Swatch the Border-Painter

Swatch had learned to see touching long before she learned to see anything else.

She'd grown up in a family of quilt-makers, in a workshop where scraps of cloth were stitched edge to edge into enormous blankets. Her mother had one unbreakable rule: no two patches that touched along a seam could be the same color, or the whole quilt went muddy and the pattern disappeared. Little Swatch spent years laying out patches, sliding them around, hunting for clashes.

The day everything clicked, she'd been stuck on a corner where four patches all crowded together. She kept running out of colors. Then she stopped staring at where the patches were and started drawing little lines — one line between every two patches that shared a seam. Suddenly the puzzle wasn't about cloth at all. It was about the lines. About which patch was tied to which.

"It's not the patches," she whispered to herself. "It's what's joined."

Once she saw the joins, the colors fell into place. She found she could almost always do it with very few colors — often just three or four, even for a quilt with hundreds of patches. The trick was never the painting. The trick was seeing what touched what.

When she was older, Swatch carried a roll of her quilts all the way to DiscreteQuest, because she'd heard it was a school that cared about how things connect.

The head of the academy, an old owl with a slow, warm voice, unrolled one of her blankets and studied it. "What is graph coloring?" he asked.

Swatch answered without hesitating. "You take a bunch of things, and you draw a line between any two that mustn't match. Then you give each thing a color so that no two things joined by a line wear the same one — and you try to do it with as few colors as you can manage." She tapped a patch. "Maps, quilts, timetables, seating plans. They're all the same puzzle wearing different clothes. It's always what touches mustn't match."

"And the fewest colors?" the owl asked.

"That's the art of it," Swatch said. "Anyone can color a map if you give them a hundred paints. The lovely part is finding the smallest number that still keeps every neighbor happy. For flat maps it's never more than four. Took the world an awfully long time to be sure of that one."

The owl's feathers settled approvingly. "You are appointed."

Swatch's favourite student problem had nothing to do with maps at all.

A harried badger named Mop came to her near tears. "The academy clubs are a mess," he said. "Chess club and choir share three members, so they can't meet on the same afternoon — those three can't be in two places. Choir and garden club share members too. And chess and garden club share one. I've got clubs clashing everywhere and I don't know how many afternoons I need!"

"Then let's not think about clubs," Swatch said calmly. "Let's think about clashes." She drew a dot for each club, and a line between any two clubs that shared a member. "Every line is a 'these two can't meet together.' Now I'll color the dots — and a color is just an afternoon. Joined dots get different colors, because clubs that share members need different afternoons."

She dabbed: chess blue, choir yellow, garden club blue again — because chess and garden club shared no member, so they could safely twin. "There. Two afternoons. Blue and yellow. Everybody fits, nobody's torn in half."

Mop stared at the little colored dots. "You turned my whole timetable into a coloring page."

"Every clash problem is a coloring page," Swatch said, grinning. "You just had to draw the lines."