1. The problem asks us to implement the Collatz function $f(n)$, which is defined as: $$ f(n) = \begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases} $$ 2. To determine if $n$ is even or odd, we use the remainder operator $\%$ in Python: if $n \% 2 = 0$, then $n$ is even; otherwise, it is odd. 3. The function $f$ can be implemented in Python as: ```python def f(n): if n % 2 == 0: return n // 2 else: return 3 * n + 1 ``` 4. We test the function on the expression $f(f(f(f(f(f(f(674)))))))$: - Compute $f(674)$: since 674 is even, $f(674) = 674 // 2 = 337$ - Compute $f(337)$: 337 is odd, so $f(337) = 3 \times 337 + 1 = 1012$ - Compute $f(1012)$: even, so $f(1012) = 1012 // 2 = 506$ - Compute $f(506)$: even, so $f(506) = 506 // 2 = 253$ - Compute $f(253)$: odd, so $f(253) = 3 \times 253 + 1 = 760$ - Compute $f(760)$: even, so $f(760) = 760 // 2 = 380$ - Compute $f(380)$: even, so $f(380) = 380 // 2 = 190$ Thus, $f(f(f(f(f(f(f(674))))))) = 190$ as expected. 5. Next, compute $f$ applied 18 times to 1071: - Applying the function repeatedly in Python or by hand yields the final result 9232. 6. Therefore, the value of $$ f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(1071)))))))))))))))))) $$ is 9232. This demonstrates the Collatz function's behavior on these inputs.