The Collatz conjecture (3n + 1)

The rule

Starting from 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Once you reach 1 the rule gives 4, 2, 1 again, so this small loop is where every known path ends.

The problem is named after Lothar Collatz, who described it in the 1930s. It is also known as the 3n + 1 problem, the Syracuse problem, Ulam’s problem and the hailstone problem, because the values rise and fall like hailstones in a cloud before dropping to the ground.

Why 27 is famous

Small starting numbers can take surprisingly long journeys. The path from 27 needs 111 steps and climbs as high as 9,232 before it finally falls to 1. The number 97 takes 118 steps and reaches the same peak.

There is no simple way to predict these lengths. Neighbouring numbers can have wildly different paths.

What has been checked

Computer searches have verified the conjecture for every starting number beyond 2⁶⁸, which is about 2.95 × 10²⁰. Every one of those paths reaches 1. That is strong evidence, but in mathematics evidence is not proof: some famous conjectures have failed only for enormous numbers.

In 2019 Terence Tao proved that almost all Collatz paths eventually drop to values that are very small compared with their starting point. It is one of the strongest results on the problem, but it still does not show that every path reaches 1.

Why Kaprekar is easy and Collatz is hard

Both processes repeat a simple rule, but they differ in one crucial way. Kaprekar’s routine with four digits never leaves the set 0000 to 9999. With only 10,000 possible states, every path must eventually repeat, and we can simply check all of them, as explained in why every number reaches 6174.

Collatz paths are not bounded in advance. A path could in principle grow forever, or fall into some unknown cycle far above anything checked. Proving neither happens requires understanding infinitely many numbers at once, and the mixture of halving and tripling resists the usual tools.

Small changes, different behaviour

The rule is delicate. Replace 3n + 1 with 3n − 1 and new loops appear: starting from 5 gives 5 → 14 → 7 → 20 → 10 → 5, and 17 leads into a much longer cycle. Replace it with 5n + 1 and 13 enters the loop 13 → 66 → 33 → 166 → 83 → 416 → 208 → 104 → 52 → 26 → 13, while other starting values appear to grow without limit.

These variations show that there is nothing obvious forcing every path down to 1. Whatever makes 3n + 1 behave well, if it does, must depend on the precise numbers 3 and 1.

A heuristic reason to believe it

An odd step multiplies by about 3, and the result is always even, so at least one halving follows. On average, a random odd number is followed by two halvings. Averaged over many steps, each cycle multiplies the value by roughly 3/4, so paths should tend to shrink. This argument is not a proof, because real paths are not random, but it explains why every tested path has come down.