Parry and Domina

She had tried absolutely everything she could think of. Dominoes laid across in tidy rows, dominoes standing on end in columns, starting from the very middle and working out, starting from an edge and working in, clever winding spirals, stubborn brute-force marches. And every single time, without exception, she would cover nearly the whole board and then find herself stranded with two lonely squares left over — sitting in different corners of the board, too far apart to ever be bridged by a single domino.

"I'm just not seeing it," she muttered, sweeping all the tiles off with the side of her hand to start over yet again. Her chest had gone tight, with that low sinking feeling underneath it — the one that leans in close and whispers everyone else could figure this out. She had counted the ways for a hundred boards without breaking a sweat. Why, then, could she not find so much as one way for this one stubborn board?

Parry drifted over, took a single unhurried look at the snipped-corner board, and — pointedly — did not reach for a tile.

"How long have you been at this?" Parry asked.

"Long enough to feel genuinely stupid about it," Domina admitted.

"Well, I've got good news," Parry said. "You're not stupid. The board is lying to you."

Parry produced two markers, one light and one dark. "Watch this. I'm going to colour the whole thing in like a proper checkerboard — light, dark, light, dark, every square taking the opposite of its neighbor." They filled it in swiftly, the familiar alternating pattern rippling out across the board. "Now hold on to this next part, because it's the key to everything. A domino always covers two squares that are touching, side by side. And here's the thing about a checkerboard: any two squares that touch are always one light and one dark, never a matching pair. Which means every domino you could ever lay down, anywhere on this board, in any direction, swallows exactly one light square and exactly one dark square. One of each. No exceptions, ever."

Domina nodded slowly. That part she knew deep in her bones.

"So here's the trap," Parry went on. "If this board could be completely tiled by dominoes, then it would have to hold exactly the same number of light squares as dark ones — because the dominoes only ever come in one-of-each pairs, and pairs can only add up evenly." They tapped the two missing corners, the empty notches where the board had been snipped. "But look very carefully at which corners got cut away. On any checkerboard, two opposite corners are always the same colour as each other. Both of these were dark. So this poor board is left with thirty-two light squares and only thirty dark ones. Two extra lights — and not a single dark square anywhere left to pair them off with."

Domina stared down at the board she had wrestled with all afternoon.

"Two extra lights," she repeated quietly. "And every domino has to take one of each."

"So no matter how clever you are, or how long you keep at it," Parry said, gently now, "you will always end up with two light squares stranded and nothing on earth to cover them with. Not because you missed some trick you should have spotted. Because there is no trick. The number of ways to tile this board isn't small, and it isn't hiding. It's zero."

Something in Domina's chest let go all at once. It wasn't the tight, sinking feeling anymore — it was very nearly the opposite of that: a clean, level, quiet certainty settling into the space where the dread had been.

"I wasn't failing," she said, half in wonder. "I was proving it — just the slow, painful way. Trying and trying and always finishing with the exact same two leftovers, over and over." She laughed a little, still shaky around the edges. "You just did it the fast way. You counted the colours."

"That's all impossibility ever really is," Parry said. "It's counting — from the other side of the mirror. You count how many ways a thing can be done: sometimes the answer is a big, friendly number, sometimes a small one. And every so often you count, and the answer comes out zero — and the colours are right there to tell you exactly why."