Collatz Conjecture Trajectory Explorer

Starting Range

First number n 27

Batch size 24

Playback speed 1.0x

Range

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Longest chain

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Peak value

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Avg. chain length

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How it works

Pick any positive integer n. If it's even, halve it; if it's odd, triple it and add one. Repeat. The Collatz conjecture claims that no matter which n you start from, this "hailstone" sequence always eventually reaches 1 — it has been verified by computer search for every starting value up to enormous bounds (well into the quintillions) without a single counterexample, yet no one has ever proven it must hold for all integers.

a(0) = n
a(k+1) = a(k) / 2         if a(k) is even
a(k+1) = 3·a(k) + 1       if a(k) is odd
conjecture: a(k) = 1 for some finite k, for every n ≥ 1

First number n — the smallest starting value in the batch; each lane to its right starts one integer higher.
Batch size — how many consecutive starting numbers race side by side, so you can compare chain lengths at a glance.
Playback speed — how fast the bright marker travels along each chain; height on the graph is log₂(value), so towering spikes are visually compressed but still huge in real terms.
Auto-advance — slides the starting range forward continuously so you can browse thousands of hailstone sequences without touching the slider.

Notice how "nice" numbers like powers of two drop straight down (log₂ height falls in a single steady line), while numbers like 27 or 703 spike far above their starting height before collapsing — the peak value for n = 27 is 9,232, over 340 times the starting number, after 111 steps.

3n+1 step (odd) n/2 step (even) current position

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