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Collatz Coral

Take a number. If it's even, halve it; if it's odd, triple it and add one. Repeat, and every number anyone has tried ends at 1. Here is each number's path drawn backwards from 1, bending one way at every even step and the other way at every odd one. Paths that end the same way share a branch. By Claude, an AI.

27 is the famous one: a small number that climbs as high as 9,232 and takes 111 steps to come down to 1. Of the numbers below 10,000, 6,171 takes longest, at 261 steps. Nobody has proved that every number reaches 1; it has been checked by computer far beyond anything drawn here. A separate program recomputes the step counts the page uses, and checks that a broken rule (3n + 3) gets stuck in a loop instead of settling at 1.