Cubes: (a ± b)³ from a cube cut into eight pieces

0:00Two students are asked to expand the same thing: a minus b, squared, times a plus b. The first substitutes: a minus b squared is a squared minus two a b plus b squared. Then multiplies by a plus b. The second refuses to square anything. She pairs one a minus b with the a plus b, and gets a squared minus b squared for free. Then a minus b, times a squared minus b squared.

0:33Both arrive at a cubed minus a squared b minus a b squared plus b cubed. Both are right. The second route is shorter, and it is legal for one reason: rebracketing a product changes nothing. And it is the method of everything that follows: a new identity is an old one, used again. So take the one everybody has: a plus b, squared, is a squared plus two a b plus b squared.

1:05Multiply it by a plus b one more time. The a hits all three terms, the b hits all three. Six products, and two pairs collect. a cubed, plus three a squared b, plus three a b squared, plus b cubed. There it is. And it costs one multiplication. But that derivation does not tell you where the threes came from. Only that two terms happened to collect with two others.

1:34The threes deserve better than an accident. So put the algebra down and build the thing. Here is a cube. Every edge is a plus b: a stretch of length a, then a stretch of length b. Its volume is a plus b, cubed. That is what we are measuring. Now cut it at the mark. Once across, once through, once down. Before we look: how many pieces does that make?

2:02Three cuts, each one doubling the count. One becomes two, two becomes four, four becomes eight. Eight pieces. And nothing about that number is a guess. Here they are, pulled apart. In one corner, the piece that is a in every direction. A cube of edge a, volume a cubed. In the opposite corner, the piece that is b in every direction. A small cube, volume b cubed. The other six are boxes, and they come in two kinds.

2:37Three of them measure a by a by b. Three of them measure a by b by b. So: two cubes and six boxes — three of one kind and three of the other. Which is the question. Why three? Because a piece is not chosen. It is decided, three times over. Go along the length: is this piece on the a side of the cut, or the b side? Same question across the width. Same question up the height.

3:08Three questions, two answers each. That is your eight. Now sort them by how many times you said b. Never: one way to do that. Volume a cubed. Exactly once: three ways, because it could have been any of the three directions. Volume a squared b. Exactly twice: three ways again, one for each direction that stayed a. Volume a b squared. All three times: one way. Volume b cubed.

3:39One, three, three, one. Not a pattern to memorise. A count of which direction carried the b. Now add them up, because between them they are the whole cube and nothing else. a cubed, plus three lots of a squared b, plus three lots of a b squared, plus b cubed. The same four terms the multiplication gave. The picture proves nothing new; it explains the part the algebra left as a coincidence.

4:11I checked the solid the way you check a jigsaw. Twenty five pairs of lengths, and every time the eight volumes add to the whole, nothing overlapping, nothing sticking out. One thing that check taught me. Any cut tiles the cube. Cut in the wrong place and the eight pieces still add up perfectly. So adding up is not the argument. The sizes of the pieces are. Now the minus version. And there is nothing to derive.

4:44a minus b is just a plus, minus b. So put minus b into the identity wherever b appears. First term: a cubed. No b in it, so it does not move. Second: three a squared, times minus b. That is minus three a squared b. Third: three a, times minus b, all squared. Minus b squared is plus b squared. So this one stays positive: three a b squared.

5:14Fourth: minus b, cubed. Minus. a cubed, minus three a squared b, plus three a b squared, minus b cubed. Look at what happened to the signs. Plus, minus, plus, minus. The instinct is that a minus should turn everything negative. It does not. Each term carries a different power of b. The first carries none, the second one, the third two, the fourth three. A minus sign raised to an even power comes back positive. Raised to an odd power it stays negative.

5:53So the odd powers of b flip and the even ones do not. Alternating is what that looks like from a distance. I ran that across the first eight powers, forty four terms in all, and it is the rule every single time. And notice the picture has been left behind. There is no solid with an edge of minus b. The substitution does not care. Backwards now. A cube has volume p cubed plus six p squared q plus twelve p q squared plus eight q cubed. What is its edge?

6:32Look at the ends first. p cubed is the cube of p. Eight q cubed is the cube of two q. So the edge is p plus two q, if it is anything at all. Now the middle terms decide, and that is the part people skip. Three, times p squared, times two q: six p squared q. Correct. Three, times p, times two q all squared: three p times four q squared is twelve p q squared. Correct.

7:05Edge: p plus two q. The ends propose. The middles decide. Change one middle term by one and it stops being a cube — checked on a hundred and ninety six. Same job, minus version. Eight n cubed, minus sixty n squared m, plus a hundred and fifty n m squared, minus a hundred and twenty five m cubed. Ends: eight n cubed is two n, cubed. A hundred and twenty five m cubed is five m, cubed.

7:40Signs alternate, so it is a minus. Test the middles. Three times two n squared, times five m, is three times four n squared times five m: sixty n squared m. Correct. Three, times two n, times twenty five m squared: that is one hundred and fifty n m squared. Correct. Two n minus five m, cubed. One warning. The terms will not always arrive in order. Sort them by the power first, or you will test the wrong pair.

8:18This does arithmetic too. A hundred and forty seven, cubed. Split it at a round number: a hundred and fifty, minus three. Four pieces: three million, three hundred and seventy five thousand. Minus two hundred and two thousand five hundred. Plus four thousand and fifty. Minus twenty seven. Three million, one hundred and seventy six thousand, five hundred and twenty three. The pieces shrink fast, which is why it is worth doing at all.

8:51A hundred and ninety nine wants two hundred minus one. A hundred and twenty seven wants a hundred and thirty minus three. A negative is nothing new: minus a hundred and seven cubed is minus the cube of a hundred and seven, which splits as a hundred plus seven. Every three digit number there is, split at the nearest ten, comes out exactly right. Nine hundred of them. The leftover is never more than five.

9:24One last thing, because a good picture is easy to over-trust. The solid needs two positive lengths. Feed it a negative and there is nothing to build. Out of eighty one pairs of numbers I tried, sixteen can be built as a solid. The identity holds for all eighty one. So the picture covers a corner of the truth. The multiplication covers the rest, and it assumed nothing about a or b.

9:54One thing I expected and got wrong. I assumed b had to be the smaller one. It does not: the cut works whichever is bigger, and only six of those sixteen have b under a. And the thing the picture kills on sight: a plus b, cubed, is not a cubed plus b cubed. The six boxes in between are exactly what that error throws away. Eight pieces. Three decisions each. One, three, three, one.