AHSME 1996 Problem 12

A function $f$ from the integers to the integers is defined as follows: $$f(n) =\begin{cases}n+3 &\text{if n is odd}\\ \ n/2 &\text{if n is even}\end{cases}$$ Suppose $k$ is odd and $f(f(f(k))) = 27$. What is the sum of the digits of $k$? $\textbf{(A)}\ 3 \textbf{(B)}\ 6 \textbf{(C)}\ 9 \textbf{(D)}\ 12 \textbf{(E)}\ 15$