What is the Collatz rule?
Start with any positive whole number. If it's even, halve it; if it's odd, triple it and add one. Repeat.
// maths › Sequences & Patterns
Collatz Trajectories
Hailstone paths for many starting numbers: even -> halve, odd -> triple and add one. They always reach 1 - but nobody has proved why.
the 3n+1 hailstone paths
A mind behind this: Lothar Collatz 1910–1990
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Frequently asked questions
Why is it famous?
Every number tested - billions of them - eventually reaches 1, yet no one has proved it must always happen. It is a famous unsolved problem.
What are 'hailstone' numbers?
The values bounce up and down like hailstones in a cloud before finally falling to 1, which is why the sequences got the nickname.
Has anyone solved it?
Not yet. It has been checked by computer to astronomically large numbers, but a general proof remains open.
Why show it to students?
It proves that maths still has simple-sounding mysteries, and it is a great introduction to algorithms and sequences.