Concept
Take any number, halve it if even, 3n+1 if odd — does it always reach 1?
Understand it in one breath
If an integer is even, halve it; if odd, triple it and add one. Must every positive start eventually reach 1? Known since the 1930s, the conjecture is easy to state but has no general proof. Starting at 27 takes 111 iterations to first reach 1 and peaks at 9,232. Computation over a vast finite range and partial “almost all” results do not prove the infinite claim.
At a glance
Step | Value |
|---|---|
| 0 | 27 |
| 1 | 82 |
| 2 | 41 |
| 3 | 124 |
| 4 | 62 |
| 5 | 31 |
| 6 | 94 |
| 7 | 47 |
| 8 | 142 |
| 9 | 71 |
| 10 | 214 |
| 11 | 107 |
| 12 | 322 |
| 13 | 161 |
| 14 | 484 |
| 15 | 242 |
| 16 | 121 |
| 17 | 364 |
Starting at 27 → the full trajectory reaches 1 after 111 steps. The table shows the first 18 steps; the sequence climbs as high as 9,232.
Key formula
Worked examples
- 1
Q.Start from 6
Key moments
Collatz first poses the problem
German mathematician Lothar Collatz found the problem while still a student. Its statement was so simple that it was sometimes treated as a mathematical joke.
Erdős offers a $500 prize
Erdős said, “Mathematics may not be ready for such problems,” and offered $500 for a solution.
Terence Tao — almost all starting values
Terence Tao proved a major almost-all result showing that nearly every starting value eventually reaches much smaller values. It was a partial advance, not a full proof.
Modern applications
A demonstration of how difficult program termination can be to prove, a teaching puzzle and programming exercise, and a candidate lens on undecidability.
Beyond MathVoyage
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