Collatz conjecture
If even, divide by 2. If odd, multiply by 3 and add 1. Does every starting number eventually reach 1? Nobody knows. Tested up to 268, unproven in general. The simplest open problem in mathematics.
n → n/2 (even) | n → 3n+1 (odd)
The conjecture
The Collatz conjecture, also known as the 3n+1 problem, states that for any positive integer, repeatedly applying the rule — halve if even, triple-and-add-one if odd — will eventually reach 1. First proposed by Lothar Collatz in 1937, it remains unproven despite extensive computational verification.
Hailstone sequences
The sequence of values is called a hailstone sequence because values rise and fall erratically, like hailstones in a thundercloud, before eventually crashing down to 1. The number 27, for instance, takes 111 steps and reaches a peak of 9232 before descending.
The tree view
The Collatz tree shows how all numbers converge toward 1. Every number has exactly one successor (its Collatz step), but each number can have multiple predecessors. The tree reveals the branching structure of the conjecture — a river system flowing toward a single ocean.