Sequences that turn out to be the same sequence in disguise

0:00At the end of section one point four, your book asks you nine questions. And six of them look, at first, like six separate tricks to memorise. Add up the counting numbers. Add up the powers of two. Pair up triangular numbers. Multiply a triangular number by six. Six unrelated puzzles. That's how they look. They're not. Four of the six are the same move, done to different rows. And when you finish all six, something happens that I don't think the book ever quite says out loud.

0:36Table one stops being a list of ten sequences, and becomes a map, with the rows joined by arrows. Let's build that map. First, the move that does most of the work. It's called taking running totals. Here's what it means. You have a sequence. You add it up as you go, and you write down each total on the way. So if the sequence is two, four, six, eight — your first total is two. Then two plus four is six. Then plus six is twelve. Then plus eight is twenty.

1:13Two, six, twelve, twenty. That's the running totals of the sequence you started with. It's a machine that eats one sequence and hands you back another one. And the question the book keeps asking is: which one? Let's warm up with the silliest row in the table. All ones. One, one, one, one, one. The row that does nothing. Take its running totals. First total, one. Then one plus one is two. Then three. Then four. Then five.

1:48One, two, three, four, five. That's the counting numbers. And it's obvious once you see it. Counting is just adding one, over and over. That's what counting is. So there's our first arrow. All ones, totalled up, gives the counting numbers. Your book also asks what happens if you go up and then back down. One. One plus one. One plus one plus one, and back. That gives one, three, five, seven. The odd numbers. Second arrow, free.

2:23Now do the same thing to the counting numbers themselves. One. One plus two is three. Plus three is six. Plus four is ten. Plus five is fifteen. One, three, six, ten, fifteen. And you know exactly what those are. The triangular numbers. Which we drew two videos ago. And here the book does the thing it always does. It asks for a picture. So here's the picture, and we've already got it. Adding one, then two, then three is stacking rows of one, then two, then three.

2:59That's just building a triangle. The running total is the triangle so far. Arrow three. Counting numbers, totalled up, gives the triangular numbers. Arrow four we proved last time, so I'll be quick. Odd numbers, totalled up. One, four, nine, sixteen, twenty-five. The squares. And the picture was the bent bands wrapping round a corner of a dot square. Notice the shape of what we've got so far. Same move, three different rows, three different landings.

3:35All ones goes to counting numbers. Counting numbers go to triangular numbers. Odd numbers go to squares. The move isn't clever. The move is always the same. What's interesting is where each row lands. Now the book changes the move, and this one is my favourite. Take two triangular numbers that are next to each other, and add them. One and three is four. Three and six is nine. Six and ten is sixteen. Ten and fifteen is twenty-five.

4:11Four, nine, sixteen, twenty-five. The squares again, arriving by a completely different door. And you can see why, if you draw it. Take a triangle of six dots. Take the next one up, ten dots. Turn one of them upside down. They fit. Exactly. Into a four-by-four square, with nothing left over and nothing overlapping. Two consecutive triangles always make a square. That's the reason, and it's a picture, not a calculation.

4:46One warning, and it cuts the other way. Not any two will do: one plus six is seven, and seven is not a square. It is the consecutive ones that always work. Next. Powers of two, totalled up. One, two, four, eight, sixteen. One. One plus two is three. Plus four is seven. Plus eight is fifteen. Plus sixteen is thirty-one. One, three, seven, fifteen, thirty-one. Hmm. Those aren't a row of table one.

5:19But look at them next to the powers of two. One is one less than two. Three is one less than four. Seven is one less than eight. Every total is exactly one short of the next power of two. So add one to each of them, and you land back on the powers of two. Two, four, eight, sixteen, thirty-two. And that plus one in the book's question isn't decoration. It's the entire content of the question.

5:50Here's why it happens. Doubling something is the same as adding it to itself. So everything you've collected so far, plus one more copy of the last thing, gets you to the next power. You're always one dot short of doubling. Now the strangest question in the set. Take a triangular number, multiply it by six, and add one. Six times one, plus one, is seven. Six times three, plus one, is nineteen. Six times six, plus one, is thirty-seven. Six times ten, plus one, is sixty-one.

6:31Seven, nineteen, thirty-seven, sixty-one. And we have met those. The hexagonal numbers, from the last video. The centre dot with rings around it. Why six? Because a hexagon has six sides — and here's the picture. Put one dot in the middle. Pack six triangular arrays around it, one against each side. Six triangles, plus the one in the middle. Six times a triangular number, plus one. Now, one thing to be careful about, because it's easy to get wrong.

7:07This construction gives you the hexagonal numbers from the second one onwards. The first hexagonal number is one — a single dot on its own, with no triangles packed round it at all. There's nothing to multiply by six. So line them up carefully. The offset is real, and it's the kind of thing an exam will check. One more arrow, and it's the best one. Take the hexagonal numbers, and total them up. One. One plus seven is eight. Plus nineteen is twenty-seven. Plus thirty-seven is sixty-four.

7:48One, eight, twenty-seven, sixty-four. Those are the cubes. Flat hexagons of dots, added up, give solid cubes. That should feel like it can't be right. One of those is a flat drawing. The other is a solid. Your book knows exactly how strange this is, and it does something lovely about it. It prints a picture. On page nine there are two drawings, side by side, and they're coloured to match.

8:19On the left, a cube built out of little unit cubes, four along each edge. Now look at it from the corner nearest you. Which little cubes can you actually see? You can see the corner one. That's one cube, and the book colours it yellow. Then three edges run away from that corner, three cubes along each. That's nine, and they're red. And then three faces, each three by three. That's twenty-seven, and they're blue.

8:49One, plus nine, plus twenty-seven. That's thirty-seven — the fourth hexagonal number. And on the right of the page is a hexagon of dots, with one yellow dot in the middle, three red arms of three, and twenty-seven blue. The same counts. The same colours. That's the book telling you they're the same object. So now peel that visible layer off the cube. What's left is a smaller cube, three along each edge.

9:21And its outer layer is one, plus six, plus twelve. Nineteen. The hexagonal number before. Keep peeling and you get seven, then one. So a cube is exactly its layers, and its layers are the hexagonal numbers. Which is why totalling them up gives you a cube. Not a coincidence. A shape, taken apart. So let's put all of that on one screen. All ones, totalled, goes to the counting numbers. All ones up and down goes to the odd numbers.

9:56Counting numbers, totalled, go to the triangular numbers. Odd numbers, totalled, go to the squares. And triangular numbers, paired up, also go to the squares. Triangular numbers, times six plus one, go to the hexagonal numbers. Hexagonal numbers, totalled, go to the cubes. Look at what that is. That's not a list of ten separate things. That's a network, and almost every row is joined to another one. And notice something about that map. One of the nodes on it isn't in table one at all.

10:34The hexagonal numbers were never in the table. They arrived as a picture on page five, and the map turns out to need them. Which is a nice hint that the table was never the point. The table was a starting sample. Your book's last question in this chapter is different from all the others, and it's the one that matters. It doesn't give you a relation to check. It asks you to find one.

11:04Find your own relation between two rows of table one, and explain why it happens. So let me tell you what makes an answer count, because this is the whole standard of the chapter. Naming the sequence you land on is half of it. Anyone can spot a landing. The other half is the reason — and for these, the reason is almost always a picture. A way the shapes fit.

11:31An arrow you can't draw is just a coincidence you happened to notice. So here's mine to get you started. What do you get if you total up the squares? Try it. It isn't in table one. And then find one I haven't done, and put it in the comments — with your picture. That's the end of the number sequences. Next time we leave numbers behind, and go looking for patterns in shapes.