The simplest math problem nobody can solve

Pick any positive whole number. Anyone you like.

If it's even, divide it by 2. If it's odd, multiply it by 3 and add 1. Then take whatever number you get and repeat the process. Keep going.

Here's the strange thing: no matter what number you start with, you always seem to end up at 1. Always. Try it with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Try it with 27 and you're in for a wild ride — the sequence shoots up to 9,232 before finally crashing back down to 1 after 111 steps.

This is the Collatz conjecture, named after the German mathematician Lothar Collatz who first wrote it down in 1937. And here's the kicker: nobody has ever proven it works for every number. We've checked it for every integer up to about 2.95 × 10²⁰ (that's roughly 295 quintillion), and it holds. But "it works for a really, really big pile of numbers" isn't the same as proof, and mathematicians don't do approximations.

The legendary Paul Erdős, a man who spent his life solving hard problems, said of the Collatz conjecture: "Mathematics is not yet ready for such problems." That was in the 1980s. We're still not ready.

What makes it so stubborn? Part of the problem is that the sequence behaves almost randomly. When a number is odd you multiply by roughly 3, when it's even you divide by 2 — and since even numbers are far more common in the sequences that emerge, you'd expect things to drift downward eventually. But expecting something and proving it are very different things. The sequence doesn't follow a neat pattern you can grab onto. It careens around like a pinball before finding its way to 1.

In 2019, the mathematician Terence Tao — arguably the greatest living mathematician — made a significant breakthrough. He proved that almost all starting numbers eventually reach a value close to 1. Not exactly 1, and not all numbers, but close enough to cause genuine excitement in the math world. It was the biggest progress on the problem in decades. And it still wasn't a proof.

The Collatz conjecture sits in a strange category of math problems: ones that a child could understand in thirty seconds but that have resisted the combined effort of the world's best minds for nearly a century. It's not alone — the twin prime conjecture and Goldbach's conjecture have similar personalities — but there's something uniquely maddening about Collatz. The rule is so simple it feels like it should be provable. Like the answer is just around the corner.

Maybe it is. Maybe someone will crack it next year. Or maybe it's genuinely beyond the tools we currently have, the kind of problem that requires mathematics we haven't invented yet.

Until then, the best any of us can do is pick a number, start dividing and multiplying, and watch it tumble — inevitably, mysteriously — back down to 1.