Pick a number. If it's even, halve it. If it's odd, triple it and add one. Repeat. Do you always reach 1?
Lothar Collatz asked in 1937 whether every starting number eventually reaches 1. Nobody knows. Computers have checked every starting number up to about 268, nearly 300 billion billion, and all of them reach 1, but checking is not a proof: some enormous number might climb forever or fall into a loop that never touches 1.
Paul Erdős said that mathematics may not yet be ready for such problems. In 2019 Terence Tao proved that almost every starting number eventually gets very small, which is the strongest result so far, but it still leaves room for exceptions. The paths are often called hailstone numbers, because they rise and fall like hailstones in a storm cloud before coming down.