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, 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.J. Estes, New York
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 molecule 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 on our side. 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, four-10,000ths of an inch. While this did not quite achieve the experimental standard we had dreamt of in sixth grade, we figured it was close enough. (OK, so maybe substituting plastic for paper was a little cheesy, but from a theoretical standpoint it shouldn’t matter. Besides, the drop cloth only cost 59 cents and we can use it when we paint the closet.)
We thereupon embarked on 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 was cheating.) Still, having easily surpassed seven folds, we felt vindicated. Obviously the wise guy who invented this bogus maxim was generalizing from insufficient evidence, and was probably stinky to boot. (We have a lot of lingering resentments from sixth grade.) Next: proving that stepping on a crack won’t break your mother’s back. Nothing against Mom, but sometimes we all have to make sacrifices for the sake of the greater good.
I have repeatedly heard that if you have a classroom of 30 children, the odds are better than even that at least two will have the same birthday. Try as I might I can’t see the logic of this. It would put me out of my misery if you could explain. –Robert McDonald, Welling, Kent, England
Your misery is going to get worse before it gets better. By my calculation you get better than 50-50 odds of matching birthdays with only 23 children. With 30 kids the probability is better than 70 percent and with 57 kids it’s more than 99 out of 100. I could demonstrate this proposition mathematically, but I got so many complaints last time I ventured into matters arithmetical that I have sworn off any concept that cannot be illustrated with apples and oranges. Then again, how about the old sock joke? You remember: You’ve got 20 black socks and 20 white socks in a drawer. Assuming it’s too dark to see and all socks feel the same, how many socks must you pick to make sure you get a pair? Sucker’s answer: 21. Real answer: 3. Lesson: getting a pair, whether it be of socks or birthdays, is a lot easier than you might think.
Art accompanying story in printed newspaper (not available in this archive): illustration/Slug Signorino.