Up and Down — That’s Collatz For Ya

The Collatz Problem goes by many names — some call it the 3n + 1 problem, though it’s also called the Hailstone Problem, Hasse’s algorithm, and others. The Collatz Problem can be stated as follows:

Let a₀ be a positive integer. Then, a_n = 0.5a_n – 1 if a_n – 1 is even,
and a_n = 3a_n – 1 + 1 if a_n – 1 is odd.

The Collatz Conjecture states that no matter what number you start with, the sequence will eventually reach 1. Originally posed in 1937 by Lothar Collatz, the problem is still unsolved.

Randall Munroe stated the following truth about the Collatz Conjecture at xkcd.com:

In line with this week’s earlier post about the MJ4MF Humorous Math Poem Contest, the following poem about the Collatz Conjuecture comes from poet and retired mathematician Joanne Growney. Growney uses a slightly different statement of the Collatz Problem; in her version, a_n = 1.5a_n – 1 + 0.5 if a_n – 1 is odd.

A Mathematician’s Nightmare
by JoAnne Growney

Suppose a general store —
items with unknown values
and arbitrary prices,
rounded for ease to
whole-dollar amounts.

Each day Madame X,
keeper of the emporium,
raises or lowers each price —
exceptional bargains
and anti-bargains.

Even-numbered prices
divide by two,
while odd ones climb
by half themselves —
then half a dollar more
to keep the numbers whole.

Today I pause before
a handsome beveled mirror
priced at twenty-seven dollars.
Shall I buy or wait
for fifty-nine days
until the price is lower?