1. The problem described is a famous unsolved problem in mathematics known as the Collatz conjecture or 3n+1 problem. 2. The rule is: for a given starting positive integer $n$, if $n$ is even, divide it by 2; if $n$ is odd, replace it with $3n+1$. Repeat this process with the new number. 3. The conjecture states that no matter what positive integer you start with, the sequence will eventually reach the number 1. 4. To test this, choose any positive integer $n$. For example, let $n = 7$. 5. Starting with $7$: - 7 is odd, so compute $3 \times 7 + 1 = 22$ - 22 is even, divide by 2 to get 11 - 11 is odd, $3 \times 11 + 1 = 34$ - 34 is even, divide by 2 to get 17 - 17 is odd, $3 \times 17 + 1 = 52$ - 52 is even, divide by 2 to get 26 - 26 is even, divide by 2 to get 13 - 13 is odd, $3 \times 13 + 1 = 40$ - 40 is even, divide by 2 to get 20 - 20 is even, divide by 2 to get 10 - 10 is even, divide by 2 to get 5 - 5 is odd, $3 \times 5 + 1 = 16$ - 16 is even, divide by 2 to get 8 - 8 is even, divide by 2 to get 4 - 4 is even, divide by 2 to get 2 - 2 is even, divide by 2 to get 1 6. It took 16 steps to reach 1 starting from 7. 7. Although it has been tested for many numbers and always reaches 1, a proof that the sequence will reach 1 for all positive integers has not yet been found. Final Answer: For $n=7$, it takes 16 steps to reach 1.