The hammer rang once. A spark jumped off the anvil and did not go out. It waited. They followed.

It was a frosty morning on the Green. Every blade of grass had a white edge, and the pond had a thin skin of ice that the ducks were walking on, looking puzzled. Lan ran after the spark, sliding on the frosty path and nearly sitting down. Inkling followed more carefully, in small steps, with her hands in her pockets.

The spark went where it always went. Past the mending shed. Past the bench with the loose plank, which had frost on it too. Right up to the door of the print shop, where it hung in the cold air, glowing orange, until they opened the door and let it in.

The print shop was warm, and its windows were steamed up. It smelled of ink and the little stove in the corner. The spark crossed the long table and settled next to a square of paper, orange on one side and cream on the other.

The square sat up.

"Fold me, I won't mind," said Kami.

"Somewhere with a puzzle," said Inkling, before Lan could say anything. "A really good puzzle. One that nobody can solve."

"That's two things," said Lan. "A puzzle and nobody can solve it."

"Every fold makes me something new," said Kami. "I can't promise nobody. Start with something simple. Start with a half. And then half of that."

Inkling folded Kami in half, edge to edge, so the orange was inside. A rectangle, two layers thick. Then she folded the rectangle in half again, the same way. A smaller rectangle, four layers thick.

"That's all?" said Lan. "That's the easiest fold in the world."

"Easy folds can hide hard puzzles," said Kami.

The last crease went flat, and the paper opened like a door.

The little rectangle opened wider and wider, until the inside of it was full of white light.

Lan stepped through first, of course. Inkling stepped through behind her, holding Kami by one corner.

It was bright. That was the first thing, before anything else. Winter sun came pouring through a row of high windows, so bright that Lan had to blink. It was not cold, though. The air was warm and dry and smelled of floor polish.

They were standing at one end of a hallway. It was the longest hallway Lan had ever seen. It went on and on, straight as a ruler, until the far end looked tiny. Metal lockers ran along both walls, blue ones, in a long neat row. There was a drinking fountain, and a fire extinguisher in a glass box, and a notice board covered in sheets of coloured paper.

Somewhere, far away, a bell rang, and then there was a lot of noise, doors opening, children laughing, and then quiet again.

"A school," said Inkling. "But not our school."

Kami crackled. "America," said Kami. "California. The month is January. The year is 2002."

"That's almost now!" said Lan.

"It's nearer than Kuwana," said Inkling. "By about two hundred years."

"Every door opens somewhere," said Kami. "This one opened in a hallway."

A boy was sitting on the floor with his back against the lockers. He was about eight, in a grey hoodie and jeans, with a backpack beside him and a pencil behind his ear. He was holding something small and frowning at it as if it had told him a lie.

Lan, who was never shy for long, sat down on the floor next to him.

"Hello," she said. "I'm Lan. This is Inkling. That's Kami."

The boy looked at Kami, who waved one corner, and then at Lan's yellow raincoat, which she was still wearing indoors. He shrugged. Lots of odd things happened at school.

"I'm Sam," he said.

"What's that?" asked Inkling.

Sam held it out. It was a small, hard, lumpy square of paper, no bigger than a matchbox. It was so thick that it did not look like paper any more. It looked like a little white brick.

"It was a whole sheet," said Sam. "A normal sheet, from my notebook. I folded it in half. Then in half again. Then again. I've done it six times."

"Why?" asked Lan.

"Tomorrow is big-question day," said Sam. "In my class, everyone has to bring a big question and try to answer it. My grandpa came for dinner last night, and he said a thing." Sam did a deep grandpa voice. "Nobody in the whole world can fold a piece of paper in half more than seven times. It can't be done. Not with any paper. Not ever."

"Is that true?" asked Inkling.

"That's my question," said Sam. "I want to fold one eight times. Just one more than Grandpa says. And show the whole class."

He pressed the little brick between his hands, trying to bend it for the seventh fold. It did not bend. It did not even creak. It stayed a brick.

"It won't go," said Sam. "It's like folding a stone. Maybe Grandpa's right."

Lan had an idea. Her ideas came out of her mouth before they had finished arriving.

"I'll do it!" said Lan. "I promise. It just needs to be flatter. Put it on the floor. I'll stamp on it really hard, and then it'll bend."

She took the brick from Sam, put it on the shiny floor, and lifted her foot high in the air, ready to stamp.

"Stamping only tears paper," said someone, and a hand caught her sleeve just in time.

Lan wobbled and turned round. Standing behind her was Mark, who drew every fold line in the world and liked nothing better than a good drawing. He had a pencil behind one ear, just like Sam.

"It's only paper," said Lan.

"That's the point," said Mark. "Paper tears. If you stamp on it, the outside layers split open. Then you'd have two pieces, or ten, and a fold of two pieces doesn't count. The puzzle is one piece of paper, folded in half." He let go of her sleeve. "Read the drawing first."

Lan put her foot down slowly, on the floor, next to the paper brick. Her ears went hot.

"I promised before I checked," she said quietly. "Sorry, Sam. I nearly broke your brick."

"It's all right," said Sam. "It's a pretty useless brick anyway."

Inkling noticed that Lan stood very still with her feet together for a while after that, as if her feet might run off and stamp on something by themselves.

"Mark," said Inkling. "What drawing? This is a hallway."

Mark pointed at the notice board.

Between the sheets of coloured paper, someone had taped up a drawing. It was neat and careful, done with a ruler. It showed a long, thin strip of paper, drawn sideways across the page. Under it, the strip was drawn again, folded in half. Under that, folded in half again. And again. Each time the strip got shorter, and fatter.

Under each drawing was a number.

Mark stood in front of it for a long time, with his hands behind his back, the way people look at pictures in a museum.

"Read the drawing before your hands move," he said at last. "Look at the numbers. Those are the layers. One sheet is one layer. Fold it in half, and it's two layers. Fold that in half, and each layer goes on top of another one, so it's four. Then eight. Then sixteen."

"It doubles," said Inkling. "Every time. Wait, I can name it. It's doubling."

This time, she had looked first. She was right.

"Every fold doubles the layers," said Mark. "So every fold is twice as thick as the one before. Sam, how many layers is your brick?"

Sam counted on his fingers. "One, two, four, eight, sixteen, thirty-two," he said. "Six folds. Sixty-four layers."

"Sixty-four layers of paper," said Mark. "Trying to bend in half. That's why it's like a stone. And here's the part that matters." He tapped the drawing, where the strip curved round the end of each fold. "When paper folds, the outside layers have to go all the way round the inside layers. The thicker the stack, the further round they have to go. So a thick fold needs more paper."

"And my square ran out," said Sam slowly.

"Your square was too short," said Mark. "It had used itself up. If you want more folds, you need paper that is long. Very long. And very thin."

"Is Grandpa's rule true for every normal sheet, though?" asked Inkling. "I want to check. Not guess. Check."

Sam tore a clean sheet out of his notebook and gave it to her. Lan found an old flyer on the floor by the lockers, bright green, about a school concert, and took that.

"Race you," said Lan.

"It isn't a race," said Inkling. "It's a test." But she started folding very fast anyway.

They counted out loud as they went, doubling each time.

"One fold. Two layers," said Inkling.

"Two layers!" said Lan.

"Two folds. Four. Three folds. Eight."

By the fourth fold, sixteen layers, the paper was the size of a playing card and starting to feel stiff. By the fifth, thirty-two layers, it was the size of a stamp and puffing up in the middle like a little pillow. By the sixth, it was a hard little lump.

Lan pushed and pushed at her green lump for the seventh fold. It bent a tiny bit, and then it sprang back open, as if it were cross with her.

"It won't," she said. "Six. That's all."

Inkling had managed seven, but only just. Her seventh fold was not a proper fold at all. It was a sort of squashed bend, with the ends nowhere near each other.

"Six and a bit," said Inkling honestly. "Not seven. Not really."

"So Grandpa's right," said Sam. "For normal paper."

"For normal paper," said Mark. "Every normal sheet runs out. That's not because the paper is weak. It's because the layers double and the sheet doesn't get any bigger. Look at your lumps. There's no paper left to go round."

Inkling held her lump up to the light. It was true. It was all thickness and no length. There was nothing left to bend.

"So we need paper that isn't normal," she said.

Sam looked down the hallway. It was very long.

"I know where there's long paper," he said, and jumped up.

Sam ran down the hallway, and the others ran after him, all the way to a door with a window in it. It was the school office. A lady with glasses sat behind a desk, typing.

"Mrs Ortiz," said Sam, out of breath. "Can I borrow a roll of the till paper? For big-question day. It's for science."

Mrs Ortiz looked over her glasses at Sam, and at Lan in her raincoat, and at Inkling, and at Kami, who had folded one corner politely. She sighed and smiled at the same time, the way office ladies do.

"Science," she said. "Bring back what you don't use."

She handed Sam a fat roll of thin white paper, the kind that comes out of a cash register, as wide as Lan's hand.

Back in the long hallway, they knelt on the floor and unrolled it. The strip ran past the lockers, past the drinking fountain, past the notice board, on and on. Lan walked one end all the way down the hall while Sam held the other. When they finally stopped, the strip was longer than two classrooms put end to end.

A class of small children came down the hallway in a line, holding hands in pairs, with a teacher at the front. They all stopped and stared at the long white strip on the floor.

"Step over it, please," said the teacher. "Big steps. Don't tread on the science."

The children stepped over it one by one, very seriously, as if it were a river. The last one in the line was a tiny boy with his shoelaces undone. He stopped and looked at Sam.

"What is it for?" he asked.

"Folding," said Sam. "In half. Eight times."

The tiny boy thought about this. "Why?" he asked.

"Because my grandpa says you can't," said Sam.

The tiny boy nodded, as if that were a very good reason, and stepped over the strip and ran to catch up with his line.

"That's a lot of paper," said Inkling.

"It needs to be," said Mark.

"Bring it all the way over," said a cheerful voice. "Corners first, always. Well. Ends first, today."

It was Cornelia, who could make any two corners meet as if they had been made for each other, and who did not mind at all that this strip had no proper corners.

"End to end," said Cornelia. "Exactly. Point to point, then hold."

Lan picked up her end and walked it all the way back up the hallway, the strip trailing behind her in a long loop, until she reached Sam. She laid her end exactly on top of his. Sam pressed the fold at the far end flat.

"One fold," said Inkling. "Two layers."

"And now the same way again," said Cornelia. "Always the same way. If you turn it, the layers get muddled."

They folded it again, end to end. Four layers. Again. Eight.

"Three folds," said Inkling, counting. "Eight layers." She checked the drawing on the notice board. "Same as the picture."

The fourth fold made sixteen layers. The strip was much shorter now, about as long as a car. The fifth fold made thirty-two, and it was as long as a bicycle.

Then came the sixth fold.

Sixty-four layers did not want to bend. When Sam brought one end towards the other, the stack bulged in the middle. The layers slid against each other. The ends did not meet. One end was ahead of the other by the width of his thumb.

"It's crooked," said Sam. "It's all crooked. It's ruined."

"Crooked is just a first go," said a gentle voice.

It was Crinkle, who was never upset by a mistake and only ever looked interested in them.

"Smooth it out and try again," said Crinkle. "The crease remembers."

Sam opened the sixth fold. A faint line ran across the stack where it had tried to bend. Crinkle looked at the stack, and at Sam's hands, which were small, and at the bulge.

"It's too thick for two hands now," said Crinkle. "It needs four."

So Sam held one end, and Lan held the other end, and they walked their ends towards each other, very slowly, keeping the edges straight, until they touched. Then Sam pressed the fold.

The ends met exactly.

"Six folds," said Inkling. "Sixty-four layers. The same as your brick."

"But this one isn't a brick," said Sam. He squeezed it. "It's still bendy. There's lots of it left."

He was right. The square from his notebook had been used up after six folds. The till paper was still as long as his arm.

"Hold it still," said a slow, calm voice. "Now press it flat."

It was Presso, who never hurried and whose folds never came undone.

Sam brought the ends together for the seventh time. It was hard work now. He had to push with both hands, and his face went red. The stack bent over in a fat round curve.

"Don't let it spring back," said Presso. "Press slowly, press firmly."

Presso knelt right on top of the packet, and pressed, and pressed, and the fat curve slowly flattened into a sharp edge.

"A fold you press is a fold that stays," said Presso.

"How many now?" asked Lan.

Inkling counted in her head, doubling as she went. "Sixty-four and sixty-four," she said. "A hundred and twenty-eight layers. Seven folds."

They all looked at the packet. It was about the size of a chocolate bar, but much thicker, and so stiff that it did not wobble at all when Sam held it out.

"Seven," said Sam. He sat back against the lockers and blew out his cheeks. "That's exactly what Grandpa said. Seven. That's the most."

"Maybe he was right," said Lan.

"Maybe," said Mark, looking at the drawing again.

"I'm still one piece of paper," said Kami.

Kami had been sitting very quietly on top of a locker, watching. Now Kami leaned over the edge.

"I'm still one piece of paper," said Kami again. "And so is that. Not torn. Not stamped. Look how long it is."

Sam looked. The packet was short, but not that short. It was still longer than it was thick. A lot longer.

"There is still enough of it to bend round," said Kami. "Every fold makes me something new. One more."

Sam knelt up. He took the packet in both hands. Lan held one end steady. Inkling held her breath.

He pushed.

At first nothing happened. Then, very slowly, the packet began to bend, a fat curve in the middle, the outside layers stretching round all the inside ones, the way the drawing showed. Sam pushed until the two ends touched.

Then he pressed it flat himself, with the heel of his hand, slowly and firmly, the way Presso had.

"Eight," said Sam. His voice came out squeaky. "That's eight folds."

"Two hundred and fifty-six layers," said Inkling, who had already worked it out. "A hundred and twenty-eight and a hundred and twenty-eight."

Sam held the packet up to the window. It was fat and small and white, about the size of a box of matches, and it was still one piece of paper.

Lan cheered so loudly that a door opened somewhere down the hallway and a head looked out, and then went back in.

Sam took the pencil from behind his ear. He tore a scrap off the end of the till roll and wrote on it, carefully, his new rule for Grandpa.

Every fold doubles. The more folds you want, the longer the paper has to be.

"That's better than Grandpa's rule," said Inkling.

"Grandpa's rule was about normal paper," said Sam fairly. "He was right about that. Normal paper runs out."

It was only then that they noticed the other people.

Far down the hallway, right at the other end, where everything looked tiny, three people were kneeling on the floor. Two of them were grown-ups. The third was a girl, a teenager, much older than Sam, with her hair tied back.

They had unrolled a strip of thin paper too. But theirs was not like Sam's. Theirs was so long that it went out of the hallway, through a pair of open doors, and on outside, further than anyone could see.

"That's even longer than ours," whispered Lan.

"Much longer," said Mark.

As they watched, the girl walked one end of her strip back up the hallway, slowly, carefully, the same way Lan had, until it met the other end. She knelt and pressed the fold. Then she picked up the new end and started walking again.

She did not look at them. She did not explain anything. She just folded, the same way every time, again and again.

"How many is she going to do?" asked Sam.

Nobody knew. But Inkling started doubling in her head. Two hundred and fifty-six. Five hundred and twelve. A thousand and twenty-four. The numbers got so big so fast that she had to stop.

"It goes up so fast," she said. "Every fold. It's like the numbers are running."

"That's doubling," said Kami. "It always runs."

Sam stood and watched the girl for a long time. She walked so carefully. She lined up the ends so slowly. Her hands never hurried. When the fold was done, she pressed it, and checked it, and only then did she pick up the next end.

"She's patient," said Sam.

"She has to be," said Mark. "With that much paper, one crooked fold and the whole stack slides. She reads her own paper before her hands move."

"Like Presso," said Lan.

"Like all of us," said Kami. "Folding is mostly waiting for the paper to be ready."

Sam put the fat little packet in the front pocket of his hoodie, where he could feel it.

"Tomorrow," he said, "I'm going to show everyone. And then I'm going to ask Grandpa for a longer roll."

Lan found a plain square in her pocket. She looked at it, and then at Kami.

"Fold me, I won't mind," said Kami. "Or rather, unfold me."

They unfolded the paper flat, and the door closed behind them.

The spark dimmed to a warm dot and settled back on the anvil. The Green was just the Green again.

Back on the Green

At the print shop, the frost had melted off the windows. Lan cut a long strip from the edge of an old newspaper. She folded it in half, and in half again, and in half again, the same way every time.

Then she opened it and counted out loud, touching each layer of the creases: one, two, four, eight.

"Eight layers from three folds," said Inkling. "What about ten folds?"

Lan started doubling. She got as far as five hundred and twelve and had to stop for a biscuit.

Tell the story back to someone. Why did Sam's square stop at six folds, but the long strip did not?