2014/061) Solve for x if p(x) is sum of digits of x and x +p(x) + p(p(x))) = 1993

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