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Collatz Conjecture (3n + 1) Calculator

Run the Collatz "3n + 1" rule from any starting number and see how many steps it takes to reach 1, and how high it climbs along the way.

Steps to reach 1
111
Peak value reached
9,232
Sequence
27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → ... → 8 → 4 → 2 → 1

How it works

The Collatz rule takes a positive whole number and repeatedly applies one of two steps: if it's even, divide it by 2; if it's odd, multiply it by 3 and add 1. Repeating this on the result, over and over, is conjectured to always eventually reach 1 — no matter which number you start from. This calculator runs that rule, counts how many steps it takes, and tracks the highest value the sequence reaches before it comes back down (these sequences are sometimes called "hailstone numbers" because they bounce up and down erratically, like a hailstone being tossed around a storm cloud, before eventually falling to the ground).

The Collatz conjecture itself is unproven for all whole numbers in general — nobody has shown it must be true, despite it having been checked by computer for enormous ranges of starting numbers far beyond what's useful to type into a calculator. Every starting value this calculator accepts reaches 1 comfortably within its step limit.

FAQ

Has the Collatz conjecture been proven?

No. It remains an open problem in mathematics — every starting number ever tested by computer (into the enormous, far beyond what any calculator input needs) eventually reaches 1, but no one has proven it must happen for every positive integer, and no counterexample has ever been found either.

Why can the sequence climb so much higher than the starting number?

Each odd step roughly triples the number (3n + 1) before any halving happens, so a run of consecutive odd steps can push the value well above where it started; the even steps (dividing by 2) are what eventually pull it back down. Starting at 27, for example, climbs as high as 9,232 before falling all the way to 1.

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