Pick any number. If even, halve it. If odd, triple and add one. Does it always reach one?
Play It
Characterization
The Collatz conjecture is the simplest unsolved problem in mathematics. Pick any positive integer. If it is even, divide it by two. If it is odd, multiply by three and add one. Repeat. The conjecture, posed by Lothar Collatz in 1937, states that this process always eventually reaches 1, regardless of the starting number. The sequence for 27 climbs to 9,232 before descending; the sequence for 7 reaches 1 in sixteen steps. Every starting number tested — up to approximately 2.95 × 10²⁰ — reaches 1. No one can prove that all numbers do. Paul Erdős said of the conjecture: "Mathematics may not be ready for such problems." The difficulty is structural: the conjecture sits at the intersection of number theory, dynamical systems, and computability theory, and the tools of none of these fields have proven sufficient. In 2019, Terence Tao achieved the strongest partial result to date, showing that the Collatz conjecture is true for "almost all" positive integers in the sense of logarithmic density — but "almost all" is not all, and the conjecture remains open. The Academy hosts the Collatz conjecture in the Mind School because it is the purest example of a game anyone can play — a child can iterate the rules — that conceals a mystery no one can resolve. The rules are the game. The proof is the wonder.
Lineage
Lothar Collatz, posed in 1937; first published treatment by Helmut Hasse in the 1950s. The problem is also known as the 3n+1 conjecture, the Syracuse problem, the Ulam conjecture, and the hailstone sequence. Paul Erdős's quoted remark appears in Richard Guy, Unsolved Problems in Number Theory (Springer, 3rd ed., 2004). Jeffrey Lagarias, "The 3x + 1 Problem and Its Generalizations," American Mathematical Monthly 92(1), 1985, provides the canonical survey. Terence Tao, "Almost All Orbits of the Collatz Map Attain Almost Bounded Values," arXiv:1909.03562, 2019 (Forum of Mathematics, Pi, 2022). Computational verification: David Bařina, 2020–2021.
Quests
The Magus
Design and explore a variant of the Collatz process — change the multiplier from 3 to 5 or 7, alter the additive constant, extend to negative integers, or work in a different number base. Compute trajectories for at least twenty starting values. Record any cycles, divergent behaviour, or unexpected regularities, and compare the variant's dynamics to the classical 3n + 1 map.
The Adventurer
Compute the Collatz sequence for every integer from 1 to 100 — by hand, by spreadsheet, or by a program you write yourself. For each, record the number of steps to reach 1 (the stopping time) and the maximum value attained (the peak). Identify the starting value with the longest stopping time and the one with the highest peak. Attend to the experience: at what point does the pattern feel predictable, and where does it surprise?
The Sage
Explain the Collatz conjecture to someone who knows basic arithmetic but no advanced mathematics. State the rule; walk through the sequence for 27 (a famously long trajectory); cite Collatz's 1937 formulation, Erdős's remark about the readiness of mathematics, and Tao's 2019 partial result on almost all orbits. Explain what Tao proved and why the full conjecture remains open.