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>.

