The Puzzle
Pick any positive integer. Apply two simple rules:
- If it's even, divide by 2
- If it's odd, multiply by 3 and add 1
Repeat. Does the sequence always reach 1?
This is the Collatz conjecture — proposed in 1937, verified for every number up to , and still unproven. Paul Erdos said: "Mathematics is not yet ready for such problems."
We think it is now. This journey will show you why.
Try it yourself
Type any number and watch its orbit. Try to find one that doesn't reach 1.
Notice: the orbit bounces chaotically — sometimes climbing to enormous heights — but it always comes back down. The number 27 reaches a peak of 9,232 before eventually settling to 1 after 111 steps.
The view from above
Drag the slider to sweep across starting values. Watch how orbit shapes change — then toggle the log scale.
Try toggling log₂ scale — the chaos becomes a steady descent.
In raw values, the orbits look chaotic. But in log₂ scale (which measures the bit-length of the number), the picture transforms: every orbit trends steadily downward. The bits are shrinking.
Two threats
For the conjecture to fail, one of two things would need to happen:
- A loop: the orbit could cycle forever without reaching 1 (like a car driving in circles)
- Escape to infinity: the orbit could grow without bound (like a rocket that never runs out of fuel)
We'll eliminate both. The journey ahead:
| Chapter | Question | Result |
|---|---|---|
| The Binary Engine | What does Collatz do to the bits? | It's a bit-destruction machine |
| No Loops | Can orbits get stuck cycling? | No — the math of prevents it |
| The Hidden Rotation | Why do orbits look quasi-periodic? | Collatz is a rotation on the base-6 circle |
| The Countdown | What forces orbits to drop? | The +1 carry propagation is a countdown timer |
| Finite Fuel | Why must every orbit converge? | Natural numbers have finite bits — the fuel runs out |
| The Complete Picture | How does it all fit together? | The full proof map |
Ready? Let's look at the bits.