Free Collatz Conjecture Explorer | 3n+1 Tree & Fractal

How it works

Open this page and watch the geometric tree grow from hailstone sequences.
Choose a starting number, or switch to the complex fractal and drag to pan, scroll to zoom.
Tune the turn angles or iterations, then save a PNG or open the image in the PictorX editor.

Rules simple enough for a child

Start with a positive integer n. If n is even, replace it with n/2. If n is odd, replace it with 3n+1. Repeat. The conjecture claims that every starting value eventually reaches 1. Once you hit 1 you enter the infinite cycle 1 → 4 → 2 → 1. Sequences bounce up and down before they collapse, which is why they are also called hailstone numbers. Start at 6 and the path is 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

Why a proof is still missing

Computers have checked every integer far past 2×10²¹, and each one reaches 1. A general proof does not exist. Paul Erdős said mathematics is not yet ready for this kind of problem. In 2019 Terence Tao proved the conjecture holds for almost all numbers, which is a major advance but not a complete solution. A Japanese company has offered a prize of 120 million yen for a definitive proof.

Two ways to see the maths

The geometric tree treats each even or odd step as a turn on the plane, so a sequence becomes a path and many sequences become a coral. The complex fractal interpolates the same rule to every point z = x + yi with a trigonometric switch that is exactly n/2 on even integers and exactly 3n+1 on odd integers. Colour records how many iterations it takes for |z| to escape.

Frequently asked questions

What is the Collatz Conjecture?

It is the claim that every positive integer reaches 1 if you repeatedly replace an even n with n/2 and an odd n with 3n+1. The rules are simple; a general proof is not.

What does this app show?

Two views of the same rule. The geometric tree turns hailstone sequences into paths on the plane. The complex fractal iterates a smooth Collatz function on the complex plane and colours pixels by escape time.

Why are the sequences called hailstone numbers?

Values often rise and fall many times before they drop to 1, the way hailstones are lifted and dropped in a storm cloud. Start at 6 and you get 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

Has the conjecture been proved?

No. Every checked integer ends at 1, and Terence Tao showed that almost all numbers eventually get very small, but a proof for every starting value is still open.

Does the maths run in my browser?

Yes. The tree, the sequence, and the fractal are computed on your device. Save a PNG or open the canvas in the PictorX editor.