Exercise 1. For any natural number $n$, we denote by $S(n)$ the sum of the digits of the decimal representation of $n$. We say that a natural number $n$ is pretty if $S(n)=S\left(n^{2}\right)$. Determine all possible values of $S(n)$ for the pretty integers $n$.
Solution to Exercise 1 Modulo $9, S(n)$ satisfies the equation $x^{2}=x$. By testing $x=$ $0,1, \ldots, 8$, we verify that the only solutions (modulo 9) are 0 and 1, so $S(n)$ must necessarily be congruent to 0 or 1 modulo 9. Conversely, we verify that if $n=10^{k}-1=99 \ldots 9$ ( $k$ times) then $n^{2}=99 \cdots 9800 \cdots 01$ (with $k-1$ digits 9 and $k-1$ digits 0), so $S(n)=S\left(n^{2}\right)=9 k$. Similarly, if $n=2 \times 10^{k}-1=199 \cdots 9$ then $n^{2}=399 \cdots 9600 \cdots 01$ so $S(n)=S\left(n^{2}\right)=9 k+1$.

We conclude that the possible values of $S(n)$ are the natural numbers congruent to 0 or 1 modulo 9.