Annie's statement relates to a famous unsolved problem in mathematics known as the Collatz Conjecture, or sometimes the 3n + 1 problem.

The process she describes is the basis of the conjecture, which proposes the following:

1. Take any positive integer $n$.
2. If $n$ is even, divide it by 2.
3. If $n$ is odd, multiply it by 3 and add 1.
4. Repeat this process with the resulting number.

The conjecture suggests that no matter which positive integer you start with, eventually, this sequence will always reach the number 1, entering a loop of 1 → 4 → 2 → 1.

Is Annie Correct?

As of now, mathematicians have not proven whether this conjecture is universally true or false. It has been verified for very large numbers (into the trillions), and no counterexamples have been found. This means that for every starting number that has been tested, the process does indeed eventually reach 1. However, without a formal proof, it remains possible (though unlikely) that there could be a number that does not lead to 1 or enters a different loop.

Summary

• Annie is mostly correct based on all known tests, but since there is no proof, we cannot confirm this is true for every possible number.
• Why not proven? The behavior of the sequence can vary significantly depending on the starting number, making it extremely complex to prove universally.

Would you like more details on the Collatz Conjecture or how mathematicians attempt to approach it?

Here are five related questions to explore:

1. What is a conjecture, and how does it differ from a proven theorem?
2. How has the Collatz Conjecture been tested computationally?
3. What is the significance of the Collatz sequence repeating the cycle 1 → 4 → 2 → 1?
4. Why do mathematicians believe the conjecture is likely true despite lacking proof?
5. How do recursive and iterative methods help in analyzing sequences like the Collatz sequence?

Tip: Studying the behavior of numbers in different sequences can be an excellent way to develop intuition in number theory and mathematical patterns!