The Rabbits That Wouldn't Settle

The rain had trapped them in the upstairs room of the fish-farm office, which smelled like old algae and warm dust. On the chalkboard somebody had left an equation and a scribble that said HOW MANY FISH NEXT YEAR?

"It's a rule for guessing," Soren said. He read it off the board. "Take this year's crowd, multiply by a number, then knock it down by how full the pond already is."

"Full ponds have less room," Maya said. "So more fish now means fewer new ones. Fine. What's the number?"

"The board just says R. The owner circled it like it mattered."

Maya found a stub of chalk and a bucket of dried-up markers. "Let's just run it. Pretend the pond is one, all the way full, is one. Start halfway. Feed the answer back in and do it again."

They picked R equals two and started chaining numbers, each year's answer becoming next year's start. Soren did the arithmetic on the back of a delivery receipt.

"Point five," he said. "Then point five. Then point five again. It stopped."

"It parked," Maya said. "The pond found a size it likes and just sits there forever." She wrote SETTLES on the board. "Boring. Turn the number up."

They tried R equals three point two. Soren ran it long, muttering, and then stopped his pencil.

"It doesn't park," he said. "Watch. High, low, high, low. Same two numbers, over and over. It's bouncing between two answers."

"Let me see." Maya leaned over the receipt. "Point seven-ish, then point five-ish, then point seven-ish. Forever?"

"Forever, as far as I went."

Maya sat back. "So one number gave one answer. A bigger number gave two. That's a weird jump." She wrote 1 then an arrow, then 2. "It didn't slide from one to two. It split."

"Turn it up more," Soren said, and he was already sharpening the pencil with his thumbnail.

At R equals three point five the pattern took longer to find its rhythm, and when it did, Soren counted under his breath.

"Four," he said. "It's cycling through four different numbers now. One, two, four."

Maya stared at the arrows. "Next it'll be eight."

"You don't know that."

"I sort of know that." She tapped the board. "One split into two. Two split into four. Everything's splitting. Eight."

Soren didn't argue . He pushed R to three point five-five and ran it patiently, filling the whole back of the receipt and then the margin of a fish-feed invoice.

"Eight," he admitted. "It takes forever to settle, but it's eight numbers."

Maya was quiet, looking at the ladder of arrows. One, two, four, eight. "The splits are coming faster," she said. "Look how close the numbers were. Three point two, then three point five, then three point five-five. Each time I have to nudge R less to make it split again."

"So the splits pile up," Soren said slowly. "They pile up and pile up and get closer together and then " He stopped.

"Then what?"

"Then they run out of room. They crash into each other." He looked up. "What happens when the splitting can't keep up? When there's supposed to be a next split but there's no gap left to fit it?"

They looked at each other. Outside, the rain changed its mind and came down harder.

"Push it," Maya said. "Go to four."

Soren set R at three point nine and started the chain. Halfway. Then his pencil began to wander all over the paper.

"It's not settling," he said. "Point three. Point eight. Point one. Point nine. Point one again, no, a different point one."

"Two, four, eight, sixteen, what number is it cycling through now?"

"There's no number." He kept going anyway, twenty steps, thirty. "It never repeats, Maya. It never comes back to where it was. I'm doing the same rule every single time and it never does the same thing twice."

Maya took the pencil from him and started a fresh chain right next to his, same R, but she began at halfway plus the tiniest bit. A hair off. One extra scratch of the pencil.

For the first few steps her numbers sat almost on top of his.

"Still matching," she said. "Almost. Point two off. Point four off." Her voice slowed. "Point nine off. They're nowhere near each other anymore."

The two columns had started as twins and ended as strangers.

"Same rule," Soren said. He said it very carefully, the way he said things he was going to write down later. "Same rule for both. I moved the start by a hair and after ten steps they've got nothing to do with each other."

"So you couldn't guess the fish," Maya said. "Not even with the exact rule. You'd have to know exactly how many you started with. Exactly. Not almost."

"Nobody can count a pond exactly."

"Nobody can count anything exactly." She looked at the board, at the neat little ladder that climbed one, two, four, eight, and then fell off a cliff into the mess on the receipt. "It's not broken. That's what gets me. It's the most obvious rule in the world. Crowded pond, fewer babies. A little kid could get it."

"And it does that." Soren nodded at his column of numbers that would never, in a hundred years, repeat.

Maya picked up the good chalk. Under SETTLES she drew the whole ladder, one splitting to two, two to four, four to eight, the rungs crowding closer and closer, and then she just let the chalk scribble, a snarl of lines going everywhere, no rung at all.

"People think you need a complicated rule to get a mess," she said. "You don't. You need a simple rule and you have to let it eat its own answer."

Soren was already writing, the pencil moving fast, copying both columns side by side so he'd have the twins that turned into strangers. He got to the bottom and stopped and looked at the gap between the last two numbers, the two that had started a hair apart.

"Turn it up a tiny bit more," he said. "I want to see if the mess ever quiets back down."

Maya wiped a clean space on the board with her sleeve and held out the chalk. The rain kept changing its pattern on the window, and not once did it fall the same way twice.