The Shapes of Numbers
I am planning to re-start reading SICP, so I was looking around for some online course structure that I could follow. I found that UC Berkeley offers a course CS 61A titled SICP which uses Python as the language of instruction rather than Scheme. I was going through the course page and found a rather interesting homework problem
Pick a positive number n
If n is even, divide it by 2.
If n is odd, multipy it by 3 and add 1.
Continue this process until n is 1.
The number n will travel up and down but eventually end at 1 (at least for all numbers that have ever been tried – nobody has ever proved that the sequence will always terminate).
The sequence of values of n is often called a Hailstone sequence, because hailstones also travel up and down in the atmosphere before falling to earth. Write a function …
I have always been fascinated by the shapes generated by equations, so
I wanted to ‘see’ this
hailstone shape for myself. I wrote a
Ruby script to generate a hailstone sequence.
1 2 3 4 5 6 7 8
I generated the sequence for the first 100 numbers.
I then used the
Grapher utility on the Mac to plot the resulting series. I found
that the shape was like a sawtooth curve, oscillating up and down for
a while before gracefully dying down to 1. This behavior was
pronounced for odd and prime numbers more than even numbers (which
tend to damp down quicker). Here’s a picture (click image to enlarge) of the
for 11, 59 and 89 (all odd and prime by choice):