4.12. Hailstone Numbers


Hailstone numbers are the numbers of a sequence which is obtained as follows. First, start with any number. Then,

if the number is even: divide by 2

if the number is odd: multiply by 3 and add 1

Using the new number apply the previous condition again. Here we illustrate this algorithm, starting with the number 10:

10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

We can see that this sequence loops into an infinitely repeating 4,2,1 sequence.

If we tray any other number, say 58:

58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

we see the same loop of 4, 2, 1.

Hailstone numbers are called like that because they typically rise and fall, resembling a hailstone inside a cloud. Every tested starting natural number has produced a hailstone sequence that eventually drops down to the number 1, then rebounding into the loop 4, 2, 1.... You can imagine that it is a hailstone heavy enough to fall from the cloud to the ground and then bounce.

The proper mathematical name for this investigation is the Collatz conjecture which was made in 1937 by the German mathematian Lothar Collatz.

The following graphic shows how different numbers (x axis) take a different number of iterations (y axis) to reach 1. We can see that some numbers take much longer than others to reach one.