The Collatz problem (3n+1)

What just happened?

The Collatz problem (also known as the 3n+1 conjecture, the Syracuse problem or the Ulam conjecture) is probably the simplest-to-state unsolved problem in mathematics. You take any natural number and repeat a single rule.

The rule (Lothar Collatz, 1937):

For an even number — you halve it; for an odd number — you multiply by 3 and add 1; then you repeat. The Collatz conjecture says that for every natural number the sequence eventually reaches 1 (and then falls into the cycle $1 \to 4 \to 2 \to 1$).

Checked for astronomically many numbers…

The conjecture has been confirmed by computer for all numbers up to nearly $3\cdot 10^{20}$ (more precisely ≈ 2.95·10²⁰) — and it always holds. The trajectories can be surprisingly chaotic: the number 27 needs as many as 111 steps and “rises” to 9232 before dropping to 1 (you can see this on the graph after choosing 27).

…but no one has proved it

Despite the simplicity of the rule and over 85 years of attempts by the finest mathematicians, no one can prove that it works for all natural numbers — nor find even a single counterexample. Paul Erdős, offering $500 for a solution, said: “Mathematics may not be ready for such problems.”

Why this matters for your exams:

It is an extension of the topics of divisibility and sequences — and a fascinating curiosity about the limits of mathematical knowledge. It shows a deep truth: simplicity of statement does not guarantee simplicity of solution. A child can understand the 3n+1 rule, and yet it remains one of the most famous open problems in mathematics. Sometimes the shortest questions are the hardest.

Remember: the Collatz rule is $n \to \tfrac{n}{2}$ (even) or $n \to 3n+1$ (odd). For all numbers checked, the sequence ends at 1 — but there is no general proof. It is still an open problem.