A rule a child can follow — halve it, or triple it and add one — has trapped every number ever tested, and no one can prove it always will

The problem

Take any number: 6 goes 3, 10, 5, 16, 8, 4, 2, 1. Every starting value ever tried — now every number up to roughly 2.36 sextillion — eventually crashes down to 1, yet the sequences bounce so chaotically (27 climbs above 9,000 before falling) that nobody can prove one doesn't shoot off to infinity or loop forever. The problem is infamous for luring mathematicians in with its childlike simplicity and then swallowing careers; Paul Erdős warned that 'mathematics may not be ready for such problems.' Even Terence Tao's landmark 2019 advance — the strongest in decades — proves only that 'almost all' numbers behave, leaving the full conjecture untouched.

Why it matters

The conjecture is a stark measure of how little we understand the interaction between multiplication and addition on whole numbers — a proof would require genuinely new mathematics, with likely spillover into dynamical systems and number theory. It is also a benchmark question about which simple iterative processes are predictable at all, a theme that echoes through computability theory.

Progress so far

• 1937Lothar Collatz poses the problem
• 2019Terence Tao proves almost all starting values eventually fall below any slowly growing threshold, the strongest result in decades
• 2021Japanese firm Bakuage offers a 120 million yen prize for a resolution
• 2025David Barina's distributed computation verifies the conjecture for all starting values up to 2⁷¹ (about 2.36 sextillion)

References

1. MathPrize (Bakuage) — official 120 million JPY prize page with terms
2. Quanta Magazine — coverage of Tao's 2019 'almost all orbits' breakthrough
3. Wikipedia — history, verification records, and partial results