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© 1999-2010 by Francis Edward Su
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From the Fun Fact files, here is a Fun Fact at the Easy level:

Hailstone Numbers

Start with any positive integer (an initial seed) and obtain a sequence of numbers by following these rules.

1. If the current number is even, divide it by two; else if it is odd, multiply it by three and add one.
2. Repeat.

Let's try a few numbers to see what happens:
n=3; 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...
n=4; 2, 1, 4, 2, 1, ...
n=5; 16, 8, 4, 2, 1, 4, 2, 1, ...
n=6; 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...
n=7; 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

Hmmn... does every initial seed yield a sequence that eventually hits 1 (and then repeats: 4,2,1,4,2,1,...)?

The Collatz conjecture says yes, but this has never been proved. However, it has been shown true for every number ever tried! The numbers in such sequences bounce up and down, which is why they are sometimes called "hailstone numbers".

Presentation Suggestions:
Students may enjoy exploring this sequence as an outside project.

The Math Behind the Fact:
A lot is known, however; see the reference for starters. Some interesting patterns emerge if you look at the number of steps it takes for an initial seed to fall to 1, or if you look at the highest numbers in a hailstone sequence. For instance, compared to other initial seeds less than 100, the seed 27 takes an unusually large number of steps to reach 1.

A field of mathematics that concerns itself with repeatedly applying ("iterating") a function is called dynamical systems.

How to Cite this Page:
Su, Francis E., et al. "Hailstone Numbers." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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References:
    J. C. Lagarias, The 3x+1 problem and its generalizations,
    Amer. Math. Monthly 92(1985), pp. 3-23.

Keywords:    unsolved problem, Collatz sequence, discrete dynamical system
Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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