Collatz’s 3x+1 chase: why minds stuck for 100 years

A simple rule about even and odd numbers leads to a tree of possibilities—and to a question mathematicians have chased for nearly a century. The Collatz conjecture, named after Lothar Collatz, has been checked up to 271, pursued through decades of partial resu

Pick any positive whole number. If it’s even, divide it by 2. If it’s odd, multiply it by 3 and add 1. Then repeat the same steps on the new number—again and again.

The trap is how quickly the rules start feeling like a magic trick. No matter what you choose, the iterations look like they’re heading toward 1. But that’s exactly where the problem lives. Whether this always happens for every possible positive whole number is the open question known as the Collatz conjecture. named after Lothar Collatz. who first investigated it in the 1930s.

Mathematicians have been unable to agree—less on the experiments than on the proof. And the human rhythm of the chase is familiar: run the chain on a calculator, see it fall neatly into 1, then imagine it must be inevitable. It’s a hope that’s kept breaking for decades.

Paul Erdős, one of the most prolific mathematicians of the 20th century, captured the mood with a warning that “mathematics may not be ready for such problems”.

The scale of the checking makes the faith fragile. Computers have been used to check every number up to 271. That feels decisive—until you remember the question includes infinitely many remaining numbers, meaning the brute-force route can’t finish the job.

The intrigue comes from how the numbers behave once the rule begins acting like a fork in a road. Starting from 1 ends immediately. Starting from 2 is just one step: halve it and you’re done. But try 3 and the chain stretches: 10, 5, 16, 8, 4, 2, 1.

From 7 it’s longer still: 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

Look closely and you notice something that feels almost like cheating—because the chain for 7 contains the chain for 3. Collatz has this property: once a chain hits a number that has already been checked, you don’t need to check it again, because you already know where that chain ends.

That same feature is part of why the conjecture became “catnip” for mathematicians, spreading like a meme that jumps from brain to brain, with many people claiming to have solved it—only for their hopes to collapse when the proof unravels.

In the middle of all that, the conjecture became tangled with internet culture and recreational math lore. The xkcd webcomic described a certain type of brain that “is easily disabled. ” and that when shown an interesting problem it drops everything else to work on it. The Collatz meme, as it spread, became the perfect embodiment of that compulsion.

Jeffery Lagarias, who has extensively surveyed the conjecture, helps explain another reason the story keeps reappearing: pinning down its origin isn’t as simple as finding one clean first publication.

In a 1980 letter, Collatz wrote that he began investigating the conjecture “almost 50 years ago”. Yet for years it seems to have stayed largely to himself, treated as idle curiosity rather than something meant for the wider math world.

The wider circulation is said to have shifted in 1950. when Collatz went to the International Congress of Mathematicians—the largest meeting in the field—and informally chatted about the problem with other attendees. From there. it moved through mathematical networks and was “rediscovered and rebadged” by others. showing up under many names: the Syracuse problem. Hasse’s algorithm. and even just the 3x+1 problem.

The conjecture didn’t appear in print until 1971, when it was described as “a piece of mathematical gossip”. A year later. it climbed into broader view when Martin Gardner wrote about it in his Mathematical Games column for Scientific American. Gardner is known as a legendary figure in “recreational mathematics”—an area serious research mathematicians often look down on publicly while secretly enjoying.

It even resurfaced with warnings aimed at the very temptation it creates. In 1983, an article titled “Don’t Try To Solve These Problems” listed the conjecture alongside others, urging mathematicians away while clearly expecting they wouldn’t be able to resist.

The first big results didn’t crack the conjecture open, but they did give the chase a map. In 1976, Riho Terras proved an important result about how quickly the sequence can drop. If you start from an even number. the chain’s first move is to halve it. which means the chain always drops below the starting number right away. But if you start from an odd number. the first step takes you above the starting number—so the question becomes how long it takes before you come back down below where you started. hopefully on the way to 1.

Terras called this quantity the “stopping time” for a number. He proved that in almost all cases, the stopping time is finite—meaning those numbers eventually go down rather than continuing to grow forever.

Still, “almost all” left too much room for doubt. A single counterexample—an unimaginably large number that never reaches 1—would be enough to disprove the conjecture. And the imprecision of the phrase made the victory incomplete, because the conjecture asks about every possible positive whole number.

More precision arrived in 2002, when Ilia Krasikov and Lagarias proved that for a given number x, at least x0.84 numbers below it will eventually reach 1. For x = 100, that means at least 47 numbers below 100 reach 1.

The proof is subtler than the concrete case: people already know every number below 100 reaches 1, but the result didn’t claim that. It instead placed an explicit cap on what remained unknown.

The closest thing to a breakthrough that felt like the finish line came in 2019. Terrence Tao—described here as arguably the world’s greatest living mathematician—took on the problem and proved a stronger version of Terras’s result. He showed that not only do “almost all” numbers eventually go below their starting point. but that effectively. the numbers can be forced “as low as you want.”.

It sounds close to a proof of the Collatz conjecture, and yet it isn’t “any closer” in a way that still matters. The possibility of a counterexample in the far reaches of the number line remains.

The sequence of progress is unusually hard to interpret because every advance narrows what could still be wrong without ever ruling out the possibility that the wrong example is hiding beyond the reach of the argument.

That’s why the question still bites, and why it still spreads. And as this story was being written. it gained a new kind of irony: the news broke that OpenAI had used a large language model to solve a major problem that had stumped mathematicians for 80 years. It didn’t solve it by proving the statement correct; it did so by finding an unexpected counterexample. The same kind of twist—an AI-generated refutation, or something similarly surprising—seems tantalizing in the Collatz setting.

Would it happen here?. No one can say. But it’s the kind of turnaround that Collatz has always invited: a puzzle that infects human minds. then waits—patiently. relentlessly—for someone to prove whether the chain always ends. or whether a single number somewhere in the infinite dark refuses to fall to 1.

Collatz conjecture 3x+1 problem Lothar Collatz Riho Terras stopping time Paul Erdős Jeffery Lagarias Ilia Krasikov Martin Gardner Terrence Tao recreational mathematics AI counterexample