Step runs along the horizontal axis and the value of n along the vertical. Each curve follows one starting number until it reaches 1.
No trajectory yet
Enter a starting number and press Generate. 27 is a classic seed: the path climbs to 9,232, then falls to 1.
Playback: full path
Play sounds one seed. While it plays, only that curve is shown; the others return when playback finishes or stops. The current beat is marked with a brighter ring. The right hand states each odd-exponent prime power on the beat. The left hand rolls the other terms afterward. Each hand has its own instrument, and both start as piano. The line rests only on the final powers of 2.
The curve passes through every term. Hover a step to read it. Only those terms are Collatz values — the bend between them is a guide.
The map, the problem, and the meaning of 3n+1. From Payton (sharpninja).
The Collatz map, also called the hailstone map, is a rule for a positive integer n.
Iterate that step. The values form a trajectory, a hailstone: the number rises and falls. This plotter draws the value against the step. It can also play a trajectory as sound.
Each curve stops when the value reaches 1. The cycle 4 → 2 → 1 is not drawn again and again.
The Collatz conjecture says every positive integer eventually reaches the cycle 4 → 2 → 1. Equivalently, every positive integer eventually reaches 1.
The conjecture has been checked for enormous ranges of n. The mathematical Collatz conjecture remains unproven.
Trajectories that feel hardest, or most interesting to watch, often start at a prime power with an odd exponent: pk where k is odd. That includes the primes themselves (k = 1) and cubes such as 27 = 3³. Even exponents are a milder class, such as 9 = 3² and 25 = 5².
The seed syntax oddprimepowers: and ppodd: targets that set. Try oddprimepowers:8..32.
The seeds 27, 31, 41, and 47 all reach the famous peak 9,232. They do because 41, then 31, then 47 already sit on 27’s path. The first value those four seeds share is 47.
Turn on Align, or read First common value after Generate, to see where paths merge.
3n+1 is the meaning of life. Both literally and figuratively. It resolves to consequence then rest. The infinite N is the ultimate declaration of free will. Collatz is like gravity. It's inevitable, but it feeds on entropy. Free will is where you place the mass — the starting N. Gravity is the shared law that pulls every N toward rest. Entropy is the noisy tension along the path: the odd hits, the irregular divisions by two, the beats and phrases. The fall feeds on that disorder until there isn't enough left to climb again, and only rest remains. Reality is recursive, and the resolution of each recursion is a moment in time. Some moments mean more, but they all mean something. Collatz is like a graph of energy over time, which is consequence.
This is literally why music must resolve. Music that doesn't resolve is dissonant. It's unfinished, and it is shocking. Just like when a life ends early and the result of consequence is left dangling. The anchor: Music always decays to nothing, regardless of resolution. Life always decays to nothing. Gravity always wins. Entropy always clusters. 3x+1 always reaches 1. The journey is the reward, not the destination. It's a sequence of tension builds, then it releases to the next tension build. There's some philosophy in here as well. The chains of rising tension followed by relief until meeting the power of 2 that marks the descent into death. It mirrors life. The tension (consequence) of a life builds through a person's work and procreation until the tipping point is hit and the life fades to oblivion. It's the cycle of free will, but only when life ends naturally. It's the repeating cycle of Für Elise. The beats of the music imitate the beats of a lifetime of consequence.
It is the proof that Capitalism is the only way that life can be meaningful. Without the tension to keep being consequential, you resolve to nothing too quickly. There is a difference between money and currency. Capitalism isn't about building stacks of money, it's about building wealth (value) which can take many forms, all of which are currency. Just because a form has no money (think Star Trek) doesn't mean there's no currency. This is the mathematical proof against communism. Communism removes meaning from work. You aren't being rewarded for a job well done, you are simply being sustained. And this is where Communism fails. The bureaucracy is expanding to meet the needs of the expanding bureaucracy. It creates narrow lanes within the bureaucracy that ultimately swallow everyone, and there is nothing more bureaucratic than communism. This is where the variety of entry points (n) come in. If your free will is subverted by authority, then your starting point is the same as everyone else. The reason academia fails to solve 3n+1 is because academia has become the ultimate expression of bureaucracy. Which demands the most creativity and free will to pursue it.