The Collatz Ring
Àlex Reyes · June 2026
In 1937, while J. R. R. Tolkien was putting the finishing touches on The Hobbit in Oxford, a German mathematician named Lothar Collatz was writing down — or so the story goes — a question of almost comical simplicity.
Take any positive integer. If it's even, divide it by two. If it's odd, multiply it by three and add one. Keep going.
Will you always reach one?
Nearly ninety years on, no one knows. Tolkien published his book that autumn. Collatz kept his question to himself.
I didn't find the conjecture until I was nearly forty. And I found it in the least heroic way possible: watching a podcast. Eduardo Sáenz de Cabezón, talking to Jordi Wild, explained the problem with the weary clarity of someone who knows it too well — and added that he was sick of receiving proposed solutions by email, that he'd stopped reading them, that people got obsessed.
I got obsessed.
There was no sign from the universe, no wizard at the door. Just a high school maths teacher with fifteen years of whiteboards behind him, staring at a screen at eleven at night. But there's something about the Collatz conjecture — once it's in, it doesn't leave. As if it had been sitting there since 1937, waiting for the wrong hobbit.
There's something Tolkien understood about impossible burdens that mathematicians rarely put into words: in the end, they leave you alone.
Finding someone willing to supervise a PhD on the Collatz conjecture is, in itself, quite a journey. The problem has a reputation — not with the general public, who find it delightful, but with mathematicians, who know the terrain. And not without reason: for decades, too many false proofs have landed in their inboxes, too many arguments that collapse under the weight of the first careful look. Sáenz de Cabezón doesn't read those emails. Many journals see the word Collatz in the subject line and already think they know what kind of story is coming.
Turning up as an unknown with a new result on Collatz is a bit like arriving at Rivendell without being an elf, a king, or a wizard. You have the question in your hands and a fragile suspicion — not yet a conviction — that you've seen something worth a closer look.
The Fellowship breaks. It always does. Everyone goes as far as they can, and some stretches of the road you walk alone. But the moment you enter Mordor is always yours.
Gandalf, though, exists.
When I first wrote to Dr. Jaume Franch, I'd spent nine months researching Collatz on my own and wanted to tell him two things: what I'd found, and that I wanted to do a PhD. He said what you'd expect — that it sounded great, that he encouraged me, that it wasn't his area. And then he did something unexpected: he started looking for a co-supervisor. We tried three times. Nobody was interested, or nobody felt they could reasonably call themselves a specialist in the subject. With this problem, that was predictable. There are no specialists in Collatz; at most, there are survivors.
So we decided to go together: Jaume as supervisor, with the door open to a co-supervisor who might show up someday. Supervising a thesis this far outside his own field weighed on him. I thought it was the best possible outcome. I'd rather have someone who genuinely throws themselves into a problem they don't own than an expert who won't even look at the map.
Gandalf wasn't a specialist in the Ring either. He was simply the only one willing to dig through the archives, ask the right questions, and start walking. That's what Gandalf is in this story: not the one who solves the problem — that person doesn't exist — but the one who tells you the road is there, even if he can't walk all of it with you.
But what is the Ring?
The One Ring isn't powerful because it destroys. It's powerful because it organises. Underneath apparent chaos it imposes an invisible order: every bearer, every journey, every kingdom ends up orbiting the same centre, whether they know it or not.
The Collatz conjecture has exactly this quality. The rule itself is trivial — that's not the disturbing part. What disturbs is that something so simple should act like an invisible gravity, pulling every number, from any starting point, toward the same place. No apparent reason. As if there were something underneath it all that we still haven't found the words for.
Bilbo knew, even if he was talking about something else:
It's a dangerous business, Frodo, going out your door. You step onto the Road, and if you don't keep your feet, there's no knowing where you might be swept off to.
That's Collatz. Every number is a door; every iteration, a step. From the starting point alone, there's no way to tell how long the journey will last or where it'll go. Twenty-six makes it home in ten steps. Twenty-seven — its next-door neighbour — climbs all the way to 9,232 before it starts coming down, and takes a hundred and eleven steps to do it. The road doesn't negotiate. You just walk it.
Tolkien loved maps. All his books have them: mountains sketched by hand, paths drawn before a single line of story existed. The map didn't illustrate the world — it was the world, and the story grew from moving through it.
The dynamics of Collatz have a map too. Mine has six regions: six territories through which numbers travel, driven always by the same rule. Some are pass-through land, moorland that a trajectory crosses without stopping. Others pull you in, set you spinning, force long detours before they let you go.
The territories of multiples of three are peculiar. They have something of a place of departure about them — a border, a door that pushes you outward. They're the Grey Havens of this map, in a sense: places you leave, but never come back to.
And there is one region that plays the role of Mordor: the numbers that leave a remainder of 4 when divided by six. Not because it's dark, but because everything points to the same conclusion — you can't close the journey without crossing it. Just as no Ring-bearer can destroy the Ring without entering Mordor, no non-trivial positive trajectory should be able to avoid that region indefinitely, if the conjecture is true. Proving that inevitability is getting to the heart of the problem.
At the end of everything, if Collatz is right, two territories remain. Just two: 1 and 2, switching between each other forever. That's the Shire — not where you started, but where you return. The place you come back to once the Ring is gone.
Frodo comes home to the Shire. That's what Tolkien's story promises: however long the road, there's a way back. But Tolkien also shows what happy endings usually leave out — whoever comes back is not the same as whoever left. The Shire hasn't changed; the hobbit has.
I haven't come back yet.
But there's something I know now that I didn't at the start: the Collatz conjecture is not the kind of problem you solve on a quiet afternoon. It's the kind of road that picks you — without asking if you're ready, without any guarantee you'll arrive. Only the strange, stubborn certainty that it's worth putting one foot in front of the other.
Bilbo was right.
It's dangerous to step out your door.
But I already have.