MHB Can you solve this equation involving the sum of digits of a positive integer?

[anemone]

If $P(x)$ represents the sum of the digits of a positive integer $x$. Solve $x+P(x)+P(P(x))=1993$.

 

[kaliprasad]

If $P(x)$ represents the sum of the digits of a positive integer $x$. Solve $x+P(x)+P(P(x))=1993$.

Spoiler

x mod 9 = P(x) mod 9 = P(P(x)) mod 9 = say k

so x + P(x) + P(P(x)) = 3 k mod 9

but x + P(x) + P(P(x)) mod 9 = 4 mod 9

as 3 is a factor of 9 there is no y such that 3y = 4 mod 9

so no solution