Numbers and numerals

Axioms and arrays

Factors and multiples

Order of operations

Positive and negative numbers

Expressions, equations and inequalities

Angles

Classifying 2-D shapes

Constructing triangles and quadrilaterals

Coordinates

Area of 2-D shapes

Transforming 2-D figures

Prime factor decomposition

Conceptualising and comparing fractions

Manipulating and calculating with fractions

Ratio

Percentages

Different number systems

Sequences

Forming and solving equations

Forming and solving inequalities

Linear graphs

Accuracy and estimation

Algebra and problem solving

Ratio (8.6a)

Ratio, real life graphs, and rates of change

Direct and indirect proportion

Univariate data

Bivariate data

Famous maths problems

Angles and parallel lines

Angles in polygons

Bearings

Volume and surface area of prisms

Area review

Circles

FDP review (9.1a)

Probability

Sets and Venn Diagrams

Solving linear simultaneous equations algebraically

Solving linear simultaneous equations graphically

Angle review (9.5a)

Constructions, congruence, and loci

Pythagoras's theorem

Ratio review (9.7a)

Similarity and enlargement

Surds and trigonometry

Quadratic expressions (9.9)

Quadratic equations (9.10)

Indices and standard form

Growth and decay

Finance

Maths

Y9 Maths

Key Stage 3

Year 9

UNIT

In this lesson, we will think about a famous, long-standing, maths problem that has been recently solved known as the four colour theorem.

Four colour map theorem

﻿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. Really enjoyed talking with you about this interesting subject and I hope you've enjoyed it too. Don't forget to go and take the end of lesson quiz so that you can show me what you've learnt and hopefully I'll see you again soon, bye!

In this lesson, we will learn about a famous unsolved maths problem called the twin prime conjecture.

The twin prime conjecture

﻿Hi I'm Miss Kidd-Rossiter. And I'm going to be taking today's lesson on the twin prime conjecture. It's a famous unsolved problem in maths. Before we get started, you're going to need something to write with and something to write on. If you need to pause the video now to get that sorted, then please do. If not, let's get going. So before I tell you what the twin prime conjecture is, I'd like you to try this activity. You put Yasmin on your screen, and she says, I'm going to investigate the difference between consecutive prime numbers up to a hundred. What's the smallest difference Yasmin could have found? what's the largest difference Yasmin could have found? And what do you notice about the differences? Pause the video now and have a go at this task. How did you do on that? What was the smallest difference Yasmin could have found? Well, that exactly the smallest difference is the difference between the first two prime numbers, which is one. So the first prime is two and the second prime is three. And their difference there is one. We call this the prime gap. What was the largest difference Yasmin could have found? Well, if you look to all of the consecutive primes up to a hundred, there were lots of gaps of six, but the largest gap was actually of eight, which was between 97 and 89, which gave us a prime gap of eight. What did you notice about the differences? We should have noticed that apart from the first prime gap, apart from the first difference, all the differences were even numbers. Why is that pause the video. Think about it. Exactly. So this counting coup whenever your finding the difference between two prime numbers, you're always doing an odd number, take away another odd number, and that gives you an even number. So the first prime gap, which is the difference between an odd and an even number will give us an odd answer, but then all the other prime gaps going forward will always be even. So the twin prime conjecture States, this, there are infinitely many primes, such that add two to that prime number. And that number is also prime. So for example, we know that 11 is prime. If we add two to that. So this is if P equals 11, then p plus two would be excellent 13. So it holds true for that. What about if a P is 17 then P plus two would be 19? So that is also true. Now it's not saying that the difference between every pair of prime numbers is two, but it's saying that there are infinitely many prime gaps of two for going on forever. So if this is true, then there would be that if it is false, then that would mean that at some point along the number line, that stops being prime gaps of two, and it increases to four or six, or who knows what it increases to now, as yet, it's not been proven that this is true, but it has been shown to hold true to numbers that have more than 380,000 decimal digits. So we can do that using computers and that's, what's been found. It's also true that they're so common, numbers greater than a hundred. You don't have to look very far numbers greater than the thousands. You don't have to look very far, not going to tell you those cause that's going to form part of your independent task. 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. How did you do on that? Did you find the first pair of twin primes greater than a hundred? You really don't have to look very far for these 101, 103. So the first two prime numbers greater than a hundred that have a gap of two, the first pair of twin primes greater than a thousand, 1,019 and 1,021, the first pair of twin Primes greater than 1 million, is there 1,000,037 and 1,000,039. So you see, we really don't have to look very far. That means that they're occurring very, very often greater than a billion, 1,000,000,007 and 1,000,000,009 and greater than 1000 billion. We get 1000,000,000,061 and 1000,000,000,063, which is incredible. It's mind blowing, isn't it? That we can go all the way up to these huge numbers and still find prime gaps too really easily. So looking at the explore task now, then Alphonse de Polignac in 1849, wrote this conjecture, "For any positive, even number n there are infinitely many prime gaps of size n" So this is the first time we actually see some sort of variation of the print of the twin prime conjecture written down. But it's thought to date back to maybe the ancient Greeks. So I want you now to think about how is the Polignac's conjecture different to the original twin prime conjecture that we've talked about. And can you find examples of prime pairs where n is less than or equal to 20, pause the video now and have a go at this. Excellent. So let's talk about it together then. So how is it different to the original twin prime conjecture? The original twin prime conjecture said that it had infinitely many prime gaps of two, and the Polignac's conjecture says that we have infinitely many prime gaps of size n where n is an even number. So that means infinitely many prime gaps of size four, six, eight, 20,200, 2000 and so on. So that's how it differs. And can you find examples of prime pairs where n is less than or equal to 20? Well, I'm going to leave that with you. Maybe you can find prime gaps where n is an even larger even number. So I will leave that with you, to think about, interestingly, quite recently, there's been a piece of research done in 2013 that did manage to prove something about prime numbers. And that proved that there are infinitely many prime gaps of an even number anywhere between two and 70 million. So it's not proven that there are infinitely many prime gaps of two or infinitely many prime gaps of 70 million, but have some even number between two and 70 million, it has been proven that those occur infinitely often. So this was a big breakthrough because it was the first time anything was proven to do with the twin prime conjecture. I really hope that you've enjoyed learning about the twin prime conjecture. I've really enjoyed teaching it to you and please, please go out there and research it some more. If it's of interest, we need more. Great mathematicians. Thank you again for watching today's lesson. Don't forget to go and take the end of lesson quiz and hopefully I'll see you again soon. Bye.

In this lesson, we will introduce the 3D coordinate planes, and solve problems involving 3D coordinates.

3D Coordinates

﻿Hello everyone, it's Mr. Millar here. In this lesson, we're going to be looking at 3D coordinates. So first of all, I hope that you are all doing well, and just in case you haven't watched any of my other videos online before, let me just briefly introduce myself. My name is Mr. Millar, I am a math teacher at a secondary school in central London and I am doing a series of two lessons for you. So the first one, this one, we are looking at 3D coordinates and the one after this, we are looking at a painted cube. They're both going to be lessons about 3D shapes and they're also going to be pretty challenging lessons so make sure that you are ready for that, make sure that you have your resilient hat on because these will be tricky but also hopefully some really interesting problems. So let's get started with the try this task. As you can see, we have got a 3 by 3 by 3 cube and each of the dots that you can see on the cube has an X-coordinate, a Y-coordinate, and a Z-coordinate. Now, you've probably all seen before, I'm sure you've all seen before, X and Y-coordinates, so you know that X-coordinates go across and Y-coordinates go up. But in this 3D cube, we have a Z-coordinate as well and you can see from the diagram underneath the cube that the Z-coordinate goes into the page. So let's have a look at our first example. If we have a look at the red dot, we can see that the coordinates are one, three, one where the first one is the X-coordinate, the three is the Y-coordinate, and the second one is the Z-coordinate. So first of all, if we look at the X-coordinate, we can see that the X-axis is going this way so anything along this line here, for example, would be an X-coordinate of one. And we can imagine that the X-coordinate is also one for all of those points up there. The X-coordinate would be two for any of these blue points here so X-coordinate of two would be here and up here as well, so yeah. That is the X-coordinates. The Y-coordinates, I've just brought these up quickly, the Y-coordinate tells us how far up we are. We can see that if we were on this first level here, we've got a Y-coordinate of one. Second level is a Y-coordinate of two and this third level, anywhere on the top of this cube is going to be three. And we can see that this red dot is on that top level where the Y-coordinate is three. Finally, the Z-coordinate, well, we can see that the Z-coordinate looks at how far into the page we are. So we can see that at this red line here, we are one unit inside the page so the Z-coordinate goes that way. So we can see that the Z-coordinate is one. Okay, so that is how this works. What I've got now is three more coordinates for you to try. So here they come, you've got a yellow coordinate, a green coordinate, and a purple coordinate. So for each of these, you need to give the X, Y, and Z-coordinates for each of these three coordinates. So pause the video, four or five minutes, try to work out these three coordinates. Pause the video now. Okay great, so let's go through these nice and quickly. So first of all the yellow coordinate, well, the X-coordinate is going to be one, the Y-coordinate is two, and the Z-coordinate is zero because we're not going into the page at all, we're kind of still on the front face. The green coordinate, the X is two, the Y is three, and the Z is two because we're two into the page. Finally, the purple coordinate is going to have X-coordinate of three, it's going to have a Y-coordinate of one and a Z-coordinate of two as well. So, those are the three coordinates and on the next slide we're going to have a look at this in more detail. So when you're ready, let's move on to the connect task. Okay great, so here we've got the same 3D cube with three coordinates, X, Y, and Z, and this time what I want you to do is I want you to copy the cube as best as you can and then write down on the cube that you have written in each of these three coordinates. So for example, zero, three, zero. I know that the first one is X, and then Y, and then Z. So I'm looking for an X-coordinate of zero, so I know it's going to be one of these coordinates up here. The Y-coordinate is three and the Z-coordinate is zero and I can see that this point here is going to be my first coordinate, zero, three, zero. What you need to do is pause the video now to copy down the cube. And on the cube, mark on the three other points, and also down here below, write the coordinates of a point that you can't see on the grid. So a coordinate that is hidden within the cube. Pause the video now to do this, four or five minutes. Okay great, so let's go through these now. So the first one, one, one, zero. Well, that is going to be here. The X-coordinate is one, the Y-coordinate is one, the Z-coordinate is zero. The next one we'll do in green, three, zero, zero. So we want this one here, and also, it's worth mentioning that this one here is on the corner, the X is three, Y is zero, Z is zero. And the final one, we will use black for the final one, three, three, two. So our X is three, Y is three, and Z is two. Well, that is going to be this one here. Okay, and the final question, write down the coordinates of a point that you can't see on the grid. So have a think, if you haven't already, of a coordinate that you wouldn't be able to see on this cube. Okay, well there's actually a lot that you could've thought of. One idea is having a think about the point one, one, one. Because if you have a look at the purple point, which is one, one, zero. If you imagine the point one, one, one, well how is that going to be different, is that it's going to be a little bit inside the cube like that because it's going to be one in the Z direction which means that we won't be able to see it on the outside of the cube in the view that we currently have. Okay, we're now going to move on to the independent task where we're going to keep the same cube but the task is going to look very different. So when you're ready, let's move on. Okay, so somewhere in the grid a piece of treasure is hidden. So we are still looking at the same grid, and you are going to get a series of clues which will lead to the treasure. So the first clue is here, the X-coordinate plus the Z-coordinate equals Y. So we can write X plus Z equals Y. Now on the right-hand side, I have given for you all possible 64 coordinates where the treasure is hidden. So it's going to be in one of these coordinates and we're going to need to work out which. The first clue is that the X plus the Z must equal to Y. So what I'm going to do is I'm going to figure out which of these 64 coordinates it could be. So for example, if I look at the first one, zero, zero, zero, so I know that my X is zero, my Z is zero, and my Y is also zero, so does this satisfy X plus Z equals Y? Sorry that should be an equal sign. Yes, it does. Because zero plus zero does equal zero, so I could circle this first one, you might want to write down zero, zero, zero, because the treasure could be in that point. If I look at the next one, zero, zero, and one, well that is zero plus one equals zero, and that is not true, so I can cross off this second coordinate here because X plus Z does not equal Y. And I can keep on going, so I know that this one doesn't work, this one doesn't work, this one doesn't work as well, but when I get to my next one, zero, one, one, well, X plus Z is going to be zero plus one equals Y, which is one, and yes, I can have this coordinate because zero plus one does equal one. So, what I want you to do is I want you to go through all of these coordinates and see which ones you end up with before we move on to the next slide. So pause the video now and go through all of these coordinates to see which ones, X plus Z equals Y. Pause the video now. Okay, brilliant. So on the next slide, I will show you the coordinates that you should have left. Okay, so on the right-hand side I have put in a box the coordinates where X plus Z equals Y. So those are the ones that you should have had, how many have you got? We've got 10 possibilities now. So here's the final slide, the explore slide, and there are five more clues, clues two through six, that will get us the treasure eventually. So here are the five remaining clues. We can go through the first one together so you see how this works. So the treasure is not to be found at any of the corners. And if you think back to that three dimensional cube, you know, any of the points on a corner will have the X and the Y and the Z-coordinate either be zero or three. So we can cross off a few coordinates here. So we know it couldn't be this one here, that's going to be a corner one. It's not going to be this one down here and it's not going to be this one down here. We've done that one, so now we've got seven left. Now we can keep on going. So pause the video and see if you can find the correct answer with these remaining four clues. Okay great, so well done if you managed to work this out and let's just talk through the remaining clues. So clue three, the Z-coordinate is less than the Y-coordinate. Well, we can cross off this next one because the Z equals the Y-coordinate. Same thing with this one, we can cross it off because the Z-coordinate equals the Y-coordinate again. And are there any more that we can cross off? Well, we can't cross off the next one because the Z-coordinate is less than the Y-coordinate, and in fact, those are the only two that we can cross off. The next one, the answer contains exactly two prime numbers. So the prime numbers here that we could have are two or three. So we need two of the coordinates to be two or three. So we can cross of this one here, we can cross off this one here, and that is all. So we're left with three candidates. Clue number five, the X-coordinate is greater than the Z-coordinate. Well, we can cross off this one here because the X-coordinate is less than the Z-coordinate there. So we are left with two possibilities. And we can use the final clue to help us find out which one it is. So only one of the coordinates is a square number and the only square number that we can have is a one. So we can cross off the two, two, zero, because there are no square numbers there, so we are left with the coordinates two, three, one. And that is where our treasure is. Really amazing job if you managed to do this yourself. I hope you've enjoyed this lesson, I bet it was fun. Next time, we're going to be looking at a similar problem involving a painted cube. So really looking forward to doing that one. Thanks so much for watching and see you next time. Have a great day, bye.

In this lesson, we will look at the painted cube problem.

The Painted Cube

﻿Hello, everyone, is Mr. Millar here. In this lesson, we're going to be looking at the painted cube problem. So first of all, I hope that you're all doing well. And in the last lesson, we looked at the problem involving 3D coordinates. And we're keeping with the same theme in this lesson, where we're going to be looking at the painted cube problem. So anyway, let's have a look at what we're doing here. So a three by three by three cube is put together using smaller white pieces. The outside is painted red and left to dry. The large cube is then taken apart piece by piece, and the puzzle is to do with what the smaller pieces look like. Okay, so let's just make sure that we understand what's going on here. So, put very simply, we are taking this cube. We are painting all around the outsides in red. And then we are looking at or we're thinking about what the smaller cubes look like. So, let's get started here. So a couple of statements that the students are making. First of all, "I know that the cubes in the corner have blank faces painted." And the other student is saying, "I know how many cubes have zero faces painted." So if we take the first statement about the cubes in the corner, so for example, if I look at this cube which I am drawing around here, that would be a corner cube, how many faces of that cube would be painted? Well, of course, three. Because all around the outside would be painted. I can paint all different as well. It's all painted. So the cube in the corner has got three faces painted. Have a think, how many corner cubes would there be in total? How many cubes in the corner would there be in total? Well, for this cube, we are going to have eight corner cubes in total. And the reason for that is because we've got four cubes on the top row, just show you that very quickly. So yeah, we've got four cubes on the top row one, two, three and four, and then four cubes in the bottom row as well. So eight in total. Now the next one, how many cubes have zero faces painted? So have a think, how many cubes would have zero faces painted? What about that? Well, if you were thinking that there is one cube with zero faces painted, then really well done. Because there's only one. If you imagine looking into the inside of this cube, and this talk is all about visualising, which is not easy. But if you think about it... If you think about the inside of the cube is essentially the cube below what I'm shading now, then that cube is totally within the shape. It's got none of the faces on the outside. But that is the only cube in the three by three by three. So we know how many cubes have three faces painted, we know how many cubes have zero faces painted. So what about the other cubes in there as well? Well, that's opening looking at any connect slides. Okay, so for the connect task, we are still thinking about the same problem. But this time, we're thinking about the number of cubes, which have got one or two faces painted. Because we've already talked about the cubes where three faces are painted, and we've already talked about the cubes where zero faces are painted. So when you think about this, I'd encourage you to, first of all, think where in the big cube are their cubes where two or one faces are painted. So for example, if you're thinking about where one face has been painted, for example, this one here, this cube here, that has only got one face painted. So that is how I'd encourage you to think about it. So pause the video and see if you can think how many cubes there are with one or two faces painted. Okay, great. Let's go for them. So first of all, thinking about where there are. There's only one face painted. So the cube that I've already highlighted, that is one cube, where only one face has been painted. And then some others we can see on the diagrams. So, this one on the sides, and this one at the top, only one face will be painted. So those are three that we can see. But are there any others that we can't see? Well, yes, there are. And in fact, there's going to be six in total. And the reason for that is because we can think about this in terms of the faces of each cube. So on each face of the cube, the cube in the middle is going to have just one of the faces painted. And we know that there are six faces. So one cube on each face is going to give us six in total. So we could say that where these cubes are in the big cube, are going to be in the middle of the faces. What about the cubes where two faces are painted? Well, one way to think about this is that in total, we know how many cubes there are. Because we have a three by three by three cube, we know that there are in total going to be 27 cubes. So because 27 is three times by three times by three, so we could do just do eight plus six plus one, which would be 15. And take that away from 27, which would give us 12. But let's actually go through it and identify where these cubes are with two faces painted. And there's a number of ways you could think about this. But first of all, it's really helpful to identify some of the cubes where two faces have been painted. So for example, this one here has got two faces painted. That is a nice starting point. See if you can identify if you haven't already, any other cubes, with two faces painted. Well, there's a few more that you can see on the diagram. So if you're just thinking about this first sort of row of cubes that I've highlighted there, then you can also see that... You can also see that this cube here will have two faces painted. So will this one on the bottom, you can't see the face on the bottom. And so will this one here. And if you think about this further, you'll find that in total, there are 12 cubes, where two faces have been painted. And there's a number of different positions. So on this space here, they tend to be not on the corners but not in the middle of the face. But then if you think about the next row along this for example, is is another cube. So that is where two faces have been painted. And in total, we can check that these add up to 27. Okay, let's now move on to the independent task where it's going to get a little bit trickier. So for the independent task, we are going a little bit trickier. So the same thing again, but this time we are looking not at a three by three by three cube but a four by four by four cube. So it's going to be more challenging, but the same principle applies. You can imagine that we are painting all around the outside of a cube. And we're thinking, how many cubes are there were three faces, two faces, one face and zero faces have been painted? So it is a tricky one, you're going to have to really think hard and really visualise here, but let's see how far you can get. I'd encourage you to start off by thinking how many cubes there are, where three faces have been painted, then the zero. Those are the easier ones. And then about two and one as well. So, pause the video now and see how far you can get. Okay, so let's go through it then. So first of all, how many cubes have got three faces painted? Well, you saw before that, where we have a cube in the corner, we're going to have three faces painted, for example, this one here, in the corner, three faces painted. So we're thinking how many corner cubes are there. And again, there are going to be eight of them, which is the same number that we had before. And that shows us that even though the cube is bigger, the number of corners that it has, is still the same. So we've done eight already, which is really good. Well done if you got that. Now let's think about where zero faces have been painted. Now this is really hard to visualise because we can't actually see any of these cubes. But if you were thinking that there are eight of them that I am really impressed. And the way to think about this is that you've got a four by four by four block of cubes, which you can see. And you can imagine that in the inside the cubes that have got no contact with any of the ones on the outside, it's going to be a nice block of two by two by two. And that is going to give us eight cubes. So well done, we got 16 already. And in total, we know that there are going to be 64 cubes that we need to work out. The next one that you might have thought about is how many cubes have one face painted, because last time we thought about this in terms of each face. So for example, this first face here, we can see that these four cubes have got one face painted. So actually should be pretty easy for you to think about how many there are in total in this cube. So well done if you thought 24, because on each face, there are four cubes, which have got one face painted, and there are six faces. So in total, there's going to be 24 cubes. So really well done if you've got that. And really well done if you thought about the final one as well where two faces are painted. So if we think about where those cubes are, well, there's a number of ways to think about this. But as I did with a three by three cube, I can think about it in terms of each layer of my diagram here. So here I'm highlighting the first layer of cubes. And if I think about this, how many have got two faces painted? Well, I've got one here, one here, one here, one here. And in total in that first row, there are going to be eight of them there. So your job now, if you haven't already, think about how many there are going to be in total. Okay, so great. So in this first row here, which we already talked about, there are eight. And the same is also true true if we think about this back row here, they're also going to be eight. But the middle row is slightly different, because the only cubes which have got two faces painted are these ones in the kind of along the edge that so one here in green, one here, one here, and then one directly below that. So there's going to be four here, and four here. And eight plus four plus four plus eight is going to give me 24. And so we have actually worked out our answer because eight plus 24 plus 24 plus eight is going to give me 64, which is the number of cubes that I have in total. So really well done if you managed to get any of these, really, really impressive, because this is a very, very hard one to visualise. But I'm afraid the explore task it's going to get even tougher. So fasten your seat belts, get ready for the next slide. Okay, so here is the final slide. And it's going to be a really, really tricky one. Well done if you're still with me, but I'm afraid it's going to get even more tricky. So this time, we're thinking about a n by n by n cube. What does that mean? Well, we've already thought about a three by three by three cube, and a four by four by four cube. So now we're thinking of cubes that are even bigger. So a five by five by five, or six by six by six, etc. But what n by n by n means is that we are generalising here. So we're going to be able to say how many of each number of faces will be painted, whatever the size of the cube. So it's going to be pretty really difficult. If you feel like you can give us a go then pause the video now. In a couple of seconds, I'll give you a clue to get you started. Okay, so the key in this question is not to think about any cubes, which are really really big, but it's actually think about the cubes that we've already thought of already. So first of all, if we start off with the number of cubes where three phases have been painted. In the first two examples, we found that in both of these, there were eight cubes in the corner. So eight cubes with three faces painted. So what do you think it would be for any cube that is bigger? Well, the answer is that however bigger the cube is, they were always going to be only eight corner cubes. It doesn't matter how big it is, it could be 100 by 100 by 100. There are still going to be only a cube from the corner. So whatever the value of n is, is always going to be eight cubes. So that's actually a nice start. We've done one of them already. Now let's have a think about where zero faces have been painted. So again, think back to the two cubes that we've done already. How did we get the number of cubes with zero faces painted? Well, in the first example, we had one kind of mini cube within that big cube. And then the second example, we had a two by two by two set of cubes, so eight cubes within that big cube. So have a think, n by n by n cube, do you see what the pattern is, that's going on here? Can you think of how many cubes there will be with zero faces painted? Well, there seems to be a pattern here, it seems to me that to get from three to one, we're subtracting by two. To get from four to two were subtracting by two. So if we had an n by n by n cube, we are actually also going to subtract by two. So it's going to be n minus two times by n minus two times by n minus two, which in total, we can say is n minus two cubes. And that is our expression. So whatever the value of n is, say n is seven, we do seven minus two, which is five. So a seven by seven by seven cube would have five times by five times by five, or 125 cubes where zero faces have been painted. So that's the next one done. The next one to think about is, the number of cubes where one face has been painted. So feel free to pause the video and have a go at this one. Okay, so again, it's worth thinking about the two cubes that we've done before. And the way that we thought about this was thinking about each of the six faces. So, for example, the four by four by four cube, we saw that if we think about the front face here, that these four cubes here would have one face painted. And then we multiply that by six to get a total of 24, because four times by six is 24. But the problem is what if we have n by n by n cubes? And actually, the way to think about this is to think how many cubes we would have on each individual face? So this front face here has got 16 cubes or n times by n, n squared. And this cube in here, the mini cube is actually n minus two squares, because we take four, we subtract by two, and we square it. So it turns out that if we have an n by n by n cube, when we think about the number of cubes where one face has been painted, we're going to have to do n minus two squares, but multiply that by six because there are six faces. Okay, we are almost there. But actually the final one is the trickiest one. So have a think if you feel like you can do this. Okay, we're going to go through it, because it is really, really tricky. But well done, if you did manage to get started. And the key to this one, I think, is at thinking about this in terms of our rows of cubes. So this front row of cubes here is going to be one thing to think about. And this front row is actually going to have the same properties as this back row here. So we can think of these two rows as separate to all of the rows in the middle. So if we think about these two rows here, well, how many how many cubes do we have with two faces painted? Well, it's this one, is this one. One, two, three, four, five, six, seven, eight. So there are eight cubes in that front row and also eight cubes in that back row. But how do we do this algebraically in terms of n? Well, we can think about it that in each of these rows we're going to have n squared cubes. And then we need to take away the four corner cubes, which have got three faces painted. And we also need to take away these middle four cubes. And we saw already that those middle four cubes while there are going to be n minus two squared, lots of them. And so if we expand that and simplify that algebraically we get four n minus eight. And there are two rows of four and minus eight. So we're going to have to do two times by four n minus 8, which is eight n minus 16. So that is for the two rows of cubes highlighted in yellow. But now we need to think about all the other rows of cubes in between those. So I will highlight them in orange. So these are all the ones that we need to think about. And there's a couple things that we need to think about. First of all, how many of these rows are there? Well, if there are n rows in total, and two that we've already thought about, there are going to be n minus two rows. And then next, we think about within each row, how many of them have got two faces painted? And the answer is going to be four. Is going to be this one here, this one down here, this one here and the one underneath that. So we're going to multiply that by four. So four times by two, n minus one, gives me up four n minus eight. And then what we need to do, of course, is need to add these two expressions together. And we do that and we get 12 and minus 24, which is a nice answer. Now, of course, to check that the answer is right, we would need to check to see whether all of these expressions add to n cubed. And it turns out that if we do the algebra, it does actually work, which is really, really nice. So, yeah, that is basically the end of this problem. We know how many cubes have got three, two, one, and zero faces painted for any n by n by n cube. So I hope that you've enjoyed that lesson. It's a really, really tricky one. So well done if you even got part of this lesson, because it is a really tricky one. But anyway, thanks very much for watching, hope you enjoyed it. And have a great day and see you for another lesson. Bye! Bye!

The four colour map theorem was finally solved in which decade?

Which of the below is a requirement of the four colour map theorem?

Fill the gap: We can think about maps in terms of _______.

Fill in the gap: The twin prime conjecture states: there are __________ many primes, 𝑝, such that 𝑝 + 2 is also prime.

In what year did Polignac publish his conjecture to build on the twin prime conjecture?

What is de Polignac's first name?

Fill in the gap: de Polignac's conjecture states that for any _________ ________ number, 𝑛, there are infinitely many prime gaps of size 𝑛.

What is the first pair of twin primes greater than one million?

Look at the 3D grid. What are the co-ordinates of A?

Imagine that the whole outside of this 3 x 2 x 2 cube was painted. First, how many cubes are there in total?

Unit Overview: Famous maths problems

Lessons:

Four colour map theorem

The twin prime conjecture

3D Coordinates

The Painted Cube