The man and the marginal note

The origin of the simplest problem nobody knows how to solve.

Let’s begin with the only simple thing in this story: the rule. The Collatz conjecture is built on a function defined on the positive integers: each number $n$ is sent to its half if it is even, and to three times itself plus one if it is odd.[1][1] · Collatz, L. (1986)From the original · English“Now we define the ‘3n+1’ function: f(n)=3n+1 for odd n, n/2 for even n. […] For each n there is an index k(n) so that f_k(n)=1.”

$$f(n)=\begin{cases} \dfrac{n}{2} & \text{if } n \text{ is even},\\[4pt] 3n+1 & \text{if } n \text{ is odd}.\end{cases}$$

Applied over and over starting from any $n$, it generates a sequence, and the conjecture claims that this sequence, wherever it starts, always ends at 1. That is how Collatz himself left it posed.

And that is how deceptive it is. No one has managed to prove it, and not for lack of evidence: it has been checked, one by one, for every integer up to 2⁷¹,[2] that is, some 2.36 · 10²¹, without a single exception. But a list of cases, however long, never amounts to a proof. It is a problem any child grasps in a minute, and one the best mathematicians in the world have failed to crack in almost a century.

Much of that difficulty lies in how capricious its sequences are. Take 27: it does not simply come down. It climbs to a peak of 9,232 before collapsing, and only lands on 1 after 111 steps.[3][3] · Guy, R. K. (1983)From the original · English“Figure 6 includes all the numbers up to 26; the branch containing 27 is a much longer one, but still comes down to 1 after 111 steps.” — a back-and-forth that has earned these numbers the nickname ‘hailstone’:

Start at or try: 6 27 97 871

Such wild behaviour from such an elementary rule is what has kept the problem alive for decades. But before it was a wall, it was a note in the margin of a notebook. And the man who wrote it would have been the first to be surprised that this — of all things — is what the world remembers him for.

Who Lothar Collatz was

One misunderstanding is worth clearing up at the outset: Collatz was no puzzle hunter. He was one of the leading figures of twentieth-century German numerical analysis — close to 250 papers, a dozen books, forty doctoral students —,[4][4] · Biener, K. (1993)Original · German„Tatsächlich hat Collatz sein gesamtes Lebenswerk nahezu ausschließlich der Angewandten, insbesondere der Numerischen Mathematik gewidmet und mehrere Algorithmen ausgearbeitet, die für einen computergestützten Einsatz zur Lösung verschiedener praxisrelevanter mathematischer Probleme gedacht sind.“Translation“In fact, Collatz devoted his entire life’s work almost exclusively to applied mathematics, and to numerical mathematics in particular, developing several algorithms meant for computer-aided use in solving various practically relevant mathematical problems.” and his name lives on where you would least look for it: in the Collatz–Wielandt formula, on the eigenvalues of a matrix,[5] and in a 1957 paper, written with Ulrich Sinogowitz, regarded as one of the founding articles of spectral graph theory.[6, 7] Sinogowitz never saw it in print: he died in the bombing of Darmstadt, in 1944.[8]

He had earned his doctorate in Berlin in 1935. His real mentor, Richard von Mises, was forced to emigrate fleeing Nazi persecution, and Collatz, like other doctoral students left without a supervisor, was passed on to Alfred Klose.[8, 9] Collatz himself joined the SA in 1933 and the Nazi party in 1937, and during the war he worked on ballistics calculations.[8, 10]

He held a view of his craft that today reads almost like a manifesto: for him, the border between ‘pure’ and ‘applied’ mathematics was nonsense; there were not two fields, but only one.[4][4] · Biener, K. (1993)Original · German„…der Verfasser wäre glücklich, wenn dieses Buch dazu beitragen würde, den unseligen Unterschied zwischen „reiner“ und „angewandter“ Mathematik ad absurdum zu führen, denn es gibt keine Trennungslinie zwischen diesen beiden Gebieten, es gibt nur eine Mathematik.“Translation“…the author would be glad if this book helped to reduce to absurdity the unfortunate distinction between ‘pure’ and ‘applied’ mathematics, for there is no dividing line between these two fields: there is only one mathematics.” Perhaps that is why there is a certain irony in a mathematician with such broad and deep work being remembered, above all, for a three-line curiosity.[11][11] · MacTutor (2006)From the original · English“In many ways it might seem a pity that a mathematician who has produced so much important and fundamental work should be most remembered for a novelty.” He drew, by the way, with notable talent — a pen study of his survives, of the mosaics in the basilica of San Vitale, in Ravenna —: the same eye for pattern and structure that, as we will see in a moment, led him to 3n+1.[4]

Where it came from

Collatz did not arrive at 3n+1 looking for a puzzle, but by doing something very much his own: drawing functions. He loved representing arithmetic functions as graphs — one dot per number, an arrow from $n$ to $f(n)$ — and classifying them by the shape they took: lines, trees, forests, cycles.[1] 3n+1 was born out of that game, while he was building simple functions in search of cycles.

His taste for this went back a long way, to the classes of Edmund Landau, Oskar Perron and Issai Schur in his student years, between 1928 and 1933.[1] In a notebook of his, dated 1 July 1932, one of these questions[12][12] · Lagarias, J. C. (1985)From the original · English“The exact origin of the 3x+1 problem is obscure. It has circulated by word of mouth in the mathematical community for many years.” already appears — though it was not yet 3n+1, but a different permutation of the integers, which Klamkin would publish in 1963 and which remains unsolved. And here a clarification is in order, one almost no one makes — myself included: in the first note of this series, The Collatz Ring, I dated the problem to 1937. That date, however, is backed by no document of the problem. The only one supported by anything written is that of the notebook — though, as Lagarias himself admits, the exact origin remains ‘obscure’.

None of this, moreover, was the centre of his work. Collatz was a serious numerical analyst; 3n+1 was always what he scribbled in the margin, while he got on with what mattered. Hence the title of this note: the man, and the note in the margin.

Nor did he publish it. He spread it by word of mouth — he described it to his colleague Helmut Hasse in 1952, on arriving in Hamburg —,[1] and, when he finally wrote it down, he posed it in two equivalent forms: that every orbit ends at 1 or, put in terms of his graphs, that the graph of the function is connected. He even specified what it would take to close the problem: prove that you always reach 1, or exhibit a cycle other than the trivial 1 → 4 → 2 → 1.[1] Those two faces — the descent and the cycles — are, precisely, the thread of the next note.

The problem that escaped the notebook

A problem that travels by word of mouth ends up collecting names along the way, and Collatz’s gathered quite a few. It is credited with an informal talk at the 1950 International Congress of Mathematicians, in Cambridge — the Massachusetts one —, where Coxeter, Kakutani and Ulam lectured; hence their names ending up attached to the problem.[12] Hasse, to whom Collatz had told it in Hamburg, spread it in turn — some called it ‘Hasse’s algorithm’ —, and it was he who, on a visit to Syracuse University, gave it the name many still know it by: the Syracuse problem.[12]

Someone even claimed authorship: the Briton Bryan Thwaites maintains he came up with it, on his own, in 1952.[13] And the sequences themselves picked up a nickname — ‘hailstone numbers’ — coined by Brian Hayes in 1984 because they rise and fall along their course like hailstones inside a cloud before they drop.[14]

What is striking is that, despite circulating since the thirties, the problem did not appear in print until 1971. In those decades one way of doing mathematics held sway — that of the Bourbaki school — which prized great interconnected theories, and an isolated problem, with flimsy results, seemed almost in bad taste: something déclassé, liable to damage the reputation of anyone who took it seriously.[15] When it finally made it into print, it did so almost in passing: in the text of a talk by the geometer H. S. M. Coxeter, who introduced it — tongue in cheek — as ‘a piece of mathematical gossip’.[16][16] · Coxeter, H. S. M. (1971)From the original · English“A more recent piece of mathematical gossip concerns the sequence of positive integers x₁, x₂, …, where x₁ is given and x_r+1 = ½x_r or 3x_r + 1 according as x_r is even or odd.” The following year, Martin Gardner brought it to his Scientific American column, and the problem stopped being a corridor secret and became a classic of popular mathematics.[17]

‘Mathematics is not yet ready for such problems’

That the problem is easy to state does not mean it is easy to tackle; rather the opposite. The phrase that best sums it up is usually attributed to Paul Erdős, one of the most prolific mathematicians of the twentieth century, who, faced with its difficulty, declared[12][12] · Lagarias, J. C. (1985)From the original · English“Paul Erdős commented concerning the intractability of the 3x+1 problem: ‘Mathematics is not yet ready for such problems.’” that mathematics was not yet ready for such problems. A variant also circulates — ‘not yet ripe for such questions’ — recorded by Richard Guy and giving the title to his famous article, Don’t try to solve these problems![3]

And yet the problem has tempted many. So much so that prizes have been offered for solving it: Coxeter promised \(\$50\) for a proof and \(\$100\) for a counterexample;[16] Erdős put up \(\$500\); and Bryan Thwaites went so far as to offer \(\pounds1{,}000\).[12] No one has collected.

It is worth getting this right, because it is the key to its fame: the Collatz conjecture is not an easy puzzle nobody has bothered to solve. It is a wall that the mathematical elite itself recognises as such. The hard part is not the rule — we have seen it fits on a single line — but what it hides underneath.

A question that survived

Almost a century after that jotting, the question still stands, unscathed. It has changed names half a dozen times and has withstood Erdős, the computers and everyone who tried.

But there is something left to tell, and it is what truly makes it formidable: Collatz himself saw it. It is not one problem but two — does it always come down to 1? are there other cycles? — and that is the story of the next note.

Related note: The Collatz Ring — the same problem, told through Tolkien.

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Bibliography

Each source links to its original document where accessible. We provide our own translation where the licence allows (CC, public domain); for copyrighted sources (©) we show only the translated fragments you see in the text, under the right of quotation.

• [1]Collatz, L. (1986). On the motivation and origin of the (3n+1)-problem. In J. C. Lagarias (Ed.), The Ultimate Challenge: The 3x+1 Problem (pp. 241–247). AMS, 2010. (Trans. by Mark Conger.)Access ↗Paywalled (AMS book) · English · fragments in the text (©)
• [2]Bařina, D. (2025). Improved verification limit for the convergence of the Collatz conjecture. The Journal of Supercomputing.Project data ↗Technical article · no translation
• [3]Guy, R. K. (1983). Don't try to solve these problems! Amer. Math. Monthly, 90(1), 35–41.Access ↗Paywalled (reprinted in The Ultimate Challenge, AMS) · English · fragments in the text (©)
• [4]Biener, K. (1993). Bedeutender Promovend unserer Universität: Lothar Collatz. RZ-Mitteilungen 5, HU Berlin, 32–33.Original (PDF) ↗German · our own translation of the cited fragments (©)
• [5]Collatz, L. (1942). Einschließungssatz für die charakteristischen Zahlen von Matrizen. Math. Z. 48, 221–226.Access: GDZ / EuDML (Math. Z. 48) · German · technical article
• [6]Collatz, L., & Sinogowitz, U. (1957). Spektren endlicher Grafen. Abh. Math. Sem. Univ. Hamburg 21, 63–77.Access: Springer · German · technical article
• [7]Cvetković, D., Doob, M., & Sachs, H. (1995). Spectra of Graphs. Theory and Applications (3.ª ed.). Johann Ambrosius Barth.Access: publisher / library · English · reference book
• [8]Althöfer, I. (2020). Lothar Collatz zwischen 1933 und 1950 – Eine Teilbiographie. 3-Hirn-Verlag.Original (pp. 1–46) ↗German · fragments in the text (©)
• [9]Mathematics Genealogy Project, id 20676 (Lothar Collatz).Original ↗Data record · no translation
• [10]Segal, S. L. (2003). Mathematicians under the Nazis. Princeton University Press.Access: Princeton UP / library · English · fragments in the text (©)
• [11]O'Connor, J. J., & Robertson, E. F. (2006). Lothar Collatz. MacTutor, Univ. of St Andrews.Original ↗English · fragments in the text (©)
• [12]Lagarias, J. C. (1985). The 3x+1 problem and its generalizations. Amer. Math. Monthly, 92(1), 3–23.Original ↗Open access (author's site) · English · fragments in the text (©)
• [13]Thwaites, B. (1985). My conjecture. Bull. Inst. Math. Appl., 21, 35–41.Access: library · English
• [14]Hayes, B. (1984). Computer recreations: The ups and downs of hailstone numbers. Scientific American, 250(1), 10–16.Access: Scientific American (paywalled) · English · fragments in the text (©)
• [15]Lagarias, J. C. (2010). The 3x+1 problem: an overview. En The Ultimate Challenge, AMS, 3–29.arXiv ↗Open access (arXiv) · English · fragments in the text (©)
• [16]Coxeter, H. S. M. (1971). Cyclic sequences and frieze patterns. Vinculum, 8, 4–7. [Reprinted in The Ultimate Challenge, AMS 2010, 211–218.]Access ↗Paywalled (AMS book) · English · fragments in the text (©)
• [17]Gardner, M. (1972). Mathematical Games. Scientific American, 226(6), 114–118.Access: Scientific American (paywalled) · English