Can paper be folded in half seven times but never eight?

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Dear Cecil: In the sixth grade, and many times since, I’ve heard it claimed that you can double over a piece of paper seven times but never eight, no matter what the paper’s size. Since, as a sixth grader, I could fold the paper in half seven times without difficulty, I felt certain an Arnold Schwarzenegger could do eight. Why not? Is there something inherent in the mathematics of doubling? Some physical limitation? Or is it simply that the eighth doubling takes more strength than most people have--meaning a sufficiently powerful machine could do it eight times? S.E., New York


Illustration by Slug Signorino

Cecil replies:

My friend Pablo and I heard this story in sixth grade, too, and we had the same thought that everybody who hears it has: “Gosh, what if you had a piece of paper a mile square and one atom thick? Couldn’t you fold that in half eight times?” Not having access to paper of these specifications at the time, we were unable to put our conjecture to the test.

Unbeknownst to us, however, powerful economic forces were working on our behalf. It has long been the aim of the plastics industry to produce sheeting so thin it only has one side, and today that aim has very nearly been achieved. We were able to purchase a plastic drop cloth measuring three yards by four yards and having a thickness of just 0.4 of a mil–that is, 4/10,000ths of an inch. This did not quite achieve the experimental standard we had dreamt of in sixth grade, and of course it meant substituting plastic for paper, with God knows what intramolecular consequences. However, we figured it was the best we could do for 59 cents, the drop cloth’s price. Besides, we could use it when we painted the closet.

We thereupon embarked upon our experimental regimen. Mrs. Adams agreed to substitute for Pablo, who, having long since lit out for the territory, was no longer available to perform the vital scientific function of holding up the other end. Since Mrs. Adams invariably wants to zig when I want to zag, our progress was initially rocky, but eventually we got our act together and proceeded with the folding. Result: not eight, not nine, but ten doublings.

Granted, on the tenth fold the finished package was a little bulbous due to trapped air. (We could have popped the bubbles with a needle but didn’t, out of a vague sense that it would be cheating.) The fact remained that we had easily surpassed seven folds. Vindication at last.

Cecil Adams

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