7.2 Find all solutions to the numerical puzzle

$$
\mathrm{AX}+\mathrm{YX}=\mathrm{YPA}
$$

(different letters correspond to different digits, and the same letters correspond to the same digits).
Solution. It is clear that $\mathrm{Y}=1$ and A is an even digit. A cannot be less than 8, otherwise the result will not be a three-digit number. Therefore, $\mathrm{A}=8$.

$+\frac{$| $8 \mathrm{X}$ |
| :--- |
| $1 \mathrm{X}$ |}{$+1 \mathrm{P} 8$}

We see that when adding, there must be a carry of 1 from the units place to the tens place. Therefore, $\mathrm{X}=9$. We get the answer: the only solution is $89+19=108$.