Collatz conjecture (3n+1)

The Collatz process is brutally simple: if n is even, halve it; if odd, triple it and add one. The conjecture — open since 1937 and called "hopeless" by Erdős — says every positive start eventually falls to 1, then loops 1→4→2→1. Despite being checked for every number up to astronomically large bounds, no proof exists. The orbits are wildly irregular: 27 climbs above 9000 and takes 111 steps before collapsing. We just trace the trajectory cell by cell so you can watch it bounce. Even steps shrink it; odd steps spike it upward.