The Sixth Point

The machine failed for the seventh time with a sound like a polite cough.

Every light-thread in the glass case went dark except three blue ones. They formed a triangle around the floating beads labeled Luna School, Mars School, Ocean School.

The exhibit computer said, "Pattern found. Reset?"

"No," said the museum designer, who had a wrench in one hand and a half-eaten breakfast bar in the other. "No pattern found. That is the opposite of what we are building. Reset."

Soren pressed reset.

The six silver beads drifted back to their places inside the clear case. Every bead had to connect to every other bead. The visitor would choose orange or blue for each light-thread. The challenge was printed on the sign already: Can you color every connection without making a single-color triangle?

Maya had disliked the sign since morning.

"It sounds smug," she said.

"Signs are allowed to be smug after the exhibit works," said the designer. Her name was Ms. Vale, and she had been awake long enough that her hair had started making its own decisions. "Before that, signs should be nervous. Soren, try a different sequence."

Soren looked at his paper notebook. On the station, almost everyone kept notes in airscreens that followed them around like obedient dragonflies. His notebook had a bent corner and a pencil loop. Two older kids from the exhibit crew had already asked if it was for historical reenactment.

He did not answer them. He had drawn six dots and fifteen tiny lines.

"If I change the middle choices, the last line still finishes a triangle," he said.

"Then don't save that line for last," Ms. Vale said. "The visiting director arrives in thirty minutes. We need a version that does not embarrass me in front of a person wearing polished shoes in orbit."

Maya put her palm against the glass. The beads trembled in their magnetic pockets.

"Use five," she said.

Ms. Vale blinked. "There are six schools in the festival. Six beads. Six logos. Six principals who will send me messages with many exclamation points if their bead is missing."

"Just to test," Maya said.

Soren had already turned off the bead labeled Cloud School. The remaining five beads made a loose pentagon.

"Orange around the outside," Maya said.

Soren touched the controls. Five orange threads lit around the rim.

"Blue across," she said.

The five crossing threads shone blue, making a star inside the orange ring.

The computer waited.

No cough.

No triangle glowed.

One of the older kids leaned over. "So the machine is fine. Six is cursed."

"Six is scheduled," Ms. Vale said.

Maya did not look away from the glass. "Six is different."

"Different how?" Soren asked.

She tapped one bead, then the dark place where Cloud School had been. "Five can dodge. Six can't."

Soren's pencil moved. Not fast. Not slow. Like a little machine that wanted permission from the world before going on.

Ms. Vale sighed and checked the time on her sleeve. "Children, I adore the poetry of doom, but I need a setting. Randomize more. Hide the triangles. Make it clever."

"Maybe clever is the problem," Soren said.

Ms. Vale stopped with the breakfast bar halfway to her mouth.

Soren's ears turned red, but he kept going. "I mean, maybe it is not failing because it is unclever. Maybe there is no way."

The older kid laughed once. "There is always a way. That's what programming is."

Maya pointed at the five-bead pattern. "Then add the sixth bead and beat it."

The older kid reached for the controls. Soren moved his notebook out of the way, but not the pencil. The kid colored Cloud School's links quickly. Blue to Luna. Blue to Mars. Orange to Ocean. Orange to Desert. Blue to Forest.

The computer coughed.

Three blue lines glowed: Cloud, Luna, Mars.

"Again," Maya said.

This time the kid tried orange, blue, orange, blue, orange.

The computer coughed.

Three orange lines glowed: Cloud, Luna, Forest.

"It is choosing favorites," the kid said.

"It is choosing math," Soren said.

Ms. Vale set down the wrench. "Show me. Quickly. With no words longer than my patience."

Soren tore a clean scrap from the back of his notebook. He drew one dot in the center and five around it.

"Pick the new bead," he said. "It has five links coming out. Each link has only two choices, orange or blue."

Maya grabbed five marker caps from the supply tray and pushed them into a row. "You cannot split five evenly into two colors. One color gets at least three."

She sorted them before anyone answered. Three blue caps on one side. Two orange caps on the other.

"So from the new bead," Soren said, "there are three links of the same color going to three other beads. Say blue."

He drew them.

Ms. Vale leaned closer despite herself.

"Now look at those three outer beads," Soren said. "If any link between them is blue, it makes a blue triangle with the new bead."

Maya took the blue marker and drew one. The computer, unasked, caught the pattern through the glass and lit a soft blue triangle in the case.

"If none of those links are blue," Soren said, "then all the links between those three have to be orange."

Maya capped the blue marker. She drew the three outer sides in orange.

An orange triangle appeared in the case.

No one talked for a moment.

The air fans hummed. Far below the transparent museum floor, Earth curved blue and white and much too large to be a school-map circle.

Maya's hand stayed on the glass.

"Five beads can keep secrets," she said. "Six tells on itself."

Ms. Vale looked at the printed sign. Then at the six beads. Then at the older kid, who had stopped smirking and was staring at Soren's scrap of paper.

"I spent three days asking you to build an impossible exhibit," Ms. Vale said.

"It's still an exhibit," Maya said.

"It is a sign that lies."

"Change the sign."

Soren picked up the printed card and turned it over. On the blank back he wrote, in block letters, Find the Triangle That Has to Be There.

Ms. Vale read it. Her face did a tired thing first, then a sharp thing, like a window catching sunrise.

"Visitor chooses the colors," she said. "Computer does not stop them. It waits until all fifteen connections are lit, then reveals the triangle."

"Or lets them try to find it first," Soren said.

"And if there are several?"

Maya grinned. "Better."

They worked fast.

Ms. Vale rewired the display mode and muttered at screws. Soren set the computer to keep quiet until the last thread. Maya tested the touch panel by coloring badly on purpose, then carefully, then in a way that made her squint because it nearly looked as if it might escape.

It never escaped.

At the preview, the visiting director arrived in polished shoes that clicked against the station floor. A cluster of kids came with him. One of them pointed at Soren's notebook poking from his pocket.

"Is that paper?" she asked.

"Yes," Soren said.

"Why?"

Before Soren answered, Maya pressed the start button.

The six beads brightened.

"Try to beat it," Maya said.

The girl with the paper question stepped up. She colored slowly at first, then faster. Orange. Blue. Blue. Orange. Orange. Blue. She paused at the last connection, glanced at the sign, and chose orange with a little jab of her finger.

All fifteen threads shone at once.

Nothing happened.

The girl smiled.

Soren counted under his breath. His finger moved from bead to bead without touching the glass.

"There," he said.

The girl frowned and leaned in.

Maya did not press reveal. She waited.

The girl traced three blue threads with her eyes. Luna to Desert. Desert to Forest. Forest to Luna.

"Oh," she said.

Then the computer brightened the blue triangle she had found.

Across the case, another orange triangle glimmered too, one she had not noticed.

Then another blue one.

The director took off his polished shoes and tucked his feet under the floor rail so he could float closer like everyone else.

"Does this always happen?" he asked.

Ms. Vale opened her mouth.

Then she looked at Maya and Soren.

Soren said, "With six points, two colors, and every pair connected, yes."

Maya said, "You can hide for five. Not for six."

The girl touched the glass where the first blue triangle had appeared. "What if I wanted four corners all the same color?"

Ms. Vale made a small sound, the kind adults make when a finished thing grows teeth.

Soren's pencil was already out.