Try Seven and You Will Stop
This is worth doing before reading any further, because the experience is the whole subject. Take an ordinary sheet of printer paper and fold it in half, then in half again, and keep going.
The first three are effortless. The fourth takes a small amount of attention. The fifth produces a thick, stubborn little packet. The sixth is a fight. The seventh, on an ordinary A4 or letter sheet, is somewhere between extremely difficult and not happening, and what you are holding by then is not a folded sheet – it is a blunt wedge about as thick as a finger and smaller than a credit card.
Almost everybody who has done this concludes the same thing: the paper has got too thick. That conclusion is wrong, and the reason it is wrong is more interesting than the experiment.
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What Is Happening to the Thickness
Each fold doubles the number of layers. One fold gives two, two folds give four, three give eight. The number of layers after any number of folds is simply two multiplied by itself that many times.
That is an entirely unremarkable statement and it has thoroughly unreasonable consequences, because doubling is the most underestimated operation in arithmetic.
Seven folds is a hundred and twenty-eight layers of paper. Ordinary paper is about a tenth of a millimetre thick, so the packet is a little over a centimetre – which matches the stubborn wedge in your hand. So far, so sensible.
Ten folds would be one thousand and twenty-four layers, which is a block ten centimetres deep. That is already a solid object. Still, nothing absurd has happened.
Then the Numbers Go Somewhere Strange
Keep doubling and the figures stop being domestic with startling speed.
Twenty folds is slightly over a million layers, which at a tenth of a millimetre each is a column a hundred metres tall – the height of a substantial tower block.
Thirty folds is over a billion layers, which is a column about a hundred kilometres high. That is past the altitude generally taken as the beginning of space.
Forty folds is a column roughly a hundred and ten thousand kilometres high, which is more than a quarter of the way to the moon. Forty-one gets you to about two hundred and twenty thousand, which is still short.
Forty-two folds is around four hundred and forty thousand kilometres. The moon sits at an average of about three hundred and eighty-four thousand. So forty-two folds does it, with distance to spare, from a single sheet of paper no different from the one on your desk.
And it does not stop being interesting there. Carry the same arithmetic to a hundred and three folds and the stack is larger than the observable universe. Not larger than the solar system, or the galaxy: larger than everything anybody can see in any direction.
The Trick Is That Nothing Is Being Added
It is worth pausing on why this feels like a cheat, because it is not one.
No paper is being added. The sheet weighs exactly what it weighed at the start, and it contains exactly as much material. All that is changing is the arrangement: the same stuff gets shorter and taller.
Which means the answer to where the height comes from is that it comes from the area. Every fold halves the footprint and doubles the height. The volume is constant, and the shape is being traded, aggressively, in one direction.
That is the key to the second half of this, because it means the two things are not independent. You cannot gain thickness without losing length, and the loss is as fast as the gain. By forty-two folds the footprint of that column would be unimaginably small – the entire original sheet, squeezed into a tower reaching past the moon, standing on a base far too small to see.
Which Is Not Why You Cannot Fold It
So back to the sheet on the desk. The usual explanation – it got too thick – sounds right and does not survive examination. Plenty of people can fold something a centimetre thick in half. A towel is thicker than seven folds of paper and folds without complaint.
The actual obstacle is geometric and it is about the fold itself, not the stack.
When you fold a single sheet, the crease is effectively a line. When you fold a stack, the crease is not a line: the layers on the outside of the bend have to travel further than the layers on the inside, so the end of the stack curls around in a rounded nose rather than a sharp corner. That nose consumes length. Paper that goes into the curve is not available to lie flat.
And the amount consumed is proportional to how thick the stack is. Thin stack, small nose, negligible loss. Thick stack, large nose, significant loss. Since the thickness is doubling with every fold, the length eaten by the rounding is also doubling with every fold – while the length still available is halving.
Two exponentials running in opposite directions converge very quickly. You do not stop folding because the paper is too stiff. You stop because there is no longer enough length left to get around the bend.
So the Limit Is Not About Paper, It Is About Proportions
Once the problem is stated that way, the famous impossibility dissolves, because it was never a property of paper. It was a property of a particular ratio: the length of an ordinary sheet divided by its thickness.
Change that ratio and the limit moves. A sheet that is very long and very thin has far more length to spend on the rounding, and can take more folds. A sheet that is short and thick can take fewer.
This is why the seven-fold claim circulated so confidently for so long and was still wrong. It was being tested exclusively on ordinary stationery, which all has broadly the same proportions, so everybody got the same answer and concluded it was a law. Nobody had asked what the answer depended on.
The Teenager Who Wrote the Equation
The person who settled it was a high-school student in the United States, working on a mathematics challenge in the early 2000s.
Rather than trying harder, she worked out the relationship: an equation giving the minimum length of sheet required to achieve a given number of folds, in terms of the paper’s thickness. The equation accounts for exactly the effect described above – the length lost to the rounded nose at each stage – and it comes in two versions, one for folding repeatedly in the same direction and one for alternating directions, because the two lose length differently.
Then she used it. Choosing a very long, very thin paper – a single continuous length of tissue, hundreds of metres of it, folded in one direction along a corridor – she reached twelve folds, comfortably past the number everybody had assured her was impossible. The result was reproduced by others afterwards, and larger attempts using machinery and sheets the size of a sports field have reached eleven and twelve with much thicker paper.
What makes this a good story is not the record. It is that the barrier everybody had accepted for decades turned out to be a straightforward consequence of one measurement nobody had thought to vary, and that the way through it was to write down why it was there.
Why Nobody’s Intuition Copes With This
The reason the moon figure is startling, and the reason the folding limit was misdiagnosed, are the same reason: human intuition handles repeated addition well and repeated doubling extremely badly.
Asked to guess the thickness of a sheet folded fifty times, people answer in metres. The correct answer is of the order of a hundred million kilometres. The gap between the guess and the answer is not a small error – it is a failure of the entire category of estimate, and it is remarkably consistent across people.
This is the same blind spot behind the old puzzles about grains of rice doubled on each square of a chessboard, and behind every description of something spreading that is habitually underestimated at the start and then described as sudden when it becomes visible. Doubling produces nothing worth noticing for a long time and then produces everything at once.
A sheet of paper is a good way to feel it rather than be told it. The whole thing fits in one hand, it is finished in under a minute, and the honest conclusion at the end – that you cannot do it because you have run out of length rather than because it is too thick – is one almost nobody reaches on their own.
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