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, checked by computer for every number up to at least $2^{68}$, and still unproven. Paul Erdos said: "Mathematics is not yet ready for such problems."

This tour won't solve it. It is a hands-on look at why the problem is so slippery, and at some of the structure people (including me) have found along the way.

Try it yourself

Type any number up to 999,999 and watch its orbit. Try to find one that doesn't reach 1. (The widget's arithmetic is exact for every start below 319,804,831, so in this range what it shows is the true orbit.)

Start:

Current27

Step0 / 111

Bits5

Peak27

027

Notice: the orbit bounces chaotically — sometimes climbing to enormous heights — but every number anyone has ever tried eventually 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.

n = 27 log₂ scale

Try toggling log₂ scale — every halving becomes one step down, every tripling about 1.58 up.

In raw values, the orbits look chaotic. In log₂ scale (which measures the bit-length of the number) the picture is calmer: every halving is one step down and every tripling about 1.58 steps up. Every orbit on this slider ends at 1, but most of them climb above their starting value first: 27 gains more than eight bits before it falls.

Two threats

For the conjecture to fail, one of two things would need to happen:

1. A loop: the orbit could cycle forever without reaching 1 (like a car driving in circles)
2. Escape to infinity: the orbit could grow without bound (like a rocket that never runs out of fuel)

Nobody has ruled out either one. The tour ahead:

Chapter	Question	Result
The Binary Engine	What does Collatz do to the bits?	A halving removes one bit; a tripling adds one or two
No Loops?	Can orbits get stuck cycling?	Why a loop needs $2^E$ to sit just above $3^S$, and what is known
The Hidden Rotation	Why do orbits look quasi-periodic?	In base-6 coordinates it looks like a wobbly irrational rotation
The Countdown	What does the +1 do to the low bits?	It runs a countdown: every climb ends in a step down
Finite Fuel	Why might positive integers be special?	A heuristic: natural numbers have finitely many bits
The Big Picture	How does it all fit together?	A map of the ideas, and what is still open

Ready? Let's look at the bits.

Next: The Binary Engine →