Drawing a sequence makes its rule visible

0:00Ten dots. Can you arrange them into a triangle? Have a real go at it in your head before I show you. Ten dots, filled in, no gaps. Now the same question for eleven dots. Triangle, yes or no? Hold on to both answers, because this is not a puzzle. This is what the word triangular means. Last video we met a row in your textbook called the triangular numbers. One, three, six, ten, fifteen.

0:32And I let the name go by as if it were just a name. It isn't. It's a claim. It says: take that many dots, and they will go into a triangle. Which is the kind of claim you can test. So let's test it, and six others, by drawing them. Your book has a whole section on this, and it's called visualising number sequences. Here's the thing to notice about where it sits. It comes before the hard work, not after it.

1:04Drawing isn't the reward you get once you've understood something. It's one of the ways you understand it. And what you get from the picture is more than a pretty version of the list. You get the next value. You get the reason it's the next value. And sometimes you get something nobody told you. So here is table two, on page four. Seven rows. Five drawings each. Every drawing labelled with its count.

1:34We'll go through them, and then we'll do the exercises the book sets, which are drawing exercises, not arithmetic ones. Start with the three easy rows, because even the easy ones say something. All ones. One dot. One dot. One dot. Five drawings, and the picture never changes. That's the row that does nothing, and now you can see it doing nothing. Counting numbers. A single row of dots, one longer each time. One, two, three, four, five.

2:09Notice what the drawing makes obvious. Adding one to the number is adding one dot to the line. Same thing, twice over. Even numbers. Now the book uses two rows of dots, always the same length as each other. One and one makes two. Two and two makes four. Three and three makes six. Four and four is eight. Five and five is ten. So an even number is a number that splits into two equal rows. That's not a new rule. That's the drawing telling you what even means.

2:46Now the odd numbers, and this one is worth slowing down for. The book draws them as two rows again — but the rows differ by one. The shorter one sits on top. One dot on its own is one. Then one over two makes three. Two over three makes five. Three over four makes seven. Four over five makes nine. There it is. An odd number is two rows that don't match, with exactly one dot sticking out.

3:18And that one dot is the whole difference between odd and even. Everything else pairs up. One warning before we move on, and it's the kind of thing that trips people up later. This is not the only way to draw an odd number, and it is not the way section one point four is going to draw them. There, each odd number gets bent round the corner of a square, like a band. Same count of dots. Completely different arrangement.

3:48Which is a small lesson in itself. The number doesn't have one picture. It has as many as you can find. Right. The triangular numbers, and our promise from the start. One dot. That's a triangle, in the way that a single dot is any shape you like. Call it one. Now add a row of two underneath. One and two. Three dots, and it looks like a triangle. Add a row of three. Now we've got six.

4:20Add a row of four. Ten. Which answers the question I opened with — yes, ten dots make a triangle. Add a row of five. Fifteen. And look at what the picture is doing. Each new row is one longer than the row above it. So the jumps are two, then three, then four, then five. Which is exactly the rule we worked out from the numbers last time. But here's the difference. Last time we spotted that rule. This time we can see why it's true.

4:54You add a longer row because a triangle gets wider as it gets taller. That's the reason, and it was invisible in the list of digits. Oh — and eleven. Eleven dots don't make a triangle. You'd need one more to finish the row. Two more rows, and they're the same idea in two different worlds. Squares. Dots in a square array. One. Then two by two, which is four. Three by three, nine. Four by four, sixteen. Five by five, twenty-five.

5:30The name is the picture. A square number is a number of dots that fills a square. Now the cubes, and here the book leaves the flat page altogether. These are drawn as solids, built out of little unit cubes stacked up. One cube. Then two along each edge, which is eight little cubes. Three along each edge, twenty-seven. Four along each edge, sixty-four. Five along each edge, one hundred and twenty-five.

6:02So don't think of the cubes as squares in disguise. A square is dots on paper. A cube is blocks in space. Same idea — fill the shape — carried from two directions into three. Now the exercise the book actually sets, and read it carefully, because it's not the exercise you expect. It doesn't ask for the next number. It asks you to draw the next picture. The number comes out afterwards, by counting what you drew.

6:34Try it with the counting numbers. The last drawing was five dots in a line. Draw a sixth. Count them. Six. Try it with the squares. The last one was five by five. Draw six by six. Count them. Thirty-six. Try it with the triangular numbers. The last one had rows one to five. Add a row of six. Count them. Twenty-one. Notice you never did any arithmetic. You built the shape, and the shape handed you the number.

7:08That's the direction the whole section is going in. Picture first, value second. And now something falls out that nobody set up for us. Look back at the last two things we drew. The square gave us thirty-six. Keep that number. Now take thirty-six dots and try to build a triangle instead. Rows of one, two, three, four, five, six, seven, eight. Count them up. One and two is three. And three is six. And four is ten. And five is fifteen. And six is twenty-one. And seven is twenty-eight. And eight is thirty-six.

7:49It works. Exactly, with nothing left over. So thirty-six is a square number and a triangular number. The same thirty-six dots. Two completely different buildings. Now, it would be easy to call that a cute coincidence and move on. Don't. The point your book is making is bigger. A number isn't one kind of thing. It's whatever its arrangement makes it. Thirty-six doesn't have to choose. It's both, and it's both at the same time.

8:23Here's my favourite thing in this section, because it happens in the other order. The book prints four dot pictures. A single dot. Then a ring of dots around it. Then another ring. Then another. The labels are one, seven, nineteen, thirty-seven. These numbers are not in table one. They were not in any list. They came from a picture, and the picture gave them their name — the hexagonal numbers.

8:53So the name came second, off the drawing. Which is the opposite of what we've been doing all video. And then the book asks: what comes next? It doesn't tell you. So let's work it out from the rings. One dot in the middle. The first ring adds six, giving seven. The second ring is bigger. It adds twelve, giving nineteen. The third ring adds eighteen, giving thirty-seven. Six, twelve, eighteen. Each ring adds six more than the ring before it.

9:27So the next ring adds twenty-four. And thirty-seven and twenty-four is sixty-one. Sixty-one. And notice — we didn't guess it, and we didn't look it up. The picture told us. Now here's something you'd never spot without laying the two tables side by side. Table one has ten rows. Table two draws seven. Three rows have no picture at all. The Virahanka numbers. The powers of two. And the powers of three.

9:58So what happened? Did the book run out of room, or is there something about those three? Well, the exercise on page five asks you to draw a way of picturing the powers of three. So the book clearly doesn't think it's impossible. And the powers of two get picked up on the very next page, by the book itself. Which leaves the Virahanka numbers with no picture anywhere in this chapter. Something to keep an eye out for.

10:29Gaps in a textbook are usually worth noticing. Sometimes they mean it's hard. Sometimes they just mean it's your turn. So, page six. Here is the book's own way of drawing the powers of two, and it's lovely. Start with a single dot. That's one. Now make a copy of it, and join the two with a line. Two dots. That's two. Now make a copy of the whole thing, and join matching corners. You get a square. Four corners. That's four.

11:03Copy the square, join matching corners, and you get a cube. Eight corners. That's eight. Do you see what the rule has become? Every single figure is two copies of the one before, joined up. Two copies means twice as many corners. That's doubling, drawn. So do it once more. Copy the cube, join matching corners. Sixteen corners. It's drawn as two cubes with lines between them — and those joining lines matter. They make it one object, not two.

11:38Now, where does that fifth shape live? Not in ordinary space. You can't build it out of sticks on your table. The book doesn't make a fuss about that here. It comes back in section one point five, which says outright that shapes can live in more than three dimensions. And notice the book's own wording. This is one way to think about the powers of two. Not the way. There are others, and yours is welcome.

12:09So what did the drawings actually buy us? The next value, without arithmetic. The reason behind the rule, which the digits hid. A number that turned out to be two shapes at once. And a whole sequence that arrived as a picture before it had a name. None of that was available from a row of numbers on a page. So here's your job, and it's the one the book sets. Draw a way of picturing the powers of three.

12:38One, three, nine, twenty-seven. Find an arrangement where each figure is genuinely three copies of the one before it. There is more than one good answer. I'd like to see yours. Put it in the comments, and next time we take the squares apart and find the odd numbers hiding inside them.