Four Colors and the Machine That Was Sure

The map on the wall was as tall as Soren and twice as wide. Forty-one counties, all penciled in by some careful person a hundred years ago, and now their job was to paint them.

"We have four," Maya said. She held up the jars. "Green, orange, yellow, blue. That's it. The purple dried into a rock."

"Mr. Adler said use as many as we want," Soren said. "He also said don't get paint on the floor, so."

"Four's not enough," Maya said. "Look at it. Some of these counties touch, like, six other counties."

"So the ones that touch can't match."

"Right. So we need more."

Soren looked at the map for a while. "Try it," he said. "Before we go ask. Just try to break it with four."

Maya liked breaking things. She started in the middle, at the fat county shaped like a boot. Green. The three around it, orange, yellow, blue. Easy. Then the next ring out.

"Blue again here," she said. "It doesn't touch the other blue."

"It touches the boot's green and the orange."

"So blue's fine."

They worked outward. Every time Maya hit a county that seemed stuck, she stopped, chewed the end of the brush, and tried a different order.

"This one," she said, near the river. "This one's the problem. It touches yellow, green, and orange all at once."

"Then it's blue."

She painted it blue. It worked.

"Okay, but that was luck," she said.

"Try the corner," Soren said. "The messy corner. If it breaks anywhere it breaks there."

The messy corner had a county like a spider, arms poking between five neighbors. Maya stared at it. She painted one neighbor, then had to go back and change it, then changed two more.

"See, it's fighting me," she said.

"It's not fighting, you just have to redo the earlier ones."

She redid them. Green became yellow. Orange stayed. The spider got blue. And it held.

Maya stepped back with paint on her wrist. "It keeps working," she said. "That's weird. Every single time I think it's impossible, four is enough."

"Maybe this map's just easy."

"No." She shook her head. "It's not the map. Draw me a worse one."

Soren tore a page from his notebook and drew. He made it ugly on purpose, blobs inside blobs, a county wrapped almost all the way around another, jagged shared edges. He slid it over.

"Break that," he said.

Maya colored it with four pencils. It took her three tries. It worked.

"Draw a worse one."

He drew a worse one. She colored it. Four.

They did this six times. Soren got meaner with every drawing. He made a county touch eleven others. He made a ring of counties around a center like a flower. He made one that spiraled.

Maya cracked every single one with four colors.

"That," she said quietly, "is not luck."

Soren looked at his row of ruined ugly maps, all of them four-colored, none of them broken.

"Somebody must have proved this," he said.

"Proved what?"

"That four is always enough. For any map. Ever. Not just these."

Maya put the brush down. "Any map that could ever exist," she said. "That's an infinite number of maps."

"So how do you prove it? You can't draw all of them."

That was the thing that stopped her. You couldn't try infinite maps the way she'd just tried seven. There would always be a worse one you hadn't drawn yet.

Mr. Adler came in with more rags. "How's my map?"

"Did anyone ever prove you only need four colors?" Maya asked.

"For maps? I think so. Ages ago." He waved a rag. "There was a fuss about it, I remember reading. Some mathematicians didn't want to accept the proof."

"Why not?"

"A computer did it," he said. "That's all I know. Ask the internet, that's what it's for." He put the rags down and left, because a museum with one worker is always a museum where the worker is leaving.

Soren already had his phone out. He read for a minute, moving his lips.

"Nineteen seventy-six," he said. "Two guys, Appel and Haken. They didn't check all the infinite maps. They found a way to boil every possible map down to a big pile of special cases. Like, a set of shapes, and if you could four-color all the shapes in the set, then every map ever had to work too."

"How many shapes in the set?"

He scrolled. "Almost two thousand."

Maya blinked. "Two thousand."

"And each one takes forever to check by hand. So they had a computer do it. Twelve hundred hours. It checked every case and none of them broke."

"So it's true."

"It's true," Soren said. Then he frowned at the screen. "But some mathematicians were mad. Because no human ever read the whole proof. No person could. It's too big. You have to trust the machine did every case right."

Maya sat down on the floor next to her jars.

"Wait," she said. "So there's a thing that's true. Definitely true. And we know it's true. But no single person has ever held the whole reason in their head at once."

"Right."

"Nobody. Not the two guys. Not anybody. The whole reason is too big to fit in a person."

Soren lowered the phone. "I hadn't thought about it like that."

"That's what made them mad," Maya said. "Not that it was wrong. That they couldn't fit it in." She looked at the map, forty-one counties, four colors, no two touching alike. "I did seven and I was already tired."

"You'd have to do two thousand. Perfectly. Twice, to be sure."

"I couldn't."

"Neither could I."

"Neither could Mr. Adler. Neither could anyone we'll ever meet." She said it slowly, like she was tasting it. "There's a true thing out there that's too big for a human mind, and we still get to know it. We just can't be the ones who hold all of it."

Soren wrote something on the ugly maps page. He didn't say what.

Maya picked up the blue jar again. In the spider county, the paint had gone tacky at the edges, but it held its border clean against the yellow and the green and the orange, four colors and not one more, on a wall, in a room, inside a fact nobody could carry whole.

She pressed the last neighbor full of orange and the map was finished.