Squares hiding inside triangular numbers

0:00Here is a shape made out of dots. One dot on the top row. Two on the next. Three on the one below that. Keep going and you have a staircase. Count the dots and you have a triangular number. One row is one dot. Two rows is three. Three rows is six. Four rows is ten. Five rows is fifteen. One, three, six, ten, fifteen. They are not a list to memorise. They are a construction.

0:30And notice one thing now, before we go anywhere. The fourth one is ten. Four rows. Ten dots. Two different numbers, and keeping them apart is going to matter later. How do you get from one staircase to the next? You add a row along the bottom, one dot longer than the row above it. Six has three rows. Put a row of four underneath and you have ten. Ten has four rows. Put a row of five underneath and you have fifteen.

1:03Fifteen, and a row of six, gives twenty-one. That is the entire rule. So you never need the list. You need the last one, and a row. Which means you can always get to the next one, however far along you are. Now take two of them and add. One and three is four. Three and six is nine. Six and ten is sixteen. Four, nine, sixteen. Those are squares. Two times two, three times three, four times four.

1:36And look at which pairs did it. One and three sit next to each other in the list. So do three and six. So do six and ten. Neighbours, every time. That is worth stopping on, because there is no obvious reason why adding two staircases should give you a square anything. Three cases, though. Three cases is a reason to go looking for an argument. It is not an argument.

2:07So before we look for one, let us see what the pattern says next. The neighbours after six and ten are ten and fifteen. Ten and fifteen is twenty-five. Twenty-five is five times five. It worked. But that is a fourth case, and we now have exactly the problem we had before, one step further along. The question is why two staircases should fill a square at all. Here is the answer, and it is a picture rather than a sum.

2:41Take the two staircases. Ten, which has four rows, and six, which has three. Leave the ten where it is. Now take the six and turn it half a turn, so it goes down where the other one goes up. Its rows used to read one, two, three. Turned round, they read three, two, one. Slide it up against the first one. It drops into the notch with nothing spare and nothing missing.

3:08Now look at the shape you are holding. And look at the join between the two pieces. It is not a diagonal. It has steps in it. That matters more than it sounds. Cut a four by four square of dots along a straight diagonal and you get six on one side, six on the other, and four dots stranded on the line itself. Six and six. Two copies of the same staircase, and the four in the middle belonging to neither.

3:39The stepped join is what hands those four to one side, and that is the difference between six and ten. Read it row by row. The first staircase has rows of one, two, three, four. The turned one has rows of three, two, one. So the top row is one dot and three dots. Four. The next is two and two. Four. The next is three and one. Four. Every row with two pieces in it holds four dots.

4:11Not roughly four. Exactly four, every time. And that is not luck. Going down the picture, one staircase gains a dot a row while the other loses one, so the total cannot move. There is one row left over. The bottom row of the larger staircase, four dots long, has nothing beside it. And it needs nothing. It is already four dots wide. So count what we have. Three rows made of two pieces each, and one row that arrived whole.

4:47And that odd row out is not a loose end. Without it you would have three rows of four, which is twelve, and twelve is not a square. It is not left over at all. It is the fourth row. Four rows. Every one of them four dots across. Four rows of four is sixteen. Which is what six and ten come to. Not because we added them. Because we counted the square they made.

5:16Now take the numbers out. A staircase of n rows, beside one of n minus one rows, turned. The turned one has rows running from n minus one down to one. The big one's row of one meets the small one's row of n minus one. That is n. The big one's row of two meets the small one's row of n minus two. n again. It keeps working all the way down, for the same reason as before.

5:46The longest row of the bigger staircase, n dots, has no partner and needs none. So: n rows, n dots each. n times n. And notice what we never once used. No formula. Nowhere did we work out how many dots a staircase holds. We only ever counted rows. Now the mistake the picture invites. Two triangles make a square. So surely two of the same size would do? Try it. Two copies of ten, four rows each.

6:20Turn one of them and slide them together. Top row: one and four. Five. Then two and three. Five. Then three and two, then four and one. Five, and five. Four rows of five. Twenty dots. That is a rectangle, and twenty is not a square number. Every row came out one dot too long, and it always will, whatever size you start from. Which is exactly why the two staircases have to be neighbours.

6:53Come back to the thing from the very beginning, because it catches people right here. The fourth triangular number is ten. Not four. Ten. Four is how many rows it has. Ten is how many dots. And when two neighbouring staircases fill a square, the side of that square is the number of ROWS in the larger one. Not the number of dots in it. Six and ten filled a four by four, and the four came from the four rows, not from anywhere else.

7:27Rows on one side, dots on the other. Mix them up and every sentence in this video stops being true. It runs backwards as well. Hand me a square and I will take it apart. A hundred. Ten by ten. So the larger staircase has ten rows, and the smaller has nine. Ten rows of dots comes to fifty-five. Nine rows comes to forty-five. Forty-five and fifty-five is a hundred.

7:58Try another. Forty-nine. Seven by seven. Seven rows and six rows, so twenty-eight dots and twenty-one. Twenty-eight and twenty-one is forty-nine. And there is exactly one way to do it, every time. Every square has one pair of neighbouring staircases that fills it. Not two ways, not none. One. One last thing, and it is the sentence to be careful about. Neighbouring triangular numbers always give you a square. That is what we argued, and it is true.

8:36It does not follow that only neighbours do. Look at one and fifteen. One is the first triangular number and fifteen is the fifth. They are nowhere near each other. One and fifteen is sixteen. Which is a square. And it is the very same sixteen we built earlier out of six and ten. So the sentence to keep is that neighbours always work. Not that nothing else ever does. Those are different claims, and only one of them is true.

9:09So, all of it in one breath. A staircase is its rows. Two neighbouring staircases, one of them turned half a turn, pair up row by row, and every pair holds the same number of dots. And there are as many rows as the square has along a side. That is the whole argument, and there was no algebra in it anywhere. Draw the next one yourself. Fifteen and twenty-one.

9:38Five rows, and six rows. Six rows of six. Thirty-six. And thirty-six is worth a second look, because it is a square and it is also a triangular number in its own right. Eight rows of dots, in a staircase.