Monika is thinking of a four-digit number that has the following properties:

- the product of the two outer digits is 40,
- the product of the two inner digits is 18,
- the difference between the two outer digits is the same as the difference between the two inner digits,
- the difference between the thought number and the number written in reverse order (i.e., the number written with the same digits but in reverse order) is the largest possible.

Determine Monika's thought number.

(L. Hozová)
The product of the two extreme digits is 40, and this is only possible as $40=5 \cdot 8$. The difference between these digits is $8-5=3$. The product of the two inner digits is 18, which is possible either as $18=2 \cdot 9$ or as $18=3 \cdot 6$. In the first case, the difference is $9-2=7$, which is different from the difference of the extreme digits. In the second case, the difference is $6-3=3$, which matches the difference of the extreme digits. From the first three conditions, it follows that the extreme digits are 5 and 8, and the inner digits are 3 and 6. Such numbers are four:

5368, $8635, \quad 8365, \quad 5638$.

To maximize the difference between the intended number and the number written in reverse, the larger of the two possible digits must be in the thousands and hundreds places, respectively. Monika thought about the number 8635.

Scoring suggestion. 1 point for the pair of extreme digits; 1 point for the possible pairs of inner digits; 2 points for determining the correct pair of inner digits and the four possibilities (1); 2 points for determining the suitable possibility.

Note. Among the numbers (1) are the numbers in the first pair and the numbers in the second pair written in reverse. The final part of the problem can be performed by comparing the four possible differences:

$$
\begin{array}{ll}
8635-5368=3267, & 5368-8635=-3267 \\
8365-5638=2727, & 5638-8365=-2727 .
\end{array}
$$