Every Road Down

The power went out at eleven in the morning, which killed Maya's computer mid-download and left Soren holding a dead phone charger like a useless fishing line.

"Well," he said.

Maya was already at the whiteboard her mom kept in the kitchen for grocery lists. She'd wiped it clean and uncapped a blue marker. "Pick a number."

"Why?"

"Because I'm bored and you're bored and I want to show you something that broke my brain last night."

Soren sat on the counter. "Seven."

Maya wrote 7. "Odd or even?"

"Odd."

"So we multiply by three and add one." She wrote 22. "Now?"

"Even. Divide by two. Eleven."

They kept going. Eleven, thirty four, seventeen, fifty two, twenty six, thirteen, forty, twenty, ten, five, sixteen, eight, four, two, one.

"One," said Soren. "So it ends."

"Pick another number."

"Fifteen."

They ran it. Fifteen became forty six, then twenty three, then seventy, then thirty five. It climbed and fell and climbed again like something alive, something restless, until it finally tumbled down through eight, four, two, one.

"One again," Soren said. He pulled his notebook from his back pocket. "Does it always do that?"

"Every single time. Every number anyone has ever tried."

"How many have people tried?"

"Billions. Trillions. Computers have tested numbers so big you couldn't write them on every whiteboard in every school on earth."

Soren was already writing numbers in his notebook, running the sequence for twenty seven. It shot up to nine thousand two hundred thirty two before it started falling. He kept going, turning the page, filling the margins. His handwriting got smaller and smaller.

"It takes a hundred and eleven steps," Maya said. She'd done it the night before by flashlight when she should have been sleeping.

"But it still hits one."

"It still hits one."

Soren looked at the whiteboard, then at his notebook. "Okay, so someone proved it. What's the proof?"

"There isn't one."

He looked up. "What do you mean?"

"I mean nobody on earth knows why it works. It's called the Collatz conjecture. Mathematicians have been trying to prove it since 1937. Some of the best mathematicians who ever lived have tried and failed. One of them said mathematics is not yet ready for such problems."

Soren put his pen down. He didn't say anything for a while. Rain hit the window in small taps.

"That can't be right," he said. "The rule is so simple. If even, divide by two. If odd, multiply by three and add one. A first grader could do it."

"I know."

"And nobody can explain why it always lands on one?"

"Nobody."

Soren picked up his pen again and started testing numbers in the forties. Maya watched him fill half a page with sequences, each one a different shape. Some shot up fast and crashed. Some barely climbed at all. Some wandered for dozens of steps before finding their way down.

"They look like mountains," Maya said.

Soren was staring at his columns of numbers. "The dividing by two part pulls everything down. And the multiply by three and add one always makes an even number, so the next step always divides. It feels like it should be obvious."

"It does feel like that."

"But feeling obvious and being provable aren't the same thing."

Maya took the red marker and started drawing the paths on the whiteboard as branching lines. One going to two going to four going to eight. Everything funneling down. She drew three going to ten, then five going to sixteen, and she started connecting them. A tree, growing upward from one.

"Every number we test," she said, "joins the tree."

Soren came and stood next to her. He traced the branches with his finger. "So somewhere out there, either every number is already on this tree, or there's a number that isn't. A number that just keeps climbing forever. Or maybe loops back to itself without ever touching one."

"And nobody knows which."

"But nobody has ever found one that doesn't reach one."

"Not in almost ninety years of looking."

Soren added a branch. Then another. The tree filled the whiteboard, and they started writing smaller, and then Maya grabbed the green marker, and they kept building. The tree grew upward and outward, branches reaching toward the edges of the board.

"You know what's strange," Soren said. He was tracing the path for one hundred and thirteen. "It isn't that every number reaches one. It's that the rule is so simple and we can't say why. Two rules. That's it. And they create something we can't see the edges of."

Maya added the path for ninety seven, which wandered for over a hundred steps before joining the tree. She connected it to the branch that held thirty one and forty seven and already looked crowded.

"My mom thinks I stayed up too late last night for no reason," Maya said. "She saw my flashlight and said go to sleep."

"What did you say?"

"I said I was counting. She said count sheep."

Soren laughed, and it was the specific laugh of someone who has been told that the thing they are doing is the wrong thing to care about. Maya knew that laugh because she had the same one.

"Here's what I keep thinking," Soren said. "Maybe the proof needs math that doesn't exist yet. Maybe someone has to invent a whole new way of thinking about numbers before this question even makes sense to answer."

"Maybe that person is alive right now."

"Maybe they're eleven."

Maya looked at Soren. Soren looked at the tree.

The whiteboard was full. Every path they'd drawn, every number they'd tested, funneled down through the branches to the same root. One. Sitting at the bottom like a drain at the center of the universe, pulling everything toward it for reasons no living person could name.

Maya uncapped the blue marker again. In the last empty corner of the board, above the highest branch, she wrote a question mark.

Soren picked up the green marker and drew a branch reaching up to meet it, unconnected to anything, hovering at the edge of the whiteboard where the tree ran out of room.

The power came back on. Neither of them moved. The apartment hummed around them, and the rain kept tapping, and the branch hung there, reaching toward something that might or might not exist.