My first step is to just to try to understand the problem by considering specific values of $n$. In the simplest case, $n=1$, and then $S(n)=1$ and $S(S(n))=1$, and the sum of these values is 3, which does not equal 1993 (as expected). Similarly for all single-digit numbers, we would have $2+2+2,\enspace3+3+3,\ldots,\enspace9+9+9$.
Then it occurred to me that $n$ cannot equal or exceed $1993$, because then $n$ by itself would equal or exceed the full sum, and the sum of the digits of $n$ cannot be zero or negative. On the other hand it seems that $n$ would need to be relatively close to $1993$ because $S(n)$ and $S(S(n))$ are relatively small in comparison to $n$. For example when $n=999\enspace S(n)=27$ and $S(S(n))=9$
I've been playing around with numbers from $1949$ and up, but can't quite get a sum of $1993$. I'm sure there's a theory behind this, but it's beyond me right now. Any thoughts?