Twoby the Pair-Matcher

Twoby had been a pairer-of-things since she was small.

Her family laid tile floors — little square tiles, set down two at a time from a double-wide trowel that always placed exactly two. As a kit she loved watching the floor fill up, two tiles, two tiles, two tiles. But one day a customer asked for a floor with an odd number of spaces, and no matter how her father laid the tiles — two at a time, two at a time — there was always, always one bare square left at the end.

"Why can't we just cover it?" little Twoby had asked.

"Because we lay them two at a time," her father said. "Two at a time can only ever fill an even number of squares. An odd floor will fight us every single time. Not because we're clumsy. Because it simply can't come out even."

That landed in Twoby like a stone in a still pond. It wasn't about trying harder. Some things couldn't be done, and you could know they couldn't, just from the way the counting worked. You didn't have to try every arrangement and fail. You could see the impossibility from the start, in the leftover.

When she was grown, Twoby waddled the long road to DiscreteQuest, because she'd heard it was a school that respected a good "that's impossible."

The head of the academy, an old owl with patient eyes, asked her, "What is a parity argument?"

Twoby answered slowly and surely. "Parity is just even-or-odd. You pair things two by two and see what's left. And an argument from parity is this: if there's something about a puzzle that's always even, or always odd, no matter what move you make — then anything that would break that pattern simply can't happen." She folded her paws. "You don't check every case. You find the one thing that never changes, and let it do the proving."

"Give me an example," the owl said.

"Each handshake involves exactly two hands," Twoby said. "So all the hands that ever shake, counted up, must be an even number. Always. That one unchanging fact rules out whole piles of impossible claims, without my checking a single party guest."

The owl's eyes gleamed. "You are appointed."

Twoby's favourite thing was using a leftover to settle an argument for good.

A stubborn marten named Cobb came to her with a checkerboard — eight squares by eight, dark and light, sixty-four squares in all — and a pile of little tiles, each one long enough to cover exactly two squares side by side.

"I want to cover the whole board with these two-square tiles," Cobb said, "but I snipped off two corners — see, the two dark corners, opposite each other. Now I can't make it work and I've been trying for hours. There must be some clever way."

Twoby looked at it and shook her head, kindly. "There's no way, Cobb. And I can prove it without you trying for one more second."

She pointed at a single tile. "Every tile covers two squares that touch — and on a checkerboard, two touching squares are always one dark and one light. Always. One of each. That never changes." She tapped the board. "So however you lay them, the tiles always cover the same number of dark squares as light ones. But you snipped off two dark corners. Now there are thirty light squares and only —" she counted — "thirty-two light, thirty dark. Wait —" she recounted gently — "the board's missing two darks, so it has more light than dark. The tiles need them equal. They never can be. So it can't be covered. Not because you're not clever. Because the leftover won't allow it."

Cobb stared at the board for a long moment, then set down the tile he'd been gripping. "Hours," he breathed. "And one leftover would've told me."

"One leftover almost always knows," Twoby said.