Hailstone Numbers

By Ivars Peterson

Muse, February 2003, p. 17.

Nothing could be grayer, more predictable, or less surprising than the endless sequence of whole numbers. Right? That's why people count to calm down and count to put themselves to sleep. Whole numbers define booooooooring.

Not so fast. Many mathematicians like playing with numbers, and sometimes they discover weird patterns that are hard to explain. Here's a mysterious one you can try on your calculator.

Pick any whole number. If it's odd, multiply the number by 3, then add 1. If it's even, divide it by 2. Now, apply the same rules to the answer that you just obtained. Do this over and over again, applying the rules to each new answer.

For example, suppose you start with 5. The number 5 is odd, so you multiply it by 3 to get 15, and add 1 to get 16. Because 16 is even, you divide it by 2 to get 8. Then you get 4, then 2, then 1, and so on. The final three numbers keep repeating.

Try it with another number. If you start with 11, you would get 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, and so on. You eventually end up at the same set of repeating numbers: 4, 2, 1. Amazing!

The numbers generated by these rules are sometimes called "hailstone numbers" because their values go up and down wildly—as if, like growing hailstones, they were being tossed around in stormy air—before crashing to the ground as the repeating string 4, 2, 1.

Mathematicians have tried every whole number up to at least a billion times a billion, and it works every time. Sometimes it takes only a few steps to reach 4, 2, 1; sometimes it takes a huge number of steps to get there. But you get there every time.

Does that mean it would work for any whole number you can think of—no matter how big? No one knows for sure. Just because it works for every number we've tried doesn't guarantee that it would work for all numbers. In fact, mathematicians have spent weeks and weeks trying to prove that there are no exceptions, but they haven't succeeded yet. Why this number pattern keeps popping up remains a mystery.

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