1996 AHSME Problem 12

A function f from the integers to the integers is defined as follows:

f(x) =\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\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.