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 · DiscoverYou are hereMeet 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 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.

Open1937

Collatz Conjecture (3n+1) — A Child Can State It, Yet No One Has Solved It

Start with any positive integer. If even, halve it; if odd, multiply by three and add one. Does every start eventually reach 1? Large computations and strong density results still fall short of a proof for every integer.

Challenge passport

First posed

1937

Time it held mathematicians

89 years open · as of 2026

Starting level

Start now

Test small cases immediately with a drawing, divisors, or colored pencils.

The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.

Jump to your first five minutes

There is no formal prize; the key is distinguishing computation, density theorems, and the universal claim.

Try a number — Collatz sandbox

Enter any positive integer n to redraw its trajectory instantly. Copy a share link to any seed so a friend can open the same view.

Steps

106

Peak

9,232

Even ratio

63%

Result

→ 1

Compare with the last run · Shared state

Same rule, different journeys

Before27Now31

Steps to 1−5

111

106

Peak0

9,232

Odd pushes−2

Longest halving run0

Why do they differ so much?

Odd values grow through 3n+1, while even values shrink by dividing by 2. The previous run had 41 odd pushes and a longest halving run of 5; the current run has 39 and 5. That rhythm of pushes and pullbacks changes the height and length of the journey.

This is an observation from two paths, not a proof that every positive integer reaches 1.

My discovery

Keep this comparison in your exploration trail

Save one observation privately, or share these exact two states with a dynamic preview.

31 surged to 9,232 before finally falling to 1 — 106 steps.

Notable seeds

Problem statement

Define f(n) = n/2 if even and 3n+1 if odd. Does iteration from every positive integer reach 1? Every start below 2^71 had been checked by 2025, but no proof covers all integers.

The story of this puzzle

Starting from 27, the sequence reaches 9,232 before arriving at 1 after 111 steps. Computation had verified every start below 2^71 by 2025, but no finite check controls all integers. In 2019 Terence Tao proved that, in logarithmic density, almost every orbit eventually falls below any chosen function f(n) that tends to infinity. That is a major typical-orbit result, not a proof that almost every orbit reaches 1 and not a statement about each individual start.

Try it yourself

Mini challenge

Try starting from 27. Track the sequence — 111 steps, peaks at 9,232 before reaching 1. Next try 871 (178 steps). Can you write a program to verify all numbers up to 10,000?

Beyond MathVoyage

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