Take a single sheet of ordinary paper, about a tenth of a millimetre thick. Fold it in half and it is twice as thick. Fold it again and it is four times as thick, then eight, then sixteen. Keep going in the mathematical ideal, and by the forty-second fold the stack would be tall enough to reach the Moon. The arithmetic is simple. The folding is impossible, and the gap between those two facts is the interesting part.

The arithmetic, fold by fold

Each fold doubles the thickness, so after a given number of folds the stack is the starting thickness multiplied by two raised to that number. Start at 0.1 millimetres and the early folds feel ordinary. Ten folds give a stack about ten centimetres high, roughly the width of a hand. Even twenty folds reach only about a hundred metres.

Then it runs away from you. By around the thirtieth fold the stack passes 100 kilometres, the altitude often taken as the edge of space. By the forty-second it is about 439,800 kilometres tall, as Live Science sets out, comfortably past the Moon’s average distance of 384,400 kilometres. Forty-one folds would not come close. Forty-two clears even the Moon’s farthest ordinary distance.

The exact number depends on what you start with. Thinner paper needs a fold or two more, which is why some versions of the claim say forty-three or forty-five. With standard 0.1 millimetre paper, the figure is forty-two.

Why you could never actually do it

The catch is that folding paper in half is far harder than the doubling suggests, because thickness is not the only thing that grows. Every fold also eats into the usable length, since the paper has to curve around the rounded edge left by the previous fold. For a long time the accepted wisdom was that no sheet could be folded more than seven or eight times at all.

That was put to rest in 2002 by Britney Gallivan, then a high school student in Pomona, California, who worked out the mathematics of the limit and then folded a single sheet twelve times. She needed a roll of very thin paper more than a kilometre long to manage it. Her formula showed something most people get wrong: each additional fold needs roughly four times as much paper, not twice, so the material required climbs even faster than the thickness does. To reach forty-two folds you would need a starting sheet larger than anything that has ever been made. You run out of paper long before you run out of folds.

The part our intuition gets wrong

Ask people how many folds it would take to reach the Moon and the usual guesses are a million, or a hundred million, or that paper cannot be folded more than a handful of times in the first place. Almost no one guesses a number in the forties. The reason is that we tend to read growth as if it adds rather than multiplies. We expect each step to contribute about as much as the one before, and doubling does not behave that way. It stays small and unimpressive for a long stretch, then erupts.

The old chessboard legend turns on the same gap. One grain of wheat on the first square, two on the second, four on the third, doubling all the way to the sixty-fourth, ends on a number of grains larger than any harvest in history. Paper reaching the Moon by the forty-second fold, and the span of the observable universe by around the hundred-and-third, are the same trick told in millimetres instead of grain.

So the folding claim is best read as a demonstration rather than something to attempt. The thickness really would reach the Moon at the forty-second fold. The paper to do it has never existed and never will. The reason the result feels wrong is the same reason doubling catches people out wherever it appears, in savings, in interest, in anything that spreads by multiplying: it is quiet for a long time, and then it is not.