Why Collatz WorksExploring the 3n+1 Problem

One amateur's field notes on the 3n+1 problem — playgrounds, explainers, a few small results, and honest notes on what didn't work. No proof is claimed.

Take the Tour

Connections

How to read this site

1. The Puzzle

Pick a number, apply the rules, and watch where it goes. Every positive whole number tested so far ends at 1. Try to find one that doesn't.

2. The Binary Engine

Watch what the two rules do to a number's bits. Halving removes one; 3n+1 adds one or two. Then meet the dropping sets.

3. No Loops?

Why a loop would need 2^E to sit absurdly close to 3^S, what is actually known about cycles, and a navigator for the near misses between powers of 2 and 3.

4. The Hidden Rotation

In log₆ coordinates, Collatz looks like an irrational rotation with a small wobble. An analogy worth playing with.

5. The Countdown

The +1 carry acts like a countdown on the trailing bits. An elementary lemma you can watch run.

6. Finite Fuel

A heuristic picture — finite bits as fuel. Not a theorem, and the page says what is missing.

INFO

This is a work in progress by an amateur. No proof of the Collatz conjecture is claimed.

What This Site Is

I am not a professional mathematician. I have been circling the 3n+1 problem for years because it is the most approachable hard problem I know: you can explain it to a child, and nobody can solve it. This site is where I keep what I find.

It has three kinds of material:

Explainers of known mathematics — dropping sets, why log₂3 keeps appearing, Terras's parity vectors. None of it is mine; all of it is worth understanding.
A few small results of my own — some proved in full, some only in outline, some checked by computer up to a stated bound. I have looked for them in the literature as far as I can; some may be folklore.
Explorations and analogies — Eisenstein lattices, transfer operators, a thermodynamic analogy. Ideas thrown at the wall to see what sticks.

Where to Start

Just curious? Take the tour — seven interactive chapters — or go straight to the playgrounds: Sturmian Bridge, Binary Shortcut, Alpha Sequence.

A mathematician? Read How to read this site first. It says what is and is not claimed, and how to tell a proof from a computation from an analogy, in one page.

Working on Collatz yourself? The one test every idea must pass: it has to fail for −1, because −1 → −2 → −1 loops forever under the same rule.

Prior Work

This exploration grew out of several years of self-published work by an amateur mathematician working in industry. The earlier writings sorted the integers by stopping time under the name dropping sets (the classification itself goes back to Terras, 1976), drew a geometric picture of it, and plotted orbits in base 6. Read more →