The Party That Couldn't Stay Random

The gym smelled like floor wax and cold pizza, and nobody was dancing yet. Maya and Soren had the sign-in table, which meant markers, name tags, and a long stretch of nothing to do.

"I want to make one where it's impossible," Maya said. She was drawing dots on the back of a name-tag sheet.

"Impossible how?"

"Six dots. Six people at a party. I draw a line between two if they already know each other. Red line, they know each other. Blue line, they're strangers. I bet I can arrange it so there's no group of three friends and no group of three strangers."

Soren leaned over. "Three friends meaning all three lines red."

"All three red, yeah. A little red triangle. Or a little blue triangle, three strangers."

"And you want zero triangles of either color."

"I want a party that stays messy," Maya said. "No clumps. Everybody sort of even."

Soren tore off his own sheet. "I'll race you. First one to finish a clean one wins the last slice."

They drew. Six dots in a hexagon. Fifteen lines to color, every dot connected to every other dot.

Maya went fast. She always went fast. Red, blue, blue, red, red, blue. She got most of the way through, then stopped, uncapped a different marker, and made a small noise.

"What," said Soren.

"These three." She tapped three dots. "All blue. That's a blue triangle. Three strangers." She flipped the sheet and started over.

Soren was slower, checking each line as he went, muttering the pairs. He got all fifteen colored. Then he traced triangles with his finger, dot to dot to dot, and every few seconds his shoulders dropped a little.

"Red one," he said. "Here. Then a blue one over here." He turned the page and started again too.

The DJ tested a speaker. A few kids drifted in and stood against the wall, being strangers to each other in the ordinary way.

"Okay, new one," Maya said. "This time I keep every dot at three red lines and two blue. Balanced. Balanced things don't clump."

Soren looked up. "Why three and two?"

"Because five lines come out of each dot. If I split them even-ish, no dot gets greedy." She drew it carefully this time, slow as Soren, counting under her breath. Three red, two blue, out of every dot.

When she finished she sat back, waiting to feel like she'd won.

"Check it," said Soren.

She checked it. Her finger stopped. "No."

"Triangle?"

"Red. Right here." She pressed her thumb flat on it as if she could hold it down. "But I balanced it. I did everything right."

Soren pulled her sheet next to his. He had four crossed-out attempts. She had three. Between them, seven parties, and every single one had a triangle somewhere. Red or blue, but always one. "Do this," he said finally. "Pick any dot. Any one."

Maya pointed at a dot.

"Five lines come out of it. Two colors. If you split five into two colors, one color has to have at least three. You can't split five evenly."

"Three and two. Yeah. So?"

"So say three of them are red." Soren touched the three dots at the ends of those red lines. "These three friends are all friends with the first person. Now look at just these three, among themselves."

Maya looked. Three dots. Three lines between them.

"If any one of those three lines is red," Soren said slowly, "that red line plus the two red lines going back to the first dot makes a red triangle."

Maya's eyes moved. "And if I want to avoid that, I have to make all three of those lines blue."

"All three blue," said Soren.

"But three blue lines between three dots is" — her mouth stayed open a second — "a blue triangle."

Neither of them said anything. The speaker thumped once and went silent.

"Do it again," Maya said. "Say it again, I want to try to break it."

Soren said it again. Pick a dot. Five lines. One color has three of them. Those three friends either connect with a matching line, and that's a triangle, or they all avoid it, and that's the other triangle. There was no third thing. There was no door out.

Maya grabbed a fresh sheet and tried anyway. She made the first dot's three red friends. She made all three of their lines blue to dodge the red triangle. She stared at the blue triangle she had just built with her own hand.

"I can't not make one," she said. It came out almost like a laugh. "Soren. It doesn't matter how I color it. Six people and it's already decided. There's a clump before anybody even shows up."

"With six," Soren said. "What about five?"

Maya was already drawing. Five dots. She colored, checked, and this time when her finger went around the triangles it kept going and found nothing. A clean five. No red triangle. No blue triangle. Messy the whole way through.

"Five stays random," she said. "Five is fine." She set the marker down like it was fragile. "Six can't. One more dot and the messiness runs out."

More kids were coming in now, filling the gym, and Maya was looking at them differently. Soren could tell because he was doing it too. Counting.

"There's like forty people in here," Soren said.

"So somewhere in this room," Maya said, "there are three people who all know each other, or three who are all total strangers. Guaranteed. Right now. We don't have to find them. They have to be there."

"They can't not be there."

Maya looked around the gym, at the clumps and the wall-standers and the DJ nobody knew, and there was something enormous in her face, the kind of thing that doesn't fit behind the eyes.

"It's not the people," she said quietly. "You could swap all of them out. It's the six. The number does it. The number was going to do it no matter who walked in."

Soren opened his notebook. His pen moved across the page: six dots, fifteen lines, and under it, small, a five with a checkmark and a six with a circle around it.

Out on the floor the crowd shifted and reformed, strangers finding friends, and somewhere in the churn a triangle nobody could see clicked quietly into place, exactly on schedule.