Cubes built from runs of consecutive odd numbers

0:00Write the odd numbers in a line and never stop. One, three, five, seven, nine, eleven, and on. Now take a pair of scissors to that line. Cut off one number. Then the next two. Then the next three, then the next four, and keep going. You have not skipped anything and you have not gone back. Watch what each piece adds up to. The first piece is just one, and one is one cubed.

0:30The second piece is three and five, which is eight, and eight is two cubed. The third is seven, nine and eleven. Twenty-seven. Three cubed. Thirteen, fifteen, seventeen and nineteen come to sixty-four, which is four cubed. Then a hundred and twenty-five, then two hundred and sixteen. That last one is six numbers: thirty-one, thirty-three, thirty-five, thirty-seven, thirty-nine, forty-one. Six pieces, six cubes, in order. That is a lovely pattern, and six lines of it is not a reason.

1:11Six lines is data. Plenty of patterns hold for six lines and then stop. So the question is not whether it happens again. Six lines is a claim about six numbers. The pattern is a claim about every number there will ever be. The question is why it cannot help happening. And the answer is not really about cubes at all. It is about where each piece sits. Before anything else, one wrong reading has to go.

1:44You have met odd numbers before, as the gaps between squares. So it is tempting to read these pieces as gaps too, as the jump from one cube to the next. They are not. The fourth piece is sixty-four, which is the cube itself. The jump from twenty-seven to sixty-four is thirty-seven, and thirty-seven is nowhere on the board. The piece IS the cube. Now, to say anything about a piece, you need to know which odd numbers it owns.

2:18And that means counting what the earlier pieces already spent. Here is the trap. The fourth piece does not start at the fourth odd number. Before it, the first three pieces took one, then two, then three numbers. That is six spent, so the fourth piece starts at the seventh odd number, which is thirteen. And nothing is dropped at the joins either: the third piece stops at eleven and the fourth starts at thirteen, with no odd number left stranded between them.

2:52One, three, six, ten, fifteen. The running totals of one, two, three, four are the triangular numbers, and that is what the scissors have used up. So take piece number n. Before it, the pieces have used one plus two, up to n minus one. Turn that into a first term and something very tidy happens. Piece n starts at n squared, minus n, plus one. And since it has n terms stepping by two, it ends at n squared, plus n, minus one.

3:28Check that against a piece you can see. The fifth: twenty-five minus five plus one is twenty-one, and twenty-five plus five minus one is twenty-nine. Twenty-one to twenty-nine — and that is exactly the piece on the board. Now look at the two ends of that piece together. Twenty-one and twenty-nine. One of them is four below twenty-five and the other is four above it. The piece is not just sitting anywhere.

4:00It is balanced around n squared. And that is not luck: n squared minus n plus one and n squared plus n minus one are the same distance out, because both distances are n minus one. Try it on the fourth piece: thirteen and nineteen, and sixteen is three below one and three above the other. Add the two ends and you get twice n squared, every time. When the piece has an odd number of terms, n squared is literally the middle one.

4:36Twenty-five sits in the middle of twenty-one to twenty-nine. But look at the fourth piece: thirteen, fifteen, seventeen, nineteen. Four terms, so there is no middle term at all, and sixteen is not even in the list. It sits in the gap between fifteen and seventeen. That is fine, and it has to be: n squared is even whenever n is, and an even number can never be one of these.

5:04Balanced does not mean present. Work inwards from the two ends instead — first with last, second with second-last — and every pair adds to the same thing. And now the whole argument is two sentences. The piece has n terms. They are balanced about n squared, so on average each one is worth n squared. Take the sixth piece from both ends at once and you can watch it happen: thirty-one with forty-one, thirty-three with thirty-nine, thirty-five with thirty-seven.

5:41Seventy-two, seventy-two, seventy-two. n terms, each worth n squared. n times n squared. Which is n cubed. No adding, no algebra beyond that, and nothing about the pattern was assumed — the balance came from the two endpoints, and the endpoints came from counting what had been spent. So here is the test. Ninety-one, ninety-three, ninety-five, and on up to a hundred and nine. Total it, without adding it. The first move is not arithmetic — it is to ask which piece this is.

6:21Its first term is ninety-one, and n squared minus n plus one equals ninety-one when n is ten. Its last term is a hundred and nine, and n squared plus n minus one equals a hundred and nine when n is ten as well. So it is the tenth piece: ten terms, balanced about a hundred. Ten times a hundred. A thousand. And if you do go and add the ten numbers, you get a thousand, which is how you know the shortcut is a shortcut and not a different answer.

6:59One more thing that falls out of the endpoints. Not every odd number gets to start a piece. The ones that do are one, three, seven, thirteen, twenty-one, thirty-one, forty-three, fifty-seven, seventy-three, ninety-one. Ninety-one starts the tenth piece. Ninety-five does not start anything — it sits inside that piece, third from the left. Ninety-one is the forty-sixth odd number, and the piece runs from there to the fifty-fifth. So if you are handed a run and asked which cube it makes, the first term is enough to tell you, and it also tells you whether the run you were handed was a real one.

7:44Here is something the pattern hands you for free, and it is worth the two minutes. Add up the first four cubes: one, eight, twenty-seven, sixty-four. That is a hundred. Now count the odd numbers those four pieces used between them. One plus two plus three plus four. Ten of them. And the first ten odd numbers add to ten squared, which is a hundred as well. So the cubes up to any point add to the square of one plus two plus three and so on.

8:18Nobody had to prove that separately — it is just what the picture already says, if you read it sideways. So step back. There is one supply of odd numbers here and two things get built out of it. Take them one at a time and the running totals are the squares: one, four, nine, sixteen. Take them in pieces of one, two, three, four and the totals are the cubes: one, eight, twenty-seven, sixty-four.

8:50Sixty-four is in both lists. It is the eighth square and it is the fourth cube, and it came out of the same line of numbers both times. The odd numbers did not change. How you cut them did.