Why the first n odd numbers add up to n²

0:00Here are the first few square numbers. One, four, nine, sixteen, twenty-five. Now do something small to them. Take each one and subtract the one before it. Four minus one is three. Nine minus four is five. Sixteen minus nine is seven. Twenty-five minus sixteen is nine. Three, five, seven, nine. Those are not random numbers. They are the odd numbers, arriving in order, with nothing skipped and nothing repeated.

0:33And they keep coming. The next gap is eleven, then thirteen. So the squares are not spaced evenly. Each step is two bigger than the step before it. That is worth stating on its own, because it is the whole video in one line. The gap from one square to the next is always an odd number, and they turn up in order. But a pattern you have checked four times is not a fact. It is a coincidence you have not yet caught out.

1:05There are infinitely many squares, and no amount of checking ever gets through them. So the question is not whether the pattern continues. The question is why it has to. And for that we stop calculating and start looking at what a square number actually is. A square number is a square. Draw one. Here is a five by five arrangement of dots. Twenty-five of them. Now make it six by six.

1:36What do you have to add? Not a whole new square. The twenty-five dots are already there, and they do not move. You add a strip along the bottom, and a strip up the side. An inverted L, wrapped round two edges of what you already had. Every square grows into the next one by having exactly one of these L shapes laid round it. So the question about the gaps is now a question about the L. How many dots are in it?

2:07Count it in three pieces. Going from five by five to six by six, there is a new row along the bottom, five dots long. There is a new column up the side, also five dots. And there is one more dot, right at the corner, where the row and the column turn into each other. That corner dot is in neither of them, and forgetting it changes the answer from odd to even.

2:35Five and five is ten. Ten and one is eleven. Eleven dots, and eleven is exactly the gap from twenty-five to thirty-six. Two equal sides, so an even number between them. Plus one corner. Even plus one is odd. The gap cannot be anything else. Now stack them. Start with a single dot. One. Lay an L of three round it and you have a two by two square. One and three is four.

3:07Lay an L of five round that and you have three by three. One and three and five is nine. Seven more gives sixteen. Nine more gives twenty-five. Eleven more gives thirty-six. Every square is built out of odd numbers, in order, starting from one. Add the first six odd numbers and you get thirty-six, which is six times six. Add up the first n odd numbers and you get n squared.

3:37Not approximately. Exactly, and for every n there is. A picture is often offered for this, and it is a good one, but be honest about what a picture can do. It shows one case: one square, of one size, growing by one L. That is not yet a proof, however convincing it looks. What turns it into one is that nothing in the argument depended on the size. Whatever the square is, the new row is one shorter than the new side. So is the column. There is always exactly one corner.

4:15Nowhere did we use five, or six. Take those two numbers away and every sentence still stands. A wordless proof has to carry not a case you can see, but a step you can see is general. So let us say it without the picture. The nth odd number is two n minus one. The first is two minus one, which is one. The second is four minus one, three. The third, five.

4:45This is where it goes wrong for almost everybody. Two n minus one is the nth odd number. Two n plus one is the step that carries you from n squared up to the next square. They differ by two, they both live here, and they get swapped constantly. Here is how to keep them apart. The L that leaves a square of side n has a row of n, a column of n, and a corner. That is two n plus one.

5:16Same three pieces. Just a bigger square underneath. Now it earns its keep. Thirty-five squared is one thousand two hundred and twenty-five. You want thirty-six squared, without multiplying. The step out of thirty-five squared is two times thirty-five, plus one. Seventy-one. One thousand two hundred and twenty-five, plus seventy-one, is one thousand two hundred and ninety-six. That is thirty-six squared, with no multiplying in it anywhere. It runs the other way too. Sixty-seven squared minus sixty-six squared is two times sixty-six plus one. A hundred and thirty-three.

6:03Forty-three squared minus forty-two squared is eighty-five. Here is one to try. One hundred and twenty-five squared is fifteen thousand six hundred and twenty-five. Which of these gives one hundred and twenty-six squared? Add one hundred and twenty-six. Add twenty-six squared. Add two hundred and fifty-three. Add two hundred and fifty-one. Add fifty-one squared. The step is two times one hundred and twenty-five, plus one. Two hundred and fifty-one. Two hundred and fifty-three is the one that catches people, and it misses by two.

6:46Everything so far has run forwards. It runs backwards as well, and that gives you a test. Take a number. Twenty-five. Ask whether it is a square, without multiplying anything. Subtract one. Twenty-four. Subtract three. Twenty-one. Subtract five. Sixteen. Subtract seven. Nine. Subtract nine. Zero. It landed exactly on nothing, so twenty-five is a square. And it had to. You have just taken the square apart into the very Ls it was built from.

7:25If they come out exactly, with nothing left over, there was nothing else in there. Now try thirty-eight. Subtract one: thirty-seven. Three: thirty-four. Five: twenty-nine. Seven: twenty-two. Nine: thirteen. Eleven: two. Two left, and the next odd number is thirteen. Subtract it and you get minus eleven. You have gone past zero without ever landing on it. That settles it. There is no way back. The numbers you are subtracting only ever get bigger, so once you are below zero you stay below zero.

8:06Thirty-eight is not a square. Notice what this test gave you that a table of squares would not. It never once asked you to know a square. Go back to the twenty-five and count the subtractions. One, three, five, seven, nine. Five of them. Five. And twenty-five is five times five. Not a coincidence either. It is the same fact once more: you built the square out of five Ls, so taking it apart takes five steps.

8:39But be careful what you are counting. It is the number of steps, not the number you started with. Eighty-one is a square. The test takes nine steps, not eighty-one. There is exactly one number where those two agree, and it is one. Which is precisely why the mistake survives the first example anybody tries. One more thing falls straight out of the gaps. Between nine and sixteen, how many whole numbers are there? Not counting the two ends.

9:13Ten, eleven, twelve, thirteen, fourteen, fifteen. Six of them. The gap from nine to sixteen is seven, but the count is six. One of those seven steps lands on sixteen itself, and sixteen is an end, not a number in between. In general, between n squared and the next square there are two n whole numbers, and not one of them is a square. The count is always one less than the gap. Always.

9:45Finish with something those widening gaps predict. Cut the whole numbers up to a thousand into blocks of a hundred. How many squares in the first hundred? One, four, nine, and on up to a hundred itself. Ten of them. In the second hundred? A hundred and twenty-one, a hundred and forty-four, a hundred and sixty-nine, a hundred and ninety-six. Four. By the last block there is one: nine hundred and sixty-one.

10:16Thirty-one times thirty-one. And the next square has already gone past a thousand. Squares thin out as you go, and now you know exactly why. The distance between them grows by two, every time. One more L each time.