3. (30 points) In math class, each of the seven dwarfs needs to find a two-digit number such that when 18 is added to it, the result is the number with its digits reversed. Can all the numbers found by the dwarfs be different?
Answer: Yes, they can.

Solution: We will write two-digit numbers in the form $\overline{x y}$, where $x$ is the number of tens, and $y$ is the number of units.

Let's find all valid numbers. According to the condition $10 x+y+18=10 y+x$. Transforming this:

$$
10 x+y+18=10 y+x \Leftrightarrow 18=9 y-9 x \Leftrightarrow y=x+2
$$

Therefore, the numbers that fit are $13,24,35,46,57,68,79$. There are as many of these numbers as there are gnomes. Thus, the gnomes could have found different numbers.