Six Strangers

Soren had drawn six dots in a hexagon on the back of his place card and was connecting them when Maya leaned over.

"What's that going to be?"

"Nothing yet. I'm bored. There are six people at this table." He pointed his pen around the circle. "You, me, that kid with the dinosaur shirt, his sister, the twins. Six."

"So?"

"So I'm drawing who already knows who." Soren tapped each line as he made it. "I'll do red for people who came together and already know each other. Blue for total strangers."

Maya watched him connect dino-shirt to his sister. Red. The twins to each other. Red. Then the twins to dino-shirt. Blue, because they'd never met until ten minutes ago when one of them stole the other's juice.

"You and me," Maya said. "Red."

"Obviously." He drew it red.

They kept going. Every pair of the six got a line. Fifteen lines, because each of six people connects to five others, and you don't count each line twice.

When they finished, Maya stared at the tangle of red and blue.

"There's a red triangle," she said.

"Where?"

"You, me, and dino-shirt. We all know each other now. Three red lines, all touching, making a triangle."

Soren checked it. She was right. Three people, every pair red, forming a closed triangle.

"Huh," he said. "Make me another one. Different table."

So Maya grabbed a fresh place card and drew six new dots. She made up six new strangers and colored the lines however she wanted, trying hard not to make a triangle. Red here, blue there, scrambling them.

When she pushed it back, Soren scanned it. He found a blue triangle in the corner. Three of her imaginary strangers who all didn't know each other.

"You made one anyway," he said.

"I wasn't trying to."

"That's the point. Do it again. Really try to stop it this time."

Maya's eyes narrowed. She liked a thing she could lose at. She drew six more dots and went slowly, checking each line before she committed. Every time she added a color, she looked at every pair of dots that could close into a triangle and picked the color that broke it.

She got to the eleventh line. Then the twelfth. Her pen hovered.

"There's no good color," she said.

"What do you mean, no good color?"

"This line. If I make it red, it finishes a red triangle over here." She pointed. "If I make it blue, it finishes a blue triangle over there." She pointed somewhere else. "Both. Whichever I pick, I lose."

Soren leaned in. He traced both threatened triangles with his pen, slow, the way he checked things when he didn't believe them yet.

"Try a different dot order," he said. "Start somewhere else."

They started over. New card. They both worked it this time, arguing over each line, hunting for the arrangement that escaped. Red, blue, blue, red. Every time they thought they were safe, three lines somewhere had quietly agreed with each other.

"It's like the triangle is hiding and waiting," Maya said. "We close one trap and it just opens in a different spot."

"Do five dots," Soren said suddenly.

"Why five?"

"Because I want to know if it's six that does it or if it's always."

Maya drew five. Just five strangers, ten lines. And this time, when she went carefully, she found an arrangement with no monochromatic triangle at all. A ring of red around the outside, a star of blue across the middle. She turned it every direction. Clean. No three same-colored lines making a closed shape.

"Five is fine," she said, almost relieved. "Five lets you escape."

Soren had gone quiet, his pen resting on the five-dot card.

"So five you can do," he said. "Six you can't. We tried six like nine times. It always shows up."

"Maybe we're just bad at six."

"No." He pulled the six-dot card back and pointed at one person in the middle of it. Dino-shirt. "Look. This kid connects to five other people. Five lines coming out of him. Two colors. Five lines, two colors."

Maya saw it before he said it. "At least three of his lines are the same color. They have to be. You can't split five lines into two colors without at least three landing on one side."

"Right. Say three of his lines are red. Going to three people." Soren circled the three. "Now look at just those three people. The triangle between THEM."

Maya's pen moved fast now. "If any line between those three is red, it makes a red triangle with him." She drew it. "And if NONE of them are red, then all three lines between them are blue, and those three make a blue triangle by themselves."

They both looked at the card.

"There's nowhere for it to go," Maya said softly. "Either way. Red triangle with the middle guy, or blue triangle without him. You can't have neither."

Soren sat back. "Six strangers," he said. "Any six people in the whole world. Three of them know each other, or three of them are total strangers. There's no way to arrange six humans where that isn't true."

"Any six." Maya looked up from the card, out at the hall. There were maybe sixty people at the party. Grown-ups laughing, kids fighting over balloons, somebody's grandmother asleep in a folding chair. "Not just here. Any six. Six people on a bus. Six people who've never been in the same room. It's already decided. It was decided before they were born."

"You can't draw it any other way," Soren said. "We tried. It's not us being bad at it. It's that order has to show up. If the thing is big enough, it can't stay messy. The pattern is forced."

Maya picked up the six-dot card and held it so the light from the window came through the paper. The red and blue lines glowed faintly, the triangle sitting there in the tangle like it had always been there.

"How big does it have to get," she asked, "before four people are forced? Five? Before a whole shape you can't even see yet is just sitting in any big enough crowd, guaranteed?"

Soren had already flipped to a clean card and started drawing dots, more than six this time, his hand moving faster than he could count them.