There we go.

It's a piece of paper now split into 16 smaller squares.

I'm going to remove the center four squares.

I'm gonna keep this ring of 12 squares intact.

I'm gonna take these middle one's out.

You could do that by stabbing the citizen, just cutting around.

But as we learned from case tickles in her fault and cut their own video, you can do anything you want in a single cot.

If you're a bit clever about how you fold it.

So one cut down there.

Come on.

And we get there's no bad.

Okay, there's the center.

Four squares gone.

We can get rid of those.

And now we got the 12 on the outside.

So this is now gonna be three easy steps and one difficult step.

So we start on the left and we go around clockwise.

First step is easy.

Take the four on the outside for them up and over.

Everything should line up.

Second step has become around.

Take three of the top fold them up in Nova and they come down like that Next step as we keep coming around, take these three for the muffin over and that is the end of the easy steps.

The only difficult step is when you take up the bottom ones.

You want to lift these up as if we did it before that first fault?

Because it has to be some metrics.

There's no beginning around.

So where you've got to do is move the original fold out of the way without ruining it.

Then lift this one up and make sure the bit next to it goes underneath.

So we kind of take all of this bit next to it and just feed through into its underneath and the rest should follow and just kind of snap into place.

And they are there is your four edged flex again, The Tetra flex again.

So what's amazing about the tent reflects again is it's got a lot of faces.

So we got face one on the front here, which I'm gonna number with ones.

And then if we fold it over, I guess ones on the front, we got twos on the backs of face one and face too.

So you got twos on that face.

Okay, you got ones on that face.

And then on the other side of the ones we've got, what's blank.

So we label that with three.

So that's face three.

This is a shit with at least three faces.

We got face three there.

We got face one there.

We got face to there.

And what happens is every time you fold it closed one way so unfolding with one's closed with the twos on the outside, I can open up on the opposite side so it closes this way.

But then I open it the other way around and I get a new face.

And so I'm gonna label that one.

Where have I gone so far?

One.

There I got through their list.

Label this one four, but and then let's see if we fold this this way.

That way, this way.

There's another one.

That's five.

We're looking for six faces because if we think of the original shape we made was marrying a 12 12 on the front.

There's 12 on the back, so I've got 24 of these little squares.

I'm using up four for each one.

So if I divide that by four, I get six.

So there are enough squares can make six whether or not it does have a slightly different question.

Four.

Is there?

Are there just this six.

Thank you for that label is when he has six.

And if we actually wanna check that, we've got all of them face to in face one folded in face.

Three.

Okay, We could pull the whole thing apart.

So if we pumped back out again, disassemble it.

There we go.

Okay, So there's or the fate was number every single square at some point to make all of them.

There are two different ways to put this back together.

And if you do it the wrong way, you will get the mirror image.

Kind of what we started with.

The numbers won't line up again.

But if I carefully put it back together the same way we folded originally we're back where we started.

Go.

And when you make one of these kind of fun, you play with it for a while.

If you've not made one before, it takes a bit to work out where all the faces are.

Obviously, you don't have to just put numbers.

I'm not very creative.

If you were one of these doodling mathematicians, you would draw pictures or, you know, the artistic.

I've just numbered it, which is a bit sad.

I'm sorry.

It is number file.

It's not doodle file, is it?

So I've numbered it, so you could be as creative as you want, and you could spend a while trying to find them.

But if you are in the numbers after, why you, like, hang on, I know they're only six faces, but what combinations of faces can I get the same time?

Obviously, I can get one opposite, too, because when I started with, I can get three opposite one.

Can I get three opposite?

Four?

Can I get five opposites like what combinations of faces can you see at the same time?

And that was the first thing I started wondering where I take one of these, and for awhile I thought I knew you could get combinations of faces, but there's always one of the two.

You never lose his original two faces, but not now.

Here's a five opposite of four, so you can you can get reasonably lost, right?

The original two faces are long gone.

We have two new faces on the outside, and I when I heard I was like, I wanna work out the logic behind this.

What is it?

Which means I can get different pairs and which one's a possible and how do I go between them?

And so I drew a diagram.

Okay.

Diagram.

We're going to use colors because that's how we were all here.

This is a close to artistic as we get.

So, uh, what face is definitely possible.

So I first did this actually didn't do colors.

Why'd you kind of lazy?

I was like, Okay, I'm gonna label that is being one too.

There's not a fraction 1/2.

It's a one opposite are using my own notation.

And then I went OK, I could go from that to 1313 of my diagram.

And then I realized, like, wait a minute.

One face always stays there whenever you flex, which everyone goes on, the outside is still there.

When you open it up, you change one of the time.

So when you go from 13 to 1 to the one doesn't change was like, Well, I can kind of imagine those ones being linked and so I can go from 13 toe one too.

If I keep the three, I can get 236 So here somewhere, I've got six and three and I've kind of lead those up like that, right?

And I started drawing the network to try and work it out.

I'm gonna show you the network I ended up with cause I'm particularly pleased with it.

First of all, I'm gonna put this back to its starting state.

One's gonna be red.

There's there's one and one can be opposite.

Let's do light Blue can be like to actually wear this.

True writes that one is opposite to Okay, so let's see, What else can I get opposite one?

Well, I can go from one that could be opposite free.

Okay.

So I could have won three.

So I'm gonna put 13 down here.

That could be purple or one can be opposite.

It's actually it's horizontal and vertical.

So if I fold it vertical fold, you get three.

If I fold a horizontal fold, you get the other ones.

That's five in there.

So, five, you're going to be an orangey color, and we'll put five up here.

So what I've done is I have run or the ones in a line down here.

So this is a ll the ones you get which have a one of them.

And so I'm still kind of got the money the size uses going okay with you two next to if we fold it horizontally, its opposite six.

So that's two opposite six that way.

So that's six with two.

And then if I hadn't folded it horizontal, I folded vertically to his opposite.

Four.

So four is gonna be done, Blues, since that is gonna be down here Now I have lined up a ll the faces with two.

And actually, we've got another line going this way.

So there are five combinations were found first and how you can go between them.

But there are other ones.

So if I go for two, let's go back over here.

There's 12 in the middle again.

Let's go out to one of these.

Let's go out to one opposite five sort of here.

If I keep five there, I can get five opposite four by fours over here.

So I can actually do is extend the five line down here, and then I could extend the fall line up here and now I've got 54 and that predicts I can pop down here to 4 to 5 preemptively locked that in.

But if I keep for on the outside, close it.

Sure enough, there it is for two.

So let's drop me down there and I predict on the opposite side over here.

If we extend six down there and we extend three up there, that predicts whenever, 63 of theirs.

Sure enough, let's get over that side.

Where should we go?

Somewhere for two.

Let's go back to 21 Go back to one.

Done that.

Let's go, Let's go down.

We got one free.

Which way's that?

There's one of three, and then we should get a pop up there to be three opposite six.

Sure enough, there it is.

There's three and six, and then that's it.

These other ones, they don't fill in, but I was able to create.

It's kind of a graph, but I've put some structure into it.

I've had straight lines, so I got six different colored lines.

And whenever two of those lines crossed that combination of faces as possible, so originally I just kind of sketched them all over the place.

And then I realized there were some logic underneath them and I was able to then arrange it to show the structure behind what combinations off faces are possible.

And that pleased me immensely.

So I've been throwing some names of Mexicans around.

The way it works is you've always got the number off faces followed by the number of edges, followed by flexible.

So this has got four edges.

So this is a tetra flex ago and because I've got six faces, it's a hex.

A tetra flex again.

Six faces for ages flex again.

So the ones you see online all the time Ahhh exa flecks of guns because they're made from hexagon.

They got six edges and then have many faces.

The easy one, the one you till the time is the try Hex of flex against.

You got three faces.

This one gives you six of the box with a lot of work, you can make a hex a hex of flex again.

So six edges six faces.

My goodness, it's a real pain.

So the try hex.

It takes a bit of work making a chain of nine equilateral triangles, technically, 10 because you want to glue.

Once you got those 10 you're gonna fold him and gloomy.

And you get three faces If you make a chain of 19 you glad for triangles?

You gotta fold it.

You gotta fold it again.

You gotta glue it right.

You're gonna realize you did that wrong.

Pull it apart.

In my experience and a lot of effort later, you get six faces.

Where is this?

No gluten, no messing around.

Cut out a square, cut out a smaller square bit of folding.