Collatz Tree
Apply the rule: if n is even divide by 2, if odd multiply by 3 and add 1. Repeat. The Collatz conjecture says every positive integer eventually reaches 1 — proven for all numbers tested up to ~2⁶⁸ but unproven in general.
T(n) = n/2 if n even | T(n) = 3n+1 if n odd | Conjecture: ∀n≥1, ∃k: T^k(n) = 1
Lothar Collatz proposed this conjecture in 1937. It is one of the most famous unsolved problems in mathematics — simple to state, trivially checkable for any specific number, yet resistant to proof. Paul Erdős said "Mathematics is not yet ready for such problems."
The sequences have surprisingly irregular behavior. The number 27 takes 111 steps to reach 1 and reaches a peak of 9232 along the way, despite being a small number. Record-holders (numbers that take more steps than any smaller number) are rare and irregular.
The "tree" view shows how numbers connect: each number n connects to 2n (which will halve to n) and, if (n-1)/3 is an integer, to (n-1)/3 (which will triple-plus-one to n). This forms an infinite tree rooted at 1 — assuming the conjecture is true.