I typed a random number into a calculator, followed one dumb rule, and couldn’t stop until it hit 1.
Here’s the rule. If your number is even, cut it in half. If it’s odd, triple it and add one. Whatever you get, do it again.
That’s the entire thing. No calculus, no formulas, nothing you didn’t already know in the fourth grade.
Try it with 27. Go on, actually try it — it’s more fun if you do the first few by hand.
27 is odd, so: ×3, +1 → 82. Even, so: halve it → 41. Odd again → 124 → 62 → 31 → 94 → 47…
It keeps going. And going. It climbs past 9,000 before it ever turns around. It takes 111 steps. And then — after all that — it lands on 1 anyway. Every single time. For every number anyone has ever tried.
Nobody can prove why.
Watch This First
Before any of the explaining — just watch a few hundred of these happen at once.
Every one of those lines is a different starting number, following the exact same rule. Halve if even, triple-plus-one if odd. Watch where they all end up.
They don’t scatter. They don’t wander off forever. They fall — every last one of them — into the same spot.
So What’s Actually Happening?
Mathematicians call this the Collatz conjecture, named after Lothar Collatz, who first asked about it in 1937. You’ll also see it called the 3n + 1 problem, or — my favorite name for it — the hailstone sequence, because the numbers bounce up and down like a piece of hail getting tossed around inside a storm cloud before it finally falls to the ground.
The rule, written the boring way:
- If n is even → n becomes n ÷ 2
- If n is odd → n becomes 3n + 1
Pick any positive whole number, apply the rule, then apply it again to whatever you get. Keep going.
The conjecture — the thing nobody has proven — is that no matter what number you start with, you always, eventually, land on 1.
Not usually. Not probably. Always. As far as anyone has ever checked.
Nobody Can Prove This Works
Here’s the part that should bother you more than it probably does.
Computers have checked this rule against every single number up to roughly 2⁷¹ — that’s a number with 21 digits. Not one counterexample. Not one number that spirals off into infinity, or loops forever without hitting 1. Every number checked, ever, comes home.
And yet: nobody has proven it has to happen. It might be true for every number that will ever exist and still be unprovable with the mathematics we have today. Paul Erdős, one of the most prolific mathematicians who ever lived, looked at this problem and said plainly: “Mathematics is not yet ready for such problems.” He offered $500 of his own money to anyone who could crack it. Nobody ever collected.
That’s the strange part of this whole thing. It’s not a hard problem because it’s complicated. A ten-year-old can compute it by hand. It’s a hard problem because simple doesn’t mean provable — and this rule has been quietly daring mathematicians to prove the obvious for almost ninety years.
Why Does It Always Fall?
Nobody can prove it, but here’s the hand-wavy version of why mathematicians believe it.
Every time your number is odd, the rule roughly triples it. But the very next step, since 3n + 1 is always even, you immediately halve it. So really, every odd step is followed by at least one halving for free.
On average, across a long enough run, a number gets multiplied by about 3 roughly as often as it gets divided by 2. And 3 loses that fight — dividing by 2 shrinks a number faster than multiplying by 3 grows it, especially once you account for the fact that most halvings chain together (even numbers often stay even for several steps in a row before they turn odd again).
So on average, a number under this rule slowly, unevenly, unpredictably — shrinks. It just refuses to prove that it always does, for every single number, with no exceptions, forever.
That gap — between “almost certainly true” and “proven true” — is where this problem has lived since 1937.
The Shape Underneath
Here’s the part that turns this from a curiosity into something beautiful.
If you don’t run the rule forward from a number, but instead ask “which numbers lead here?” and trace it backward from 1, you get a tree. 1 connects to 2. 2 connects to 4. 4 connects to 8 and also to 1 (wait, 4 → 1 doesn’t apply forward, but plenty of odd numbers feed into every even one). Numbers branch off in both directions, some paths short, some absurdly long, all of them eventually rejoining the same trunk.
Drawn out, it looks like coral. Or a lightning bolt, frozen mid-strike, with hundreds of forks that never quite recombine until they all reach the base. Nobody designed that shape. It just falls out of a rule you could teach a child in thirty seconds.
The Bigger Picture
This is the part that keeps mathematicians up at night, in a good way.
Rules this simple are usually either trivial or completely understood. The Collatz conjecture is neither. It sits in a strange, narrow gap where a problem is too simple to ignore and too slippery to solve. Entire branches of number theory have been built trying to get a handle on it, and it keeps refusing to fully cooperate.
It shows up as a favorite test case for distributed computing projects — volunteers around the world donate spare processing power just to push the “checked every number up to here” boundary a little further out, the same way people hunt for the largest known prime. Computer scientists use 3n+1-style rules to stress-test random number generators, because a sequence this unpredictable is a good way to catch a machine that’s cutting corners. And it’s become something of a rite of passage: if you’ve ever taken a first programming class, there’s a decent chance “write a program to compute the Collatz sequence” was one of your very first assignments — most people who write that code have no idea they just typed out an unsolved problem in mathematics.
Nobody was trying to make coral. Nobody was trying to build a storm of falling hail. It’s just what happens when a simple rule gets applied, over and over, without anyone checking in on where it’s headed.
Try It Yourself
You don’t need a computer. You need a number and a few minutes.
Pick any number. Your age. Today’s date. The last four digits of your phone number. It doesn’t matter — that’s the whole point.
Even? Halve it. Odd? Triple it, add one. Write down every number you get, in order, until you hit 1.
Count your steps. Compare with a friend who picked a different number. See whose number took the longest, wandered the highest, or dropped the fastest. There’s no wrong number to pick — every single one, eventually, comes home.
I still think about how ordinary this rule is. Multiply, add, divide — the same three operations from a third-grade worksheet. And yet it’s been standing quietly unproven for almost ninety years, daring anyone who looks closely enough.
Every number you will ever pick, for the rest of your life, wants to become one. Nobody can tell you why it has to. Just that, so far, it always has.
— Keep noticing.
Tagged: mathematics · patterns · unsolved