This article is on a math problem that sounds like a riddle your younger sibling could understand. But then, after hours (or decades) of trying, thinking, and computational simulations, you are still far away from a solution. Below you will find all about the Collatz conjecture.
This deceptively simple problem has been called “the simplest unsolved problem in mathematics”. It’s not about stochastic processes, high-dimensional regression, or heteroskedasticity. Instead, it starts with just one number and a couple of rules. Yet it has puzzled both amateur tinkerers and professors alike.
The Conjecture
Take any positive integer, n. Now, follow these rules:
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If n is even, divide it by 2.
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If n is odd, multiply it by 3 and add 1.
Repeat this process with the resulting number and keep going. For example, start with n = 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Eventually, we reach 1. Once we hit 1, the sequence enters a loop: 1 → 4 → 2 → 1 → ...
The Collatz conjecture, named after German mathematician Lothar Collatz, simply states: “No matter what positive integer you start with, you will always eventually reach 1”. That’s the whole conjecture, and no one has been able to prove it.
Lothar Collatz introduced the problem to his students. At first, it seemed like an odd little number game. But the more people looked into it, the more it resisted any formal proof. In the decades that followed, mathematicians verified the conjecture for larger numbers using computers. As of today, it has been verified for numbers up to 2^68 (over 295 quintillion), and still there is no counterexample.
Why It’s So Difficult
Part of the Collatz conjecture’s mystique comes from its unpredictable behaviour. The rules are deterministic, but the path from a given number to 1 can be wild. For instance, start with 27. It takes 111 steps to reach 1, and along the way, the sequence soars up to 9,232 before eventually falling back down. Why does it spike so high? Why does it take so long? No one knows.
Mathematically, the problem lies at the intersection of number theory and dynamical systems. The sequence defines a discrete dynamical process; essentially, a function that maps integers to integers. But while most dynamical systems we study in econometrics are continuous and behave nicely (or at least predictably), the Collatz function is discontinuous, nonlinear, and chaotic.
Attempts to analyse it involve everything from modular arithmetic to Markov chains, from stopping times to heuristic probability arguments. Yet none of these have been able to crack the problem.
Why Should We Care?
It’s a fair question. This isn’t going to show up on your econometrics exam or help you improve the efficiency of your GLS estimator. But the thing is, mathematics is full of connections that seem obscure at first and then turn out to be profound.
Problems like the Collatz conjecture remind us that even the simplest systems can produce complexity beyond our current understanding. That’s relevant not just in pure math but in economics, econometrics, and the modelling of human behaviour. After all, our field is built on the premise that we can explain complex patterns using models, and here is a model with only two lines of code that no one can fully explain.
There’s also an aesthetic value here. It’s a reminder that math still has secrets, that we haven’t solved it all, and that sometimes the most profound mysteries hide in plain sight.
Final Thoughts
So next time you’re staring down a long night of matrix algebra or wrestling with a tricky problem in your thesis, take a break and try running the Collatz sequence on your favourite number. Watch it twist and turn, sometimes shrinking, sometimes exploding, always moving toward that inevitable 1.