Another "Unsolvable" Math Problem!

In CALLIOPE's July/August 2010 issue on numbers, we included an article on mathematical problems that remain unsolved today. When Jakob Coles received his "Who's Counting?" issue, he showed his father, Professor Drue Coles, who is in the department of mathematics, computer science, and statistics at Bloomsburg University of Pennsylvania. They then sent CALLIOPE an e-mail that told about other "unsolvable" problems.

Thanks, Jakob, it was great to hear from you, and we really enjoyed receiving examples of other "unsolved" problems. Now, why not try your hand at the problem from Jakob's father that we included below?

Hailstone Numbers:

Choose any natural number (that is, a positive whole number). Call it n. Now play this game: If n is even, cut it in half. If it is odd, multiply it by 3 and the add 1. Repeat this game until n = 1. For example, if you start with n = 3, the process will go as follows:

 3 - 10 (multiply by 3 and add 1)

10 - 5 (divide by 2)

 5 - 16 (multiply by 3 and add 1)

16 - 8 (divide by 2)

 8 - 4 (divide by 2)

 4 - 2 (divide by 2)

 2 - 1 (divide by 2)

The sequence 3-10-5-16-8-4-2-1 is called a hailstone sequence, since the numbers go up and down but eventually reach 1, like hailstone going up and down in the wind before crashing to the ground.

The open question, first asked by the Hungarian mathematician Lothar Collatz in 1937, is whether every hailstone sequence eventually ends. The answer seems to be yes. Computers have checked the hailstone sequence for every starting number up to about quintillion (that's a 5 followed by 18 zeros) and each one eventually stops. There is also other evidence suggesting that every hailstone sequence comes to an end sooner or later. But nobody has been able to demonstrate that this must always, absolutely, be the case. Perhaps there is some very large and strange number that leads to a loop in a hailstone sequence or causes it to keep increasing forever.

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Updated: 5/27/12 03:42 am
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