Value at every step, on a logarithmic scale — up on the 3n+1 gusts, down on the halvings, like a hailstone tossed in a thundercloud until it finally falls out at 1.
The conjecture: every positive whole number, fed through this rule, eventually reaches 1. Proposed by Lothar Collatz in 1937, it has been checked by computer for every starting number up to beyond 268 — about 295 quintillion — without a single escape. And yet no proof exists that it must always happen, which is the entire scandal: a problem a child can state and no one can close.
Visualize the Collatz conjecture — enter any starting number, plot its hailstone sequence step by step, race two numbers side by side, and see total stopping time, peak value and the longest dive.
Total steps to reach 1, for every starting number — the famous Collatz scatter. The chaos has texture: the banded streaks are families of numbers whose flights merge early, and the glowing point is the champion.
Champions are strange: 837,799 — the longest flight below a million at 524 steps — is famous as Project Euler's Problem 14, solved by hundreds of thousands of programmers who mostly never learn its name. And the highest flyer below a million, 704,511, climbs to 56,991,483,520 — eighty thousand times its starting altitude — before landing like everything else. So far, like everything else.
Collatz records computed live — the longest hailstone flights and highest peaks below 100, 1,000, 10,000, 100,000 and 1,000,000, with the full total-stopping-time scatter plot.
The distribution of flight durations, computed live. Most numbers land within a couple hundred steps — a tidy, almost bell-shaped crowd — and then there's the long tail. Proving that the tail never becomes an escape is the entire problem.
Erdős's verdict: Paul Erdős — the century's great problem-solver, who priced open problems in cash — looked at 3n+1, said "Mathematics may not be ready for such problems," and offered $500 for a solution. The bounty stands, technically, uncollected. The trouble isn't that the problem is deep in any known direction; it's that it belongs to no direction at all — number theory's tools grip multiplication or addition, and Collatz braids both into something neither field can hold.
Why the Collatz conjecture is unsolved — the probabilistic drift argument, why heuristics aren't proofs, Terence Tao's almost-all result, Conway's undecidability theorem, and Erdős's standing $500.
The binary length of every value in your number's flight (from The Flight tab). An odd step adds about 1.6 bits; the halvings that follow strip them away — averaging out to a slow, jagged descent of roughly a fifth of a bit per step. The chart is the drift argument, drawn.
The machine's-eye view: in binary, the Collatz rule is almost embarrassingly simple — halving is deleting a trailing zero, and 3n+1 is a shift-and-add. Every question about hailstones is secretly a question about how trailing zeros are born, which is why our binary converter is the right lens: a number's fate is written in its bits, if only anyone could read that far ahead.
Patterns in Collatz sequences — the binary bit-length sawtooth, parity vectors that encode entire flights, and how trajectories merge into rivers flowing to 1.
Computed live, including the current year — which will quietly change every January. The Answer to Everything lands in eight steps; the Project Euler champion takes 524.
Why it endures: the Collatz conjecture is mathematics' perfect campfire story — statable to a ten-year-old, checkable on any laptop, resistant to every professional. It has eaten uncountable afternoons (Erdős's warning was partly a public-health notice to young mathematicians), and it keeps recruiting: every generation rediscovers it, plots exactly the charts on this page, feels the pull, and learns why the pull is famous. Consider our sequence pattern detector and Fibonacci generator for sequences that behave — and the full story of how humans write numbers at all in the numeral systems guide.
Collatz conjecture lore — Lothar Collatz, the Syracuse and Ulam names, Kakutani's Cold War joke, verification past 2^68, xkcd, Project Euler 14, and famous numbers' flight times.