Why adding the odd numbers gives the squares

0:00One. One plus three. One plus three plus five. And keep going — one plus three plus five plus seven, then plus nine, then plus eleven. Six sums, all made of odd numbers. Now look down the right-hand side at what they come to. One. Four. Nine. Sixteen. Twenty-five. Thirty-six. Those are the square numbers. Every single one. Add up odd numbers from the start, and you land on a square. Six times out of six.

0:34So that's the pattern. Now, what does your book do next? This is the bit I want you to notice, because it's easy to read straight past. The book does not move on. It stops, and it asks two things. Why does this happen? And does it keep happening for ever? Think about what that means. The book has just shown you six confirmations, and it is telling you that's not enough.

1:02It's not being fussy. It's being honest. Because those two questions are exactly the difference between spotting something and knowing it. We met this idea in the very first video of this chapter, and here's where it earns its keep. Checking cases can never finish the job. However many you check, there's always a next one you haven't. Remember the circle and the chords? One, two, four, eight, sixteen — and then thirty-one.

1:34Five confirmations in a row, and the sixth broke it. So we have six confirmations here. That is genuinely good evidence. It is not a reason. What we need is something that settles all of them at once. Every size, including the ones nobody will ever draw. And here's the remarkable thing. For this one, a Grade six student can produce that reason. Completely. Today. It isn't algebra. It's a picture.

2:07So let's go back to what a square number actually is. Not a number you multiply by itself — though it is that. A number of dots that fills a square. Twenty-five dots. Five rows of five. That's what makes twenty-five square. And thirty-six is six rows of six. Here it is. Now hold that picture in your head, because we are going to do exactly one thing to it.

2:34We are going to cut it up. Here is the question that does all the work in this video. Can that square of dots be cut into groups of one, three, five, seven — the odd numbers, in order, with nothing left over? Because look at what that would mean. If the thirty-six dots split exactly into groups of one, three, five, seven, nine and eleven, then those six odd numbers add up to thirty-six.

3:06Not because we added them. Because they are the same dots, counted a different way. So: can it be done? Have a go at it before I show you. It's a real puzzle, and it's the whole thing. Here's the cut, and once you see it you can't unsee it. Start in the corner. One dot. That's a one-by-one square, and it's our first odd number. Now wrap a band around it. Not a row — a band, bent round the corner. One dot along the top, one up the side, one in the corner.

3:42Three dots. And what have we got now? A two-by-two square. Wrap another band. Two along the top, two up the side, one in the corner. Five dots. And it completes a three-by-three. Again. Seven dots, and we have a four-by-four. Nine dots, and we have five-by-five. Eleven dots, and the square is full. Six by six. Thirty-six. One, three, five, seven, nine, eleven. Every dot used, none used twice, and the answer is the whole square.

4:18Now, why does that settle it for ever? Look at what the bands are actually made of, in general. You have a square with some number of dots along its side. Call that side length whatever you like. To grow it by one, you need a new row along the top. You need a new column up the side. And you need one dot in the corner where they meet.

4:42So the band is a row, plus a column, plus one. The row and the column are the same length as each other. So together they're an even number of dots. And then the corner dot makes it odd. Every time. It cannot come out even. That's the whole argument. And notice what it never mentioned. It never mentioned six. It never mentioned any particular size at all. Which means the cut works on a ten-by-ten square, and a hundred-by-hundred, and any square you like.

5:17That's what a reason buys you that six lines of addition don't. Six lines cover six cases. That picture covers all of them. So now let's spend it. Your book asks two questions here, and doesn't answer either. First: what do the first ten odd numbers come to? That's one plus three plus five, all the way up to nineteen. Ten of them. And you already know, because ten odd numbers build a ten-by-ten square.

5:49Ten times ten. A hundred. No adding. Second question, and this is the one that makes the point. What do the first hundred odd numbers come to? A hundred odd numbers build a hundred-by-hundred square. Which is a hundred times a hundred. Ten thousand. Stop and appreciate that. You just did ninety-nine additions in one step, and you didn't guess. A pattern would have let you guess. The reason let you know.

6:21Your book then does something rather beautiful. It comes at the same squares by a completely different road. Watch these sums. One. Then one plus two plus one. Then one plus two plus three plus two plus one. Up to a number, then back down again. One plus two plus three plus four plus three plus two plus one. And keep going up to five, and then up to six.

6:49Now the totals. One. Four. Nine. Sixteen. Twenty-five. Thirty-six. The same six squares. Again. And the book asks you the same question it asked before. Why? Then it does something I love. It prints a six-by-six square of dots, with no lines drawn on it at all. That's not a diagram. That's a blank worksheet, and it's for you. So let's finish it. Same thirty-six dots. Same square. Different cut.

7:23Last time we cut in bands around a corner. This time, cut along the diagonals. Start at the top-left corner. That diagonal holds one dot. The next diagonal holds two. The next holds three. Then four. Then five. Then six — that's the long one, right through the middle. And now they shrink again. Five. Four. Three. Two. And one, in the far corner. Read that off. One, two, three, four, five, six, five, four, three, two, one.

7:57That is the up-and-down sum, sitting inside the square, and every dot is in exactly one diagonal. So it has to come to thirty-six. Not by adding — by looking. And the same cut works at any size, for the same reason as before. Nothing in it cared that the square was six. So one plus two, all the way up to a hundred, and back down again? A hundred-by-hundred square. Ten thousand.

8:28Now put the two pictures next to each other, because this is the real lesson. That's the same square, both times. The same thirty-six dots. Cut one way, it says the odd numbers add up to a square. Cut the other way, it says the up-and-down sums add up to a square. Two different facts, and they weren't two different facts at all. They were two ways of slicing one object.

8:55That's what happens when you have a picture instead of a list. The picture holds more than you put into it. So let's be exact about what just happened, because it matters. At the start of this video, we had a pattern. Six lines of addition, six squares, and a strong feeling. Now we have a reason. And a reason is a different kind of thing. The six lines entitled us to say: it works for the first six. That's all they ever entitled us to.

9:29The cut entitles us to say: it works for every square there is, and every square there ever could be. That's not a bigger pile of evidence. That's a different category of knowing. And you can do this. That's the part I want you to take away. You just did. So here's one for you. What do the first twenty odd numbers add up to? And tell me how you knew, without doing twenty additions.

9:56Put it in the comments. Next time, we go looking for patterns in shapes instead of numbers.