Petr said to Pavel: "Write a two-digit natural number that has the property that when you subtract the same two-digit natural number just written in reverse, you get a difference of 63."

Which number could Pavel have written? Determine all possibilities.

(L. Hozová)

Hint. What is the difference between the digits of Pavel's number?
The task can be solved as an alphametic:

$$
\begin{array}{rr}
a & b \\
-b & a \\
\hline 6 & 3
\end{array}
$$

Since the difference is positive, it must be that $a > b$. Additionally, since the difference in the units place is 3, the calculation must involve a carry-over from the tens place. Since the difference in the tens place is 6, it must be that $a - b = 7$. Furthermore, since both numbers are two-digit numbers, it must be that $b > 0$. Altogether, this gives us two possibilities:

$$
\begin{array}{r}
81 \\
-18 \\
\hline 63
\end{array} \quad \begin{array}{r}
92 \\
-29 \\
\hline 63
\end{array}
$$

The number that Pavel could have written was 81 or 92.

Note. A two-digit number written as $\overline{a b}$ can be expressed as $10a + b$. The previous notation is therefore equivalent to the equation

$$
(10a + b) - (10b + a) = 63
$$

which, after rearrangement, leads to $a - b = 7$.