The Collatz conjecture · 3n + 1
Pick any positive whole number. Follow a two-rule recipe long enough, and — so far as anyone has ever checked — you always land on 1. A single exception would end the conjecture. None is known.
A short history
The conjecture is also called the 3n + 1 problem, Hasse’s algorithm, Kakutani’s problem, Ulam’s problem, the Syracuse problem, and Thwaites’s conjecture. The names track the people who carried it, not rival claims that have displaced Collatz.
1937
Lothar Collatz, then a young German mathematician, wrote down a simple iteration on the positive integers. He did not publish it. Years later he explained why: he could not solve it, so he kept it in his notes.
1950
At the International Congress of Mathematicians in Cambridge, Massachusetts, Collatz circulated iteration puzzles among colleagues. The 3n + 1 question began to travel by word of mouth rather than by journal.
1952
Collatz told Helmut Hasse about the map when they were colleagues in Hamburg. Hasse carried it to Syracuse University, where listeners named it the Syracuse problem. Stanisław Ulam spread it at Los Alamos and tested it on early computers — hence “Ulam’s problem.” Shizuo Kakutani lectured on it at Yale and Chicago, and it picked up his name as well. Bryan Thwaites later claimed an independent 1952 discovery and offered a cash prize for a proof.
1972
Martin Gardner’s column introduced the puzzle to a wide public. The same year, John Horton Conway proved that some Collatz-like rules are formally undecidable: there is no general algorithm that can say, for every similar recipe, whether every starting value returns to 1. That does not settle the original 3n + 1 map — it only shows how treacherous this family of questions can be.
1985
Jeffrey Lagarias published the first major survey of the 3x + 1 problem. Paul Erdős, who offered $500 for a solution, is widely quoted as saying that mathematics was “not yet ready for such problems.” The remark still fits. The question is elementary to state and has resisted every standard tool.
2019
Terence Tao showed that almost all starting values, in a precise density sense, eventually drop below any slowly growing bound. It is the strongest general theorem so far. It still leaves room for rare exceptions. Computers, meanwhile, have checked every integer up through at least 2⁶⁸ — about 295 quintillion. Checking a huge finite range is evidence, not a proof: infinity always has more numbers.
The rule
Start with any integer greater than 1. Apply one of the two rules below, then apply them again to whatever you just got. Keep going. The Collatz conjecture says that you will always reach 1, no matter which positive integer you began with.
If n is even
Cut the number in half. Even numbers always shrink, so this step feels like falling.
If n is odd
Triple it and add one. Odd numbers jump upward — often a lot — before the even rule pulls them down again.
The sequence rises and falls the way hailstones are lifted and dropped inside a thundercloud before they hit the ground. That is why the terms are often called hailstone numbers. The number 27 is the classroom favourite: it climbs to 9,232 before it finally rains down to 1, taking 111 steps to get there.
Once you reach 4, the rules force 4 → 2 → 1 → 4 → 2 → 1 … forever. Hitting 1 is the same as falling into this tiny loop. The conjecture is the claim that every starting integer greater than 1 is sucked into that loop, eventually.
One integer that never reaches 1. That number would either grow without bound, or enter some other loop that avoids 1. Finding either would end the conjecture. Checking a million examples cannot prove it — but a single counterexample would be enough.
The experiment
Even numbers fall; odd numbers leap. The graph uses a log scale so the leaps do not crush the landing.
Sample numbers
Current
—
Steps so far
—
Largest so far
—
Vertical axis · log scale