The Collatz Conjecture, also known as the 3n + 1 problem or the hailstone sequence, is one of the most famous unsolved problems in mathematics. Proposed by German mathematician Lothar Collatz in 1937, the conjecture revolves around a deceptively simple iterative sequence.

The conjecture is based on the following rules:

Start with any positive integer n.

If n is even, divide it by 2.

If n is odd, multiply it by 3 and add 1.

Repeat the process indefinitely.

The conjecture posits that, regardless of the starting integer, the sequence will always eventually reach the cycle 4→2→14→2→1, after which it will continue in a repeating loop of 4→2→14→2→1.

Despite its simplicity, the Collatz Conjecture remains unsolved. While computer simulations have verified the conjecture for vast ranges of starting integers, no mathematical proof has been found to confirm its validity for all positive integers. The conjecture’s apparent simplicity belies its complexity, as the behavior of the sequence becomes increasingly unpredictable with larger numbers.

The Collatz Conjecture has captivated mathematicians for decades due to its accessibility and the tantalizing possibility of a simple solution. Its persistence as an unsolved problem highlights the enigmatic nature of mathematics and the allure of unresolved mathematical mysteries.