Find the number of all 3 digit numbers $n$ such that $S(S(n))=2$ For any natural number $n$ ,let $S(n)$ denote the sum of the digits of $n$.Find the number of all 3 digit numbers $n$ such that $S(S(n))=2$ 

Write $n=abc=100a+10b+c.$ Then, $S(n)=a+b+c.$ Note that $S(n)$ has two digits. Then,
if $a+b+c<10$ then $a+b+c=2$ (since $S(S(n))=a+b+c);$
if $10\le a+b+c<20$ then $a+b+c=11$ (since $S(S(n))=a+b+c-9);$
if $20\le a+b+c$ then $a+b+c=20$ (since $S(S(n))=a+b+c-18).$
