Undo

Undo's brother, whose name was Joist, was a carpenter. He was nine years older than Undo. From the time Undo was small, she would sit on a little stool in Joist's workshop and watch him build. He built chairs, tables, shelves, doors. His favourite things to build, though, were cabinets. Cabinets with drawers. Cabinets with hinges. Cabinets with doors that swung and latched and stayed shut and opened cleanly when you wanted them to.

Joist's cabinets were beautiful. Customers came from three towns over. He was, the local carpenters' guild had told their mother, "unusually patient."

What Undo noticed, sitting on her little stool, was that Joist did something at the end of every cabinet that no other carpenter she had watched did.

He unbuilt it.

Not completely. Not destructively. He would, after spending two weeks building a perfect cabinet, take it almost entirely apart again. He would pull each drawer out and slide it back in. He would unscrew each hinge and screw it back on. He would test every joint by pulling on it in the opposite direction it was meant to hold. He would, in short, undo every piece of his own work, and then put it all back together.

Undo, who was seven, asked him why.

Joist said: "I do not trust a joint I have not undone."

She thought about this for the rest of the day.

What she eventually understood — and she understood it slowly, over several years of watching — was that every step in building a cabinet was the opposite of some step in taking one apart. If you nailed a board, the opposite was pulling the nail. If you glued a joint, the opposite was separating the joint with a thin blade. If you screwed in a hinge, the opposite was unscrewing the hinge. For each action, there was an action that exactly reversed it. And Joist insisted on knowing all of them.

This was, Undo realised much later, exactly the principle of inverse operations.

In algebra: if you added five to one side of an equation, the operation that exactly reverses it is subtracting five. If you multiplied one side by three, the operation that exactly reverses it is dividing by three. If you squared a quantity, the operation that exactly reverses it (for positive numbers) is taking the square root. Every step has an undo. Every step. If you do not know the undo, you cannot trust the step.

When Undo arrived at the EquationQuest academy at nineteen — with a small toolbox her brother had given her containing six different small implements, each of which was a reverse of some carpentry action — the academy master watched her for a few minutes and said, "You will be teaching inverse operations. Welcome."

Undo has been teaching ever since. She still keeps the toolbox in her classroom. She uses it as a demonstration. She holds up a hammer. She says: "This puts a nail in." She holds up the back end of the hammer, which has a small claw. She says: "This takes a nail out. They are the same tool. They are the same idea, in two directions."

Children remember this. Children draw little hammers in their notebooks.

There is a story she tells, every year, in kit 8. It is a true story. Joist built a cabinet for a customer once — a beautiful corner cabinet — and the customer, after taking it home, sent a letter back saying the cabinet's main door would not open. Joist was, briefly, panicked. He could not understand. He had tested the door. He had opened and closed it a hundred times. He went to the customer's house. He looked at the cabinet.