There's been an advance in a sixty-year-old
Math's problem, a maths problem which was
Related to the four color theorem
Which is something I've mentioned before, perhaps I should do a quick recap of what the four color theorem was.
It's a similar sort of thing really. Before color theorem said, imagine you had a map
Right, and I'm just going to make up some sort of map here. And the idea was, I want to color in this map
So that neighboring countries are not using the same color
So if I color this in, it might be kind of this bit here, the idea being that any map can be done
Using four colors or fewer, or in other words, there aren't any maps that need five colors. So that's true
That's a fact right, that was proven in the 1970s. But this problem that I'm going to talk about today, comes from the
1950s and it's not something that's been solved yet. It's called the Hadwiger-Nelson problem, catchy name
And I'm going to try to quote you what the problem is
How many colors are needed to color the plane?
So that no two points at unit distance are the same color. That's very formal
So let me just break that down. What it really means is imagine,
I'm somewhere on this map here. So there I am and I'm going to go for walk
I'm gonna take a step and my step is going to be one unit
So these might not be countries anymore because I'm changing the scale of this. So we're taking a walk and I want to go
One unit. Maybe that's one unit. Whatever that means and then take another step and I'll take one unit step
But every time I take a step, I want to step on to a different color. I want the color to change each time
I step so I'm gonna go for a walk. I want to change the color change the color, okay change the color again
I could step back to here, change the color again
Change the color again. I could go over here, but then for example
I couldn't walk this way. Should I take a one unit step over here, I wouldn't be changing the color
So the question is, is there a way to color the plane. So imagine this as an infinite map
Is there a way to color it so that I can take a walk always changing the color?
Can I do it so I can take any walk?
So any walk that I care to do, I will always be changing the color with each step I take, and that's the question
brady: No matter what direction you go. No matter what direction you pick no matter what walk that I choose to do
Every step lands on a different color. Well for starts
There is a solution to this, as seven colors work
We can do it with seven colors, and we can prove that really really quickly. You take the infinite plane
We're going to tile it with hexagons. So I've got now a little unit of hexagons
I got seven hexagons there and I've used seven colors and now I'm just going to repeat this over and over
So I'm going to repeat this
So I'm creating a pattern of colors with my hexagons.
Now All I need to do to solve this problem, is imagine the diameter of those hexagons was less than one
Slightly less than one, that means that every step I take would be on to a different hexagon
So I could start here, and every step I take would have to be on a different hexagon, which is a different color
I just add one thing though. We wouldn't want to step from this purple hexagon on to the next purple hexagon
which means I want this distance to be greater than one, which means the diameter of your hexagons needs to be less than 1,
and greater than .76
Fine. That's all you need so take the hexagons to be .9 and
The problem is solved you can walk around that hexagon
Infinite plane and change the color with each step
Brady: and even if I wanted to step on one of the same color, couldn't do it
Even if you wanted to, even if you tried you can't because those hexagons are chosen so that you can't step onto
The color that you're on. Brady: Cool. What's the problem? Exactly, What's the problem?
So we definitely have a solution, and it can be done for seven
So the question is can it be done for six colors, or can we do it with five colors?
Four, three, two, right?
How many colors, what's the fewest colors we can do this with?
So we can eliminate some of the answers really quickly. You can't do it with two, I'll show you that
So imagine we try to do it with two, so all I'm going to draw out is the walk
I'm just going to draw a walk. So I'm walking in a triangle an equilateral triangle. now, let's try and color this walk
So say I start in the green region, and then take a step and I want it to change color
So now I've stepped onto a blue region then I want to take another step, and I want to change color now
I've only got two colors available
So I'm gonna have to step onto a green
Region here. But that means if I want to step back to where I started, and there's no reason why I can't do that
I'm not changing the color
I'm stepping from green to green, that's not allowed. this is an example of a walk that can't be done with two colors
It can be done with three colors, but it can't be done with two. What about three?
Well again, there's an example it shows you can't do it with three
So same sort of idea. Here's a specific example, every step there in that shape is one unit. So you're walking around that path
Great, but can I do this with three colors? Let's start with green and then I move on to the next step
Well, it might be blue. Now, I want to step here. This has to be a different color. So let's have a different color
let's have red, now if I want to step on to the next one
Where it can't be red
So let's make it... oh it can't be green as well because I don't want to step back to the green
So it has to be blue. So that one's pretty okay. Now what if I take another step?
I think I've got a choice here.
I now think I can choose my color
I'm gonna choose green here for the middle, now for this if I take one step up, oh it can't be green
It can't be blue. So it's red. And if I step down to here, it can't be blue, green or red
So I need a fourth color. I can't do it with three.
It can be done with four and then, that's where we got to. 60 years ago
That's as far as we got, we showed that it can be done with seven
It can't be done with two, it can't be done with three
So maybe it could be done with four, five, six, and seven we know is definitely true. And that's where we were
60 years, until recently.
So in 2018 an amateur mathematician found a walk, an example of a walk that can't be done with four
Which means it can only be done with five, which means four is not a solution. So we've eliminated four colors from this problem
So I call him an amateur mathematician. I think that's a bit of a disservice
This is a person called Aubrey de Grey who is a biologist who's famous in his own right in his own field
He works in anti-aging, he's a very interesting guy
Himself, but he does maths for fun. And this was a problem he was trying to do, for fun
And so he came up with a walk that can't be done with four colours and what he did is, he took
This example, it's got a name
It's called the Moser spindles, strange name, and he made copies of that which he fused together
So he fused together lots of copies of this, and he created this monster, this monster Network
20,000 points
Huge thing, and then he got a computer to show
that
Four colours are not enough. It can't be coloured in with four, and it can be coloured in with five
Fantastic, we eliminate four
For the first time in sixty years. He then improved that, he wanted to find a smaller example
Smaller than this monster and he got an example that's I think
1581 points so that's much better right so a much smaller example, but it's the same thing
It's an example that can't be colored with four colours
Can be done with five
so then he published this and
Mathematicians got stuck in they said well, can we do better? Can we find a smaller one the smallest we found so far is
This one here. So this is a walk found by Martin hooli
Who we have mentioned before, we talked about this guy and he colored in the Pythagorean triples
We did a video about
Pythagorean triples and coloring them red and blue, it's a similar sort of problem and he found this example
It's a walk with 533 points that needs five colors
So we've got red, green, blue and yellow
And then the fifth color is white, which is used only once, right there in the center
This episode has been brought to you by brilliant, the website that's full of
Questions and quizzes and puzzles and courses and all sorts of things to make you smarter
You're seeing some examples on the screen right now
They have all sorts of new things every day
But they're not just dumping stuff on the website, all sorts of weird conundrums for you to solve
Everything on brilliant has been designed with a purpose in mind, and that's to really make the people using it
Smarter, to change the way you think, to make you a bit of problem solver
It's really about enriching the people who use brilliant. Not just giving them questions and then marks out of 10
There's loads of stuff on the brilliant site You can look at for free,
So go ahead and do that.
If you sign up for their premium subscription,
Which gives you access to everything, You can get 20% off that by going to brilliant.org/numberphile .
20% off and they know you came from here. Now if I've done my job correctly,
The URL should be down below me right now, and down in the video description.