Definition
The Collatz conjecture says that if you pick any whole number and keep repeating one rule, you will always end up at 1. The rule: if the number is even, cut it in half. If it is odd, multiply by 3 and add 1. The rule is simple enough for an elementary school kid, yet modern mathematics still cannot prove whether the claim is true or false. It is one of the best-known unsolved problems.
A Roller Coaster of Numbers, Like Hailstones
Let's follow the number 6 step by step. 6 is even, so we halve it and get 3. 3 is odd, so we multiply by 3 and add 1 to get 10. 10 is even, so it drops to 5. 5 is odd, so it shoots up to 16. Now 16 is even, so it falls fast: 8, 4, 2, 1. It lands on 1.
Numbers that go up and down like this look like hailstones, which rise and fall inside a storm cloud, growing bigger, before dropping to the ground. That is why these sequences are also called hailstone numbers. Even an ordinary number like 27 climbs as high as 9232 and bounces up and down 111 times before it finally reaches 1.
Some numbers reach 1 in just a few steps. Others take a dizzyingly complicated path. But here is the surprising part: every single number people have tested so far has ended up at 1, without a single exception.
The Loop You Can't Escape Once You Reach 1
What happens after a number reaches 1? 1 is odd, so multiplying by 3 and adding 1 gives 4. 4 is even, so it becomes 2. 2 is even too, so it goes back to 1. In other words, the number is trapped forever in an endless 4-2-1 loop.
For the conjecture to be true, two things must not happen. First, there must be no escape route, where a number grows bigger and bigger forever and never reaches 1. Second, there must be no other loop, where strange numbers circle among themselves instead of falling into 4-2-1.
Mathematicians have used several supercomputers to check every number up to 2 to the 71st power (about 2.4 × 10²¹), one by one (reported in 2025). They found no number that falls into a different loop or grows forever. Still, in math you can never plug in every number, so a rigorous logical proof is still needed.
To Be More Precise: Why Is It Still Unsolved?
The rule is so simple. So why have even the world's greatest mathematicians thrown up their hands? At first glance, it seems the numbers should shrink in the end, because halving an even number happens more often than tripling an odd number. But the order in which odd and even numbers appear is as irregular as random noise, so it is hard to capture with a general rule.
The mathematician Paul Erdős said, "Mathematics may not be ready for such problems." It looks like plain arithmetic, but it is believed to be tied to tough problems in number theory and dynamical systems.
In 2019, Terence Tao, one of today's leading mathematicians, proved an important result: almost all numbers, if you keep going, come down to values far smaller than where they started. But the perfect finish, "every whole number, 100 percent, no exceptions," is still an unfinished homework problem.
Common misconceptions
Supercomputers have checked enormous numbers and every one of them reached 1, so it is already proven.
Whole numbers go on forever, so no matter how many you check with a computer, you cannot declare the claim true. A single counterexample would make it false, so a complete logical proof is needed.
Where you meet it
Start with 7 and you get 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1. It reaches 1 in 16 steps.
Powers of 2, like 16, 32, and 64, never meet an odd number. They are only divided by 2 again and again, so they race straight down to 1.
In one sentence
Halve even numbers, and multiply odd numbers by 3 and add 1. Every number seems to end up at 1, but this simple rule is still an unsolved math problem.