The Collatz Conjecture: The 3n+1 Problem No One Can Solve

The Collatz conjecture is the most famous unsolved problem in mathematics that a ten-year-old can understand. The rule is almost laughably simple: pick any whole number, and if it's even, halve it; if it's odd, triple it and add one. Repeat forever. The conjecture claims that no matter which number you start with, you always eventually reach 1. It sounds like it should be easy to prove, and yet, despite enormous effort, no one has. This guide explains the 3n+1 problem, walks through examples, and explores why such a simple-looking question has resisted the world's best mathematicians.

The rule

Start with any positive whole number, n. Then apply this rule over and over:

• If n is even: divide it by 2.
• If n is odd: multiply it by 3 and add 1 (that's the "3n + 1").

The Collatz conjecture says that whatever number you start with, repeating this process always leads you, eventually, to 1, after which you fall into the loop 4 → 2 → 1 forever.

Worked examples

Let's try n = 6:

6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Six is even, so halve it to 3. Three is odd, so 3 × 3 + 1 = 10. Ten is even, halve to 5. Five is odd, 3 × 5 + 1 = 16. Then 16 halves down: 8, 4, 2, 1. Reached 1 in eight steps.

Now a longer one, n = 7:

7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Seven takes sixteen steps and climbs as high as 52 before coming back down. That up-and-down behaviour is why these are nicknamed hailstone numbers: like hailstones tossed up and down in a storm cloud, they rise and fall unpredictably before finally landing at 1.

Why it's so hard

The Collatz conjecture is maddening precisely because it's so simple to state. Here's what makes it resist proof:

• The path is wildly unpredictable. Some numbers reach 1 in a few steps; others wander for hundreds of steps and soar to values thousands of times larger before descending. There's no obvious pattern to how long a number takes or how high it climbs.
• Tripling fights halving. The odd step makes numbers grow, the even step shrinks them, and which one dominates depends on the sequence of odd and even values you hit, which is itself unpredictable.
• "Always reaches 1" is a statement about every number. Checking examples, however many, can never prove it; you'd need an argument that covers all infinitely many starting numbers at once. No such argument has been found.

The problem is named after Lothar Collatz, who proposed it in 1937. It also goes by the names the 3n+1 problem, the Ulam conjecture, and the hailstone problem.

What we know so far

The conjecture has been verified by computer for every starting number up to astronomically large bounds, well beyond a quintillion, and not a single counterexample has ever turned up. Every number anyone has ever tested does reach 1. But verification is not proof. It remains possible (if widely doubted) that some enormous number either grows forever or falls into a different loop.

The great mathematician Paul Erdős said of it, "Mathematics may not be ready for such problems," and offered a prize for a solution. Decades later, it's still open, a humbling reminder that simplicity and difficulty are not opposites.

Try it yourself

The best way to appreciate the Collatz conjecture is to run a few numbers by hand. Pick a starting value, apply the rule, and watch the hailstone bounce. You'll quickly see both the eerie reliability (it always seems to reach 1) and the unpredictability (you never know the path in advance). It's the perfect example of how the number challenges world can take a one-line rule and turn it into something genuinely deep.

If you enjoy this kind of number behaviour, our harder number challenges explore the same territory of divisibility, parity, and surprising structure, with the reassuring difference that those puzzles do have answers.

Frequently asked questions

What is the Collatz conjecture?

The Collatz conjecture states that if you take any positive whole number and repeatedly apply a simple rule, halve it if it's even, or triple it and add one if it's odd, you will always eventually reach 1. Despite the simple rule, no one has been able to prove it's true for every number.

What is the 3n+1 problem?

The 3n+1 problem is another name for the Collatz conjecture. The "3n+1" refers to the rule applied to odd numbers: multiply by 3 and add 1. Even numbers are halved. The claim is that repeating these steps from any starting number always ends at 1.

Why are they called hailstone numbers?

Because the sequence rises and falls unpredictably, often climbing far above the starting value before descending, like a hailstone carried up and down inside a storm cloud before it finally falls to the ground. The number eventually "lands" at 1.

Has the Collatz conjecture been solved?

No. It remains unsolved. It has been checked by computer for every number up to extremely large limits without finding a counterexample, but that isn't a proof, and a general proof that works for all numbers has never been found.