Parry and Domina

She had tried everything. Dominoes across, dominoes down, starting from the middle, starting from an edge, clever spirals, stubborn rows. Every single time, she'd cover almost the whole board and end up with two lonely squares stranded in different places, impossible to bridge with one domino.

"I'm just not seeing it," she muttered, sweeping the tiles off to start again. Her chest had that tight, sinking feeling — the one that whispers everyone else could do this. She'd counted the ways for a hundred boards. Why could she not find one way for this one?

Parry wandered over, took one look at the snipped-corner board, and did not pick up a single tile.

"How long have you been trying?" Parry asked.

"Long enough to feel stupid," Domina admitted.

"Good news," Parry said. "You're not stupid. The board's just lying to you."

Parry pulled out two colors of marker. "Watch. I'm going to colour it like a real checkerboard — light, dark, light, dark, every square." They filled it in, the familiar alternating pattern spreading across the board. "Now. A domino always covers two squares that touch, side by side. And on a checkerboard, two touching squares are always one light and one dark. Every domino, no matter where you put it, eats exactly one light and one dark. One of each."

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

"So," Parry went on, "if the board could be fully tiled, it would have to have the same number of lights as darks — because the dominoes come in one-of-each pairs." They pointed at the two snipped corners. "But look which corners got cut off. Opposite corners of a checkerboard are always the same colour. Both dark. So this board has thirty-two lights and only thirty darks. Two extra lights, with no darks to pair them."

Domina stared at the board she'd fought all afternoon.

"Two extra lights," she repeated. "And every domino needs one of each."

"So no matter how clever you are," Parry said gently, "you'll always finish with two lights left over and nothing to cover them with. Not because you missed a trick. Because there is no trick. The count of tilings isn't small. It's zero."

Something in Domina's chest unclenched all at once. It wasn't the tight sinking feeling anymore. It was almost the opposite — a clean, quiet certainty.

"I wasn't failing," she said. "I was proving it, the slow way — trying and trying and always getting the same two leftovers." She laughed, a little shaky. "You just did it the fast way. You counted the colours."

"That's all impossibility ever is," Parry said. "It's counting — from the other side. You count how many ways: sometimes it's a big friendly number, sometimes it's small. And sometimes you count and it comes out zero, and the colours tell you why."