Here's a conjecture that relates a process to the original number selected:

Conjecture: Given any non-negative integer $n$, define a process as follows:

1. If $n$ is even, divide it by 2 to get a new number.
2. If $n$ is odd, multiply it by 3 and add 1 to get a new number.
3. Repeat the process with the new number.

The conjecture is that no matter what initial positive integer $n$ is chosen, the process will eventually reach the number 1.

This conjecture is known as the Collatz Conjecture (also called the 3n + 1 conjecture).

Would you like more details or have any questions?

Here are five related questions:

1. What is the significance of the Collatz Conjecture in mathematics?
2. Has the Collatz Conjecture been proven for any specific cases?
3. How does the process behave for different starting numbers?
4. What are some other famous unsolved conjectures in mathematics?
5. Can this process be modified to explore other mathematical patterns?

Tip: When dealing with conjectures, it's often helpful to experiment with small cases to observe patterns and gather insights before attempting general proofs.