Departure question
Find order without counting everythingPort 13 of 16Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
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.
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.
Take any number, halve it if even, 3n+1 if odd — does it always reach 1?
Q.Start from 6
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
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.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearErdős said, “Mathematics may not be ready for such problems,” and offered $500 for a solution.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearTerence 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.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearA demonstration of how difficult program termination can be to prove, a teaching puzzle and programming exercise, and a candidate lens on undecidability.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Collatz Conjecture
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
