Solve (1)

By casting out nines , the remainder of a number divided by $9$ is the same as the remainder of a number's digital sum divided by $9$ . In other words, $x \equiv S(x) \equiv S(S(x)) \pmod{9}$ , which means that $x + S(x) + S(S(x)) \pmod{9}$ must be divisible by $3$ .

However, $1993 \pmod{9} = 4$ , but $4$ is not divisible by $3$ , so there is no solution to $x + S(x) + S(S(x)) = 1993$ .