Today we have a puzzle, so I'm gonna give you a ll the numbers from 1 to 15.

I suspect you're gonna shorten this dramatically in the challenges.

Can you take numbers from 1 to 15?

And can you rearrange them into a new warder?

Such that any two adjacent numbers add to give you a square number.

So, for example, you could put one.

Let's give it a go.

We're going to try.

We can put one there and then next to it, we could put three.

You go.

Okay.

That's pretty good.

Because of the three is for that's a square number.

I could chuck six here because a group of six is nine.

That's it.

Square number.

Oh, why am I gonna do, uh what can I add to that?

To make it a square number 10 work, right.

And then someone So we'll give you a second if you want to try it.

So we have to use all of them.

Yeah, fuse all of them in one long line.

Any order you want?

Any two that are adjacent Have to add to a square number way.

Welcome back.

Thank you for trying the puzzle if you got it, could it could be days later.

My goodness, how you have aged.

But, you know, you gave it a go.

You saying Brady you didn't think it was possible?

It feels unlikely, you know, But it can be done.

And I gave you a full start.

It doesn't work if you have these numbers in this order.

So I deliberately and very meanly gave you a, um starting boy, that does not work.

Why would you do that?

Because I'm a jerk.

I want people to work it out for themselves.

But why doesn't it work?

Because this doesn't.

This fulfills the requirement.

Every two of these do add to give you a square number.

But then you just you can't join up the rest of them, right?

You run out of the room.

So the way I went about solving this when I first did, it was I thought, OK, I'm gonna draw a ll the possible links because you start looking at this and you're like, Okay, what?

10 works with 15.

I get 25 but 10 also works was six.

And so, actually, I've got other options for how I put them in the same order.

And some of them have more possible links or neighbors.

They could go with them.

You can actually put in a long line.

So I was, like, Right.

Right.

I'm gonna do this.

There's wine, but one over here.

You live in order.

That's one.

Okay, too.

One and two.

Don't add to give you a square numbers.

I'm gonna put two over there.

Out of the way.

Right?

Three.

Nothing I can add.

Think toe one, because I guess before that is a square number now, Ford doesn't work with anything so far, but does work with five foot five over there.

Six works with three.

So you can see if I only had the numbers from 1 to 6, it wouldn't be possible to order them.

So every single payer adds to a square number seven into the mix.

Seven goes with tooth.

That gives you 98 Adds on to one.

Gives you 99 fits in off seven because I get 16.

10 links into 6 16 11 and still, Now, I've got these serious I could use.

In fact, if this looks familiar that there is the original Siri's I gave 8136 10 is one possible ordering, but there's no way so far to link it up with the rest of them.

Will keep filling these in 12 to 4 13 will also linked to 12 because I could get 25 but we got to get 16 so that links to half.

Okay, so now we're trying to join this all together.

That's put in 14 and 14 is the first number where now everything holds together cause 14 25 in 16.

We've now got a connected network or graph whatever you wanna call it, but there's no path.

Right said this would work, But you've missed one in eight.

That would work, but you've missed days.

So at 14 it's all joined together because work 15 is the one you need to hold all together because 15 fills in this gap.

One.

There's 25 16 and now there's our solution.

81 15 10 63 13 12 And then away along that tail.

And so that is the order.

You've gotta put those numbers so that every single pair between them adds to a square number, and I annoyingly I picked this bit here of this curve, knowing that that is not a valid part off the path that goes all the way through this square sums network.

That makes 15 pretty special number that makes between a prisoner's number.

It's the first number for which you can order the numbers.

So everything.

So you you're right, Brady.

To think that doesn't feel likely.

And it's because I picked 15 because up until then doesn't work.

I couldn't pick 16 actually, because 16 you can add on to nine to get 25.

Okay, well, 17.

Okay, so 17 goes over there.

And so now you think that you guys put in this apart 16 all the way through to 17.

You have a perfectly valid solution.

But why stop there?

Why not do it from 1 to 18?

18 fits in.

Over here.

There's 18 0 now, now we've got a problem, because if you put 18 over there, there's no longer a way to go through the whole network going through every single number once and once.

Only because we got a tail here sticking off when we got little tail sticking here.

And if we start here, go.

That way we could never get back to 18.

And if we start here, we could never get back for these guys.

So we've broken it.

So it works for all the numbers from 15 to 17 and then 18 breaks it.

But what if it works again after that?

So I would call this a graph in mathematics and graph theory is what we're kind of using to solve this puzzle.

And we're looking for a path.

A path is any kind of journey, which doesn't go through the same Vertex more than once.

And if we have a part that goes through every single Vertex, then it's called a Hamiltonian path.

And so to solve this puzzle, you're looking for the Hamiltonian path, which goes all the way through the graph.

So let's try adding on a few more numbers and see if we can get to work for a bigger value.

If I had 19 over here, that doesn't solve our two tails problem over there.

A CZ.

Long as this exists, there are no solutions.

20 doesn't help.

Oh, actually, let's now solved one tael problem so we can come in here.

All right, but if we come in here.

We either go around that Well, we go around that way because we hadn't have to come off into the rest of this.

And because of this bottleneck here and here, you do One way.

If you look around, you can't get out.

So that's still not solvable.

We need another way out of this loop before we can do it.

21 start to get a crowd of 20 1 Could go down here because it links 15 which I'm gonna run under there 24 So these are crossing.

But that's not a new Vertex.

Just drawing the links.

22 links up 3 to 14.

Does that get us out of our problem?

Because we can come through here and out.

But then we can't get I still still not doable.

It can be done once we get to 23.

So if we put 23 in, put it down there.

Now it is possible to trace all the way through this network show.

Shall we give it a girl?

I have spoken too soon.

I can We can work this out so I can see if we can do this.

You go start from 18.

Because that's like a little tail over here.

If we come in here and we go away around here, I suspect we need to mop up these Because if we leave any of these behind, we're gonna be in troubles.

That's come down.

Take those Get into here.

Come up here.

Oh, yeah, You have a good.

Then we go across 2115 10 6 1917 81 And there you go.

So it's possible to find a Hamiltonian path on the graph as soon as we add node 23.

Very sadly, when we had No.

24 it breaks again.

24 joins 12 12 24 goes there and then one thanks to 12th and now it's broken again.

But when you add 25 it's fixed again and it then works for 26.

That works with 27 it works away up.

So we think as far as we're aware from 20 five onwards, it works for every possible value.

You can somehow find a path through this network.

Well, it's not over yet, People.

If you'd like to see even bigger graphs with more nodes and their Hamiltonian paths.

Have a look at our extra footage over on number file to.

You'll also hear more stories from Matt, including how he made a bit of a Parker Square of this issue when he was doing his book, and you'll hear from a very special guest.

Are put links down in the description and in all the usual places.

Thanks for watching.