Fold a sheet of paper in half 42 times and, on paper at least, the stack would reach past the Moon. That sounds wrong, but it is ordinary arithmetic. Assume the sheet is 0.1 mm thick. Each fold doubles the thickness, so 42 folds multiply it by 2⁴² = 4,398,046,511,104. That gives 0.1 mm × 2⁴² ≈ 439,805 km, or about 440,000 km. The Moon is about 384,000 km away. This is exponential growth, and it is why doubling beats everything: anything that doubles on a schedule eventually outruns anything that only adds.
The legend of the chessboard
The oldest popular version of this lesson is a legend with many tellings. The inventor of chess (in some versions Sessa, an Indian minister) asks his ruler for a modest-sounding reward: one grain of wheat on the first square, two on the second, four on the third, doubling up to the 64th. The ruler mocks the request until his treasurers report that the grain would outstrip everything he owns. The story is first known in writing from Ibn Khallikan in 1256, and the endings differ: the inventor becomes an advisor, or is executed.
The 64th square alone holds 2⁶³ = 9,223,372,036,854,775,808 grains, and the whole board holds 2⁶⁴ − 1 = 18,446,744,073,709,551,615. At 65 mg a grain, that is about 1,199,000,000,000 metric tons of wheat, more than 1,400 times the world’s yearly wheat production.
Exponential growth: why doubling beats adding
Add up every square before the last one: 1 + 2 + 4 + … + 2⁶² = 2⁶³ − 1. So the 64th square alone holds one grain more than all 63 before it combined. Every doubling step is bigger than the whole history before it.
The early steps look harmless, which is why the video starts with a plain bar chart of 1, 2, 4, 8, 16, 32. Now race two machines: one adds 1,000 every step, the other starts at 1 and doubles. After 10 steps it is 10,000 against 1,024. After 14 steps it is 14,000 against 16,384, and the doubler never falls behind again.
The general rule: in exponential growth the rate of growth is proportional to the current size, so the bigger the quantity, the faster it grows. Linear growth adds the same amount every step. In the long run, exponential growth of any kind overtakes linear growth of any kind.
The paper follows the same script. Ten folds: 0.1 mm × 1,024 ≈ 10 cm. Twenty folds: about 105 m. Thirty folds: about 107 km. Forty-one folds: about 219,902 km, still well short of the Moon. The 42nd fold adds another 219,902 km by itself, and the stack sails past 384,000 km.
Can you really fold paper 42 times?
No. Every fold spends part of the sheet’s length on the curved edge, and that loss grows exponentially too. A long-standing myth held that paper cannot be folded in half more than eight times. In December 2001 Britney Gallivan, a California high-school student, worked out the minimum length L a strip of thickness t needs for n folds in one direction:
L = (πt/6)(2n + 4)(2n − 1)
In January 2002 she folded a 4,000-foot (1,200 m) strip of toilet paper in half twelve times. For t = 0.1 mm and n = 42, her formula asks for a strip about 10²¹ meters long. The trip to the Moon stays a thought experiment.
Doubling in the real world
Anything that grows by a fixed percentage has a fixed doubling time. The rule of 70 estimates it: divide 70 by the growth rate in percent. At 7% a year, a quantity doubles about every 10 years (the exact figure is 10.24).
Doubling looks quiet for most of its life and then finishes all at once. Splitting growth into ever smaller steps leads somewhere else: to the number e, in the story behind compound interest.
Doubling looks boring, until it doesn’t.
