Math Hysteria

Posted on December 11, 2010

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One day my father, for no apparent reason, began a hypothetical discussion with me about pennies and how quickly they could add up to something. It’s possible that he was using a diversionary tactic after I’d requested an increase in my allowance. He said that if we were to put a single penny into a jar, at the end of the day we would have one cent. I don’t remember how I responded but I must have agreed, because I was a brilliant child. On the second day we were to put two pennies into the jar. He asked me how much money we’d have then. I said we’d have two cents. He said, no, that we’d have three cents, because we had put in a penny on the first day. He was no doubt growing concerned here, having lost me so soon. (I lied earlier when I said I was a brilliant child. The truth is, I existed in a perpetual cloud of befuddlement.)

On the third day we’d drop four pennies into the jar — giving us seven cents — and on each successive day we would double the number of pennies we’d added in the previous deposit.

“How much money do you think we’ll have at the end of one month?” he asked.

“A dollar?” I answered stupidly.

He told me that we’d have more than ten million dollars. I couldn’t believe it. He had written down the first few entries and I could see that five days in, we’d have only thirty-one cents. I was pretty sure my father had lost his mind, but then, realizing there was a slight chance that he might be right about this penny thing, I said, “Let’s do it!” As usual, I had missed the point and no doubt left him feeling frustrated as I ran off to find an empty jar.

Many years later I heard a related story about a servant who had done his emperor a big favor. To thank him, the emperor asked the servant what he would like as a reward. The servant said that he had modest needs, and that he would happily accept a chessboard filled with rice. He asked that the emperor place a single grain of rice on the first square, two on the second, four on the third, and so on, doubling each time for all sixty-four squares. It turns out that this process would have required 461 billion tons of rice to meet the servant’s request. That would be the equivalent of the entire world’s rice production for about nine hundred years. I can imagine the servant going home and telling his son how he had tricked the emperor, and them both having a good laugh over it. I can also imagine the emperor having the servant decapitated the next day, then telling him, “There. There’s your chessboard filled with rice.” Emperors did things like that all the time.

Here’s one more. It involves my first experience with a microwave oven. The year was 1984, and I had just made myself a grilled cheese sandwich. I was terrified to take a bite, sure that when I went to bed I would see my stomach glowing in the dark right through the blankets. That didn’t happen, of course, and the next day I got out the microwave cookbook to learn how to make a baked potato. One medium-size potato, I found out, took four minutes on HIGH. In order to bake two potatoes, I had to give it eight minutes. This was my first clue that the traditional oven had certain advantages; for one thing, it didn’t seem to care how many of something you crammed into it. The cooking time would remain about the same. But this new microwave appliance took twice as long to make two lousy potatoes. Upon further reading, I discovered that three potatoes required sixteen minutes. Do you recognize the pattern here? Apparently, each additional potato doubled the cooking time.

I won’t go through all of the arithmetic again, but here’s enough to give you the general idea. Baking ten potatoes in the microwave would take thirty-four hours. Twenty would need just under ninety-six years, and thirty would require forty-one centuries. You may want to print out this information and put it into some kind of handy chart form. It could save you a lot of time and embarrassment if you’re ever called upon to bake potatoes for the neighborhood block party.

Okay, that last example was ridiculous, but I hope you’re still reading because I’m just now getting to the point. I once tried to apply this doubling concept to my family tree, and it left me with a big headache. Here’s the problem:

Like most people, I had exactly two parents. Each of them also had two parents, and those were my four grandparents. Each of my four grandparents had two parents — my eight great-grandparents. So say I’m the first penny, or the first grain of rice. My parents were the next two pennies, or the two grains of rice on the second square. From there, the math is exactly the same. If I count back five generations, I had thirty-two great-great-great-grandparents, just as you must have. All thirty-two of those people had to have been contemporaries, more or less. But then it starts to get weird.

Going back ten generations, the number reaches 1,024. By thirty generations, it’s jumped to an incredible 536,870,912. That’s 536 million. Another ten generations back and we’re suddenly at 549,755,813,888 ancestors, all of whom should have been living at roughly the same time. Five hundred forty-nine billion. The catch is, that’s more people than have ever lived on Earth since the beginning of human existence. And forty generations is only about a thousand years. Something is wrong with the logic. There’s a major flaw in the theory, but what is it?

I’ve tried asking a few people to help me with this dilemma, people whose minds I respected, and who would, I was certain, help me see how to approach this question correctly. But the best they could come up with was some lame guess about long-ago incest, or the plague. All right, I’ll grant you an occasional inappropriate relationship, but I doubt that would have a significant impact on the numbers. And death by plague doesn’t explain anything: if some of my ancestors died from the plague they still lived long enough to have children — and if they didn’t, they weren’t my ancestors.

There must be a more reasonable explanation, but I can’t figure out what it is. Apparently, the cloud of befuddlement has still not lifted. If you can solve this puzzle, I’d be eternally grateful. And there may even be a reward — as long as it doesn’t involve pennies, a chessboard, or baked potatoes. (I wish I could be more generous, but I never even got that raise in my allowance.)

 

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