The cleaner was cleaning the stairs in a skyscraper. To make her work go more smoothly, she counted the cleaned steps. When she had cleaned exactly half of the steps, she took a break. After a while, she got back to work and wanted to continue counting the steps. However, when she tried to recall the number of steps already cleaned, she made a mistake. She "read" the correct three-digit number backwards, which resulted in a lower number, and continued counting from this number. After cleaning all the steps, she reached the number 746. How many steps could she have actually cleaned?

(L. Šimünek)
Let $100a + 10b + c$ be the number of steps the cleaner washed before the break. From the problem, it follows that

$$
100a + 10b + c + 100c + 10b + a = 746,
$$

i.e.,

$$
101(a + c) + 20b = 746.
$$

The sum $a + c$ is therefore even, so $a + c = 6$. From this, it follows that $b = 7$. Since $a > c$, the number of steps in the skyscraper could be either 571 or 472. Therefore, the cleaner could have washed either 1142 or 944 steps.