Weir and Rewind

Weir went first, and set up the forward chain.

"Here is a rule," she said, writing g(x) = 2x + 3. "It doubles, then adds three. Watch it flow. Pour in 5: double to 10, add three, 13." She drew the water stepping down two weirs. "So g(5) = 13. Out the bottom comes thirteen. A clean forward chain."

The children copied it. Thirteen sat at the foot of the hill.

Then Rewind stepped up, uncoiling her rope, and did not rub anything out. She simply took Weir's thirteen and began to walk it home.

"Now I bring it back," she said. "Weir's last step was add three, so my first undoing is subtract three: thirteen becomes 10. Her step before was double, so my next undoing is halve: ten becomes 5." She wrote the inverse beside it: g⁻¹(x) = (x − 3) ÷ 2. "And there it is. Five. The very number Weir started with."

There was a small, delighted stillness in the room — because the number had gone all the way down the hill and all the way back up, and landed on its own doorstep.

The academy master smiled and handed a fresh piece of chalk to a boy in the second row. "You run the whole chain — Weir's rule, then Rewind's — on a number of your own. Start with 8."

The boy took a breath. First he did Weir's job. "Pour 8 into g: double to 16, add three, 19." He drew the water stepping down. "So g(8) = 19."

"Now hand it to me," said Rewind gently.

He walked it back the way she'd shown him. "Rewind's rule undoes it: subtract three, 16; halve it, 8." He stopped, and looked up, and his eyes widened. "It's 8 again. I sent it out through the rule and back through the undo, and it came home. Exactly. Not a pace off."

"There it is," said Rewind, warm with pride.

"A rule chained to its own inverse," said Weir, "is the one chain in the whole world that changes nothing. You can send any number down it and it lands right back where it began. f⁻¹(f(x)) = x — undoing follows doing, and the two cancel to a clean, honest home."

Weir and Rewind looked at each other — the doer and the undoer — with the particular fondness of two people who spent a season thinking they were opposites and have just watched a child prove they were partners all along. "I could chain rules forward all day," Weir admitted, "and never be sure I could get back. Rewind is how I know I can."

"And I could walk any rule home," said Rewind, "but Weir is the one who shows me the road out in the first place. Doing and undoing. Neither of us is the whole of it."