Boxer Builds the Room for Every Piece

Boxer is a badger, and badgers are builders. Give a badger a muddle and the first thing it wants to do is put up walls and doors and make the muddle into rooms - because a thing with rooms can be walked through one at a time, and a muddle cannot.

The muddle people bring Boxer is always the same kind. Somebody wants to multiply (a + b) by (c + d), and they try to do it all in their head at once, and they end up sure they have missed something. They usually have.

“Stop,” Boxer says, rolling up his sleeves. “Let’s build you a box.”

He draws a rectangle. Down the left side he writes the two parts of the first factor - a on top, b below. Across the top he writes the two parts of the second factor - c on the left, d on the right. Two rows and two columns make four rooms, and Boxer walks into each one in turn.

The top-left room sits under c and beside a, so it holds a times c, which is ac. The top-right room, under d and beside a, holds ad. Down in the bottom-left is b times c, or bc. And the bottom-right, under d and beside b, is bd. Four rooms. Four pieces. ac, ad, bc, bd.

“Here is why the box is honest,” Boxer says, patting a wall. “Every part of the first factor has to meet every part of the second factor - a meets c, a meets d, b meets c, b meets d. In the box, each of those meetings gets its own room, and there is no way to leave a room out, because you can see the empty space where it would be. Try to skip one and the box just stares at you with a hole in it.”

That is the whole idea, and Boxer trusts it completely: to multiply two sums, give every pair of parts its own room, fill each room with its little product, and add up all the rooms. The box does not let you forget a pairing, the way doing it in your head does. It holds every piece in plain sight until you have counted it.

The badger has a favorite feeling, and it comes at the end - when all four rooms are filled and he steps back to look at the finished box. It is the calm feeling a builder gets standing in a house with no gaps in the walls: everything has its place, and every place is filled. No draft. No hole. Nothing forgotten in a corner.

“Add them up,” he says. “ac, plus ad, plus bc, plus bd. That’s your product. Not because I said so - because you can point at every single piece of it.”

Then he grins, because he knows a secret the box has been hiding. “And if the two factors happen to be the same - if you’re squaring (a + b) - watch what happens.” He relabels the top with a and b too. The top-left room becomes a². The bottom-right becomes b². And the other two rooms? One is ab, the other is ba - the same thing, twice. Miro and Moro, right there in the box.

“Two of the same room,” Boxer says, delighted. “That’s your 2ab. The box knew all along.”

And Boxer stands in his neat four-room box, every wall square, every room filled, exactly the way a badger likes a finished thing to be.