# Lesson video

In progress...

Hi, I'm Miss Kidd-Rossiter and I'm going to be taking today's lesson on the four colour map theorem, which until recently was an unsolved problem in mathematics.

For today's lesson you're going to need a pen and something to write on and you're also going to need at least four colours, maybe more.

So, if you need to pause the video now to get any of that stuff, then please do, if not let's get going! So, we're going to start today's lesson with a try this activity.

Binh is colouring this map of the US, she applies the below condition, "No bordering states can have the same colour." So, for example, if I colour this one here red, then this one and this one cannot be red, nor can this one or this one.

Okay, but this one here could be red, because it's not bordering the one that already is coloured red.

So, pause the video now and have a go at this task, if you've done it, find another map and see how many colours you need to use this time, off you go.

Excellent, what did you find? You should've found that for any map you tried, the maximum number of colours that you needed was four, so that is the four colour map theorem.

The four colour map theorem states that any map can be coloured using a maximum of four colours, such that neighbouring countries or states or regions are coloured different colours.

So, this is a really exciting problem that went unsolved for hundreds of years and then it was finally solved in the 1970s.

We're going to look at one way that they thought about it today.

Now in order to prove the four colour map theorem, you either have to prove that all maps in existence only need four colours or less, which is going to be really tricky, or you need to find one map that needs five or more colours.

So, people did this, they researched, they found all the different maps in the world and they found that the maximum was four colours.

So then they had too get a bit creative and they came up with their own maps, so maybe not maps of the world, but just random maps to see whether it would still hold true, so here's a couple of examples.

There's one, there's another and it still holds that the maximum colours needed is four.

So draw yourself a map now, maybe something like this, that's my dodgy drawing, so I apologise and see whether it's the case.

So pause the video now and have a go at that.

Excellent, you should've seen that something along those lines does still need four colours, so we still haven't proven it yet, 'cause we can't possibly draw every single map that could ever exist and we still haven't found one that needs five or more colours, so we've still not proven it.

So, what mathematicians did was they used something called network, so let's look at this example together.

This is an example of a map that needs four colours, so for example the centre could be red, the top could be blue, the right hand side could be green and the left hand side could be purple.

Now, instead of colouring in the full sections, mathematicians thought, "Well, what if I just coloured in a dot and then what about if I link these dots, so any bordering country would be linked with a straight line?" So, this country borders this country, my blue country and my purple country.

This blue country also borders my green country.

My purple country borders my green country.

My blue country borders my red country.

My purple country borders my red country and my green country borders my red country.

So, I've connected with a straight line any countries that are bordering.

So the question now becomes, I've got this network here, which I'm going to try and draw a little bit more neatly for you.

This is a network, so we've made the problem a little bit more abstract, which for mathematicians means often that problems become a little bit easier to solve, because they're now talking about something abstract, rather than something like a map that's real.

So the problem now becomes is it possible to colour every network using a maximum of four colours? So this is what you're now going to have a go at, you're going to have a go at drawing some networks for yourself, so pause the video now, navigate to the independent task and when you're ready to go through some answers, resume the video, good luck! Excellent, how did you do? So, let's have a think about it, if we've got Country A bordering no other countries, then we're just going to have Country A on its own, aren't we? If we've got Country A bordering two other countries, well there's different ways to do this, so if you drew it slightly differently to me, that's okay.

I'm going to draw it in kind of a linear fashion, so Country A here is bordering another country here and another country here.

Obviously, could've been down here, could've been up there, those two could've been connected as well, that's fine.

Country A bordering three other countries, so you'd have Country A and then it's got to border three other countries, so I could draw my network something like this.

Four other countries, again it's going to be very similar to my three network, but instead of bordering three countries, it's bordering four.

Again, yours might've looked different and that's absolutely fine, where I've got A in the middle.

Or it could've been bordering five other countries, so I could have something like this, where it's bordering one, two, three, four, five countries.

Now interestingly, with the exception of Country A bordering one other country, which I didn't ask you to draw, which would be something like this, then every single map that's in existence will have one of these networks on it.

Draw a network to represent Great Britain and Northern Ireland.

Well this is quite an interesting one I think, because Northern Ireland clearly is not connected and then we've got England, Scotland and Wales or something like that.

What's the most possible complex network that you represented? Well, I'd really enjoy seeing those if you managed to draw one.

So, for the explore task then, Antoni is thinking about the four colour map theorem and he says, "All maps can make a network.

Therefore, all networks could make a map." Do you agree with Antoni? Can you find some examples that either show or don't show this to be true? Pause the video now and have a go at this task.

Excellent, well done, what did you think? I'm not going to go through too many answers here, because I don't want to limit your thinking and I really want you to go away and think about this some more and maybe puzzle on it or do your own research.

So, it's not true, not all networks could make a map, for example if I had a network that looked something like this, this would not make a map and I'm going to leave you to think about why, but there are lots of different networks that couldn't make a map, but all maps can make a network, so I'm going to leave that with you to think about.

If you've enjoyed this lesson and you've drawn some really lovely networks, then please ask your parent or carer to share your work on Twitter tagging @OakNational and #LearnwithOak.