How many times can you fold a piece of paper?

The short answer: about 12 or 13 times, and only with a very long, very thin strip of paper. A normal sheet of printer paper gives up at about 7. But if you could keep going, something amazing happens: after 42 folds, your paper would reach the Moon.

The real-world record

For years people believed paper couldn't be folded in half more than 7 or 8 times. In 2002, a high school student named Britney Gallivan proved them wrong. She worked out the math of how much paper each fold uses up, then folded a strip of toilet paper about 1.2 km (4,000 feet) long in half 12 times.

Others have pushed it a little further since. The TV show MythBusters folded a sheet the size of a football field 11 times (with help from a steamroller), and a group of students in Massachusetts reportedly reached 13 folds with an even longer roll of toilet paper.

Why is it so hard?

Every fold doubles the thickness and halves the size of the sheet. After 7 folds, a sheet of paper is 128 layers thick and only a few centimeters across, so it's thicker than it is wide, and there's nothing left to bend. Worse, the paper on the outside of each fold has to wrap around everything inside it, and that curve eats up more paper with every fold. Gallivan's formula shows the length you need grows roughly four times with each extra fold.

How much paper does each fold need?

Gallivan didn't just fold; she worked out a formula for the shortest strip that can be folded n times in one direction. For paper of thickness t, the length you need is

L = (πt ⁄ 6) × (2ⁿ + 4) × (2ⁿ − 1)

You don't need to follow the algebra to see the problem: the length depends on 2ⁿ twice over, so each extra fold needs about four times as much paper as the one before. Using ordinary 0.1 mm paper, 12 folds already needs a strip of roughly 900 m. A 13th fold would need about four times that, well over 3 km, and a 14th more than 14 km. That's why the record has barely moved since 2002: the paper runs out long before the folding does.

Try the 7-fold experiment at home

You only need one sheet of printer paper (A4 or US Letter).

1. Fold it in half, crease it firmly, and keep folding the same block in half, turning it a quarter turn each time.
2. Count the layers as you go: 2, 4, 8, 16, 32, 64, 128.
3. Around fold 6 or 7, the block becomes a hard little brick about 1.3 cm thick. Try pressing it as hard as you like: it won't bend again.

Now try a sheet of tissue paper, or a long strip cut from a newspaper and folded only lengthwise. Thinner and longer paper lets you go a fold or two further, exactly as Gallivan's formula predicts.

How thick would paper get?

A sheet of paper is about 0.1 mm thick. Every fold doubles it. Doubling doesn't sound like much, but it's the fastest kind of growth there is (it's called exponential growth). Here's what happens:

Notice how the jumps get bigger. Going from 1 fold to 27 takes you from a sliver of paper to higher than Everest. Just 15 more folds and you're at the Moon. Every single fold adds as much height as all the folds before it put together.

A shortcut: every 10 folds is about ×1,000

Doubling 10 times multiplies by 1,024, which is very close to 1,000. That gives an easy way to estimate any fold in your head:

• 10 folds: about 10 cm, the width of your hand
• 20 folds: about 100 m, a 30-story building
• 30 folds: about 100 km, the edge of space
• 40 folds: about 100,000 km, more than a quarter of the way to the Moon
• 50 folds: about 100 million km, three-quarters of the way to the Sun

Each step is a thousand times the last. Ten more folds never means “a bit taller”; it means a thousand times taller.

42 folds to the Moon

This is a famous puzzle: if you could fold a piece of paper 42 times, how thick would it be? The answer: 0.1 mm × 2⁴² ≈ 440,000 km (about 273,000 miles). The Moon is about 384,400 km away, so a 42-fold stack would reach right past it. Fold it 41 times and you'd only be about halfway there.

103 folds… or 102?

You may have heard that 103 folds would be as wide as the observable universe. That's about its whole width. To reach the edge of the observable universe, 46.5 billion light-years away, you need 102 folds: 0.1 mm × 2¹⁰² is about 54 billion light-years.

Doubling in real life: the rice and the chessboard

An old story tells of a king who offered a reward to the inventor of chess. The inventor asked for one grain of rice on the first square of the board, two on the second, four on the third, doubling on every one of the 64 squares. It sounded like a modest request. But the last square alone would hold 2⁶³ grains, about 9.2 quintillion, and the whole board more than 18 quintillion: more rice than the whole world grows in several hundred years.

Folding paper is the same story told in height instead of rice. Exponential growth shows up in many real things: savings that earn interest on their interest, bacteria dividing in a dish, or news shared from person to person. The early steps always look harmless, and the later ones are enormous.

Folding that really does reach space

Engineers can't fold paper to the Moon, but they do use folding to get big things into space. In 1995 a Japanese satellite, the Space Flyer Unit, unfolded its solar panels using the Miura fold, a pattern invented by astrophysicist Koryo Miura that opens and closes in a single smooth pull. The James Webb Space Telescope launched in 2021 with its tennis-court-sized sunshield and its golden mirror folded up to fit inside a rocket, then unfolded on its way to its spot 1.5 million km from Earth. The same ideas help fold car airbags, medical stents and maps. You can trace some of the classic crease patterns yourself on our origami page.

Questions

Can you fold a piece of paper 8 times?

With an ordinary sheet of printer paper, usually not: most people stop at 7, because the folded block is thicker than it is wide. With a much bigger or thinner sheet, 8 is possible, and the record is 12 or 13 folds with a very long strip.

How thick is paper after 10 folds?

About 10 cm (4 inches): 0.1 mm doubled 10 times is 1,024 layers, roughly the width of your hand.

How many folds would it take to reach the Moon?

42. A 0.1 mm sheet folded 42 times would be about 440,000 km thick, past the Moon at about 384,400 km. After 41 folds it would only be about halfway.

Why does each fold make such a big difference?

Because every fold doubles the stack, each new fold adds as much thickness as all the folds before it put together. That is exponential growth, the same kind of growth as the old story of rice on a chessboard.

Try it yourself

You can't fold real paper 102 times, but you can in The Great Paper Expansion, a free idle game you play in your browser. Start with a single sheet, build paper mills, and fold your way past a house cat, Mount Everest, the Moon, the Sun and the stars, with a real fact at every landmark, all the way to the edge of the universe.

▶ Fold your own paper (free)

New to idle games? Read How to play or the FAQ.