3. A two-digit number has the following property: if it is added to the number with the same digits but in reverse order, the result is a perfect square.

Find all such two-digit numbers.
Solution. If $x$ is the unit digit and $y$ is the tens digit of the desired two-digit number, then according to the condition of the problem, we have

$$
(10 y + x) + (10 x + y) = m^{2}
$$

i.e.

$$
11(x + y) = m^{2}
$$

According to this, 11 is a divisor of $m$; if $m = 11k$, we get

$$
x + y = 11k^{2}
$$

From $1 \leq x \leq 9$ and $1 \leq y \leq 9$, it follows that $2 \leq x + y \leq 18$, which means that

$$
x + y = 11
$$

From (3) it follows that the desired numbers are: $29, 38, 47, 56, 65, 74, 83$ and 92.