Lesson video

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!.