One great problem, three kinds of immersion

From a story to your own conjecture

Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.

1 · DiscoverMeet the questionLearn when it appeared and why it still holds people’s attention.
2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
3 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.

Week 1 · ClassicalOpen

Collatz Conjecture — The Simplest Unsolved Problem

Simple rule · open problem· Posed 1937A006577

Problem

For any positive integer $n$ , iterate $f(n) = n/2$ if $n$ is even, $f(n) = 3n+1$ if $n$ is odd. Does the sequence always reach $1$ ?

Why it matters

The rule is grasped in a minute by a child. Yet $n=27$ rockets up to $9{,}232$ and takes 111 steps to descend to 1. Erdős said "Mathematics is not yet ready for such problems" and offered $100. A deterministic-system's chaos in its purest form.

Progress so far

David Barina's public verification project confirmed in January 2025 that every $n<2^{71}$ reaches 1. This remains a finite computation. Tao's 2019 theorem says that for every function $f(N)\to\infty$ , the orbit minimum is at most $f(N)$ for almost all $N$ in logarithmic density. Those quantifiers and the density notion matter; the full conjecture remains open.

27 and 31 start apart—so why do both pass through 9,232?

Instead of dressing an empty workshop up as community activity, the editors have left one reproducible Collatz comparison you can test in a minute.

Compare 27 and 31 Start from this observation

1 · Read one observation
Changing the starting value from 27 to 31 moved the peak from 9,232 to 9,232.
This is an observation from two cases, not a general law or proof.
2 · Touch the same state
Inspect the trajectories of 27 and 31 side by side, then change one starting number to see whether the observation survives.
3 · Leave my one-line conjecture
Only the calculated observation and reproducible URL are carried over. Your conjecture stays blank until you write it.

The larger question in this workshop: Will every starting number eventually reach 1?

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

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Related concepts

Collatz Conjecture