Problem 8.6. Six princesses have a magic chest. Every minute, a dress of one of 10 colors and one of 9 styles can be taken out of it. However, within one hour, it is impossible to take out two dresses from the chest that match both in color and style. What is the minimum number of dresses the princesses will have to take out of the chest to guarantee getting 6 dresses of different colors in one style?
Answer: 46.

Solution. Note that 45 dresses would not be enough, as the chest can issue exactly 5 dresses of each of the 9 styles.

We will prove that if the princesses take out 46 dresses, there will definitely be 6 dresses of the same style, and thus of different colors (identical dresses could not have occurred over the 46 minutes). Indeed, if this were not the case, then there would be no more than 5 dresses of each style, meaning there would be no more than 45 dresses in total; a contradiction.