These are record holders — each takes longer to fall than every smaller number before it. 27 is the famous one: a tiny number that climbs to 9,232.
Scanning upward in the background. Each row is a number that beat every smaller starting point.
The whole rule fits on one line:
Start anywhere. Apply it forever. The Collatz conjecture says you always end up at 1 (and then loop 1 → 4 → 2 → 1 forever). Computers have verified this for every starting number up to about 268 — roughly 300 quintillion — without a single exception.
Nobody has proved it. Paul Erdős, one of the most prolific mathematicians who ever lived, said of
it: Mathematics is not yet ready for such problems.
He offered $500. It is still unclaimed.
Because the rule pushes both ways. Halving shrinks a number by a factor of 2; the odd step grows it by about 3. But an odd step always produces an even number, so on average each odd step is followed by at least one halving — net factor about 3/4 per pair. So numbers tend to fall. "Tends to" is not a proof. To prove it you'd have to rule out both an escape to infinity and any hidden loop other than 4→2→1, and nobody knows how to do either.
Also called the hailstone sequence, because the values rise and fall like a hailstone in a thundercloud before finally hitting the ground.