In the sum below, the same letters represent the same digits and different letters represent different digits.

$$
\begin{array}{ccc} 
& & X \\
+ & & X \\
& Y & Y \\
\hline Z & Z & Z
\end{array}
$$

What digit does the letter $X$ represent?

#
Solution

Initially note that $X+X+Z \leq 2 \cdot 9+8=26$. Therefore, at most 2 units will be added to the tens place. Since $Y \neq Z$, when we add 1 or 2 units to $Y$, we should obtain the two-digit number $\overline{Z Z}$. As $Y+2 \leq 9+2=11$, the only possibility is to have $\overline{Z Z}=11$, thus, $Y=9$. For the sum $2 X+Y$ to end in 1 and have $X \neq 1$, we must have $X=6$. Indeed, these values satisfy the sum:

$$
\begin{array}{r} 
\\
6 \\
+\quad 6 \\
\\
99 \\
\hline 119
\end{array}
$$