The Rabbit Number

The rabbits next door had a number. That was the whole point of the project. Count this year, predict next year, win a ribbon.

Soren had the rule written at the top of his notebook. Take how full the field is, from zero to one. Multiply by a growth number. Multiply again by the room left over. That gives you next year.

"It's almost too simple," he said. He punched it into the calculator. Growth number two. Field half full.

"Run it again," Maya said. "Feed the answer back in."

He did. The number climbed, then steadied. Point five. Point five. Point five forever.

"Boring," Maya said, delighted. "It just sits there."

"That's good. Boring means we can predict it." Soren tapped the growth number up to three. Ran it. The number wobbled, then settled again, a little higher.

Maya leaned in. She had her elbows on the workbench and her chin on her hands, watching the little gray screen the way some kids watch a fire.

"Push it more," she said.

Growth number three point two. Soren ran the loop. The number climbed, dropped, climbed, dropped.

"It's stuck on two answers," he said. "High year, low year. Like the rabbits eat all the grass, then there's less grass, so fewer rabbits, so more grass."

"Feast and famine." Maya nodded. "That makes sense."

"It does make sense." He wrote it down. High, low, high, low. The same two numbers, over and over, no matter how long he ran it. He liked that. It meant the rule was honest.

"More," Maya said.

Three point five. He ran it. Wrote the numbers as they came. Then stopped writing.

"It's not two anymore," he said slowly. "It's four."

Four numbers, repeating. A loop of four. He ran it again from a different starting field, just to check, the way he always checked. Same four numbers. He believed it.

Maya was very still now. "Soren. It doubled. Two became four."

"By itself." He stared at the screen. "I only changed one number. The growth number. Everything else is the same rule."

"Push it again. Just a little."

Three point five five. He ran the loop. Counted the repeat.

"Eight," he said. His voice had gone quiet. "Eight numbers now."

"It's doubling faster," Maya said. She was already ahead of him, he could tell, leaning so close her breath fogged the corner of the screen. "Two, four, eight. The jumps are getting closer together. The growth number barely moved."

Soren wrote the steps in a column. Two at three point two. Four at three point five. Eight at three point five five. The gaps between the doublings were shrinking. Each new doubling needed a smaller and smaller push.

"They're going to pile up," he said. "The doublings. They're going to run out of room."

"Push past it." Maya's eyes were huge. "Push past where they pile up."

Soren set the growth number to three point nine. He didn't run it yet. His finger hovered.

"This is supposed to predict rabbits," he said. "It's the same rule. The exact same rule we started with. Multiply, multiply, feed it back."

"I know," Maya whispered.

He ran it.

The numbers came out and there was no pattern. None. Not two, not four, not eight. He filled a line of the notebook and there was no repeat in it. He ran it longer. He filled another line. The numbers jumped high, low, middle, almost the same as before but not, never landing in the same place twice.

"Run it again from the start," Maya said. "Same growth number. Same field. Exactly the same."

Soren typed point five zero zero for the starting field. Ran it. Wrote the numbers.

Then he changed the start by the tiniest amount he could. Point five zero zero one. One thousandth. He ran it again, right next to the first column, so they sat side by side.

The first few numbers matched. Then the fourth was a little off. The fifth was further off. By the tenth the two columns had nothing to do with each other. One said high. The other said low. Same rule. Same growth number. A start that was almost exactly the same.

"It split," Soren said. He looked at the two columns of his own handwriting like they belonged to a stranger. "A start that's basically identical, and ten steps later they're completely different."

Maya sat back on the stool. For once she didn't say push it again.

"So we can't predict the rabbits," she said.

"Not if real rabbits are anywhere near here." He pointed at three point nine. "Even if we knew the field exactly, we'd never know it exactly enough. One thousandth off and we're wrong inside ten years."

The garage was quiet. Outside, through the open door, the actual rabbits were doing whatever they were doing in the actual field, eating, multiplying, not knowing they were a number.

"It's the same rule the whole time," Maya said. "It didn't get more complicated. We didn't add anything. We just turned one knob."

"Three point two, it was boring. Predictable." Soren ran his finger down his column of doublings. "Three point nine, nobody could ever predict it. And there's no wall between them. No moment where it breaks. It just doubles and doubles until doubling turns into never."

Maya picked up his calculator. She didn't change the rule. She set the growth number right at the edge, in the place where the doublings piled up, the place neither of them had a number for.

"There's a spot in here," she said. "Between the part you can predict and the part you can't. A spot where it changes over."

"Yeah."

"Does the spot have a number?"

Soren opened his mouth to say he didn't know. Then he stopped, because something was standing up in his chest the way it did when a thing was too big for the inside of his head.

"It has to," he said. "It has to have a number. The doublings keep happening at growth numbers that get closer together. They're squeezing toward something. There's an exact place where boring ends and chaos starts." He grabbed the notebook back. "And it's the same place for the rabbits and for anything. Any rule shaped like this one. The doublings squeeze toward the same edge."

Maya stared at him. "How could it be the same number? For different things?"

"I don't know." Soren's pencil was already moving, measuring the gaps between his doublings, dividing one by the next, hunting for the number hiding in the squeeze. "Let's find out where the edge is."

Maya pulled her stool closer and read out the first gap.