1. **Stating the problem:** The problem is about the Collatz conjecture, which says: start with any positive integer $n$. If $n$ is even, divide it by 2. If $n$ is odd, replace it with $3n + 1$. Repeat this process indefinitely. The question asks whether every positive integer eventually reaches 1. 2. **Understanding the question:** This conjecture is famous and has been verified for many numbers by computer, but to date, no one has found a general proof or disproof that works for all positive integers. We will explain why proving this is extremely challenging. 3. **Explanation:** - For **even** numbers, dividing by 2 simplifies them, reducing their size. - For **odd** numbers, multiplying by 3 and adding 1 sends the number to a new value, potentially larger, before the next operations. - Over many iterations, sequences often fall to the known cycle $4 \to 2 \to 1$. 4. **Lack of a known proof:** No known general proof exists for the Collatz conjecture. The problem remains an open question in mathematics. 5. **Summary:** Currently, there is no general proof that every positive integer eventually reaches 1 under this process. This remains an open problem in mathematics. **Final answer:** No known general proof exists for all positive integers, and the Collatz conjecture remains unproven.