Rick Robinson

Dec 28th 2022

The Collatz Conjecture: Closing in on a “Dangerous” Challenge


If you want to give a mathematician a brain teaser, give them a simple puzzle or problem that no one has ever solved. The more (seemingly) simple, the better.

In fact, some mathematical brain teasers are so potent that mathematicians warn each other against getting hooked on them. As Quanta Magazine reports, one notorious example of unsolved problems in mathematics is known as the Collatz Conjecture. One leading expert on the conjecture, mathematician Jeffrey Lagarias, calls it “a really dangerous problem,” adding that “people get obsessed with it.”

A Hailstorm of Numbers

So, what exactly is the Collatz Conjecture? Start by picking any positive whole number, such as 7, 42 or even 3,092,721. If the number is even, divide it by 2. If the number is odd, multiply it by 3 and add 1.

Next, take the new number that you just got and repeat the same process: if even, divide by 2, if odd, multiply by 3 again and add 1. Rinse and repeat, and keep repeating. To save on time and figuring, there are a variety of online calculators that will do the grunt work for you.

Also, to save you from going over ground already covered, Wolfram Mathworld informs us that the Conjecture has been tested for every positive whole number up past five quintillion.

The actual Conjecture is not this process itself, but a guess made by the mathematician who came up with it. His conjecture was that, if you pick any positive whole number and follow the process, eventually, the resulting sequence or “orbit” of numbers will reach 1. (Once it does, the sequence will keep going, but follow the simple loop 1, 4, 2, 1, 4, 2, 1 … )

The number sequences that this process creates have been called hailstone numbers, because they tend to alternately rise and fall, as hailstones forming in a storm cloud were once believed to do. UCLA mathematician Terence Tao notes that meteorologists have changed their theory of how hailstones are formed, but mathematicians still describe Collatz sequences or orbits as hailstone numbers.

Polling the Positive Integers

But Tao has done a great deal more than distinguish between hailstone numbers and hailstone weather. He has also made the most progress than anyone else in solving the Collatz Conjecture — determining whether sequences really all converge on 1.

Tao’s work on the Conjecture was inspired by an anonymous commenter on his personal blog. They suggested that he try to prove it for “almost all” numbers, without yet attempting to prove that there were no exceptions at all.

Tao took the suggestion and followed up on it, developing a statistical process not unlike the process of choosing a suitable cross-section of voters for a pre-election poll. Per Quanta, Jeffrey Lagarias, who warned of the obsessive nature of the challenge, calls it “a great advance in our knowledge of what’s happening on this problem.”

The Collatz Conjecture and Dynamical Systems

For mathematicians, solving unsolved problems in mathematics does not need to be “about” anything but itself, any more than a piece of music needs to be “about” something.

But as Quanta notes, and Tao himself points out, mathematical sequences of the sort produced by the Collatz problem have much in common with dynamical systems found in the real world that can be described by differential equations.

So, while Tao’s partial solution of the Collatz Conjecture is unlikely in itself to lead to new technology, whatever furthers our understanding of mathematics will probably, sooner or later, lead to greater understanding of the world around us — in much the same way that Collatz sequences themselves seem to converge on 1.