Problem 7.5. In each room of the hotel, no more than 3 people can be accommodated. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. In one room, only men or only women can be accommodated; also, fans of different teams cannot be accommodated together. How many rooms need to be reserved to definitely accommodate all the fans?
Answer: 37.

Solution. All fans are divided into six groups: men/women, supporting the first team; men/women, supporting the second team; men/women, supporting the third team. Only people from the same group can be housed in one room. We divide the number of people in each of the six groups by 3 with a remainder. If the remainder in any group is 1, we will house one person from that group in a separate room. If the remainder is 2, we will house any two people from that group in a separate room. Now, in each group, the number of unaccommodated fans is divisible by 3.

After we have accommodated the remainders of each group, some number of unaccommodated people remain. This total number is no more than 100 and no less than 88 (since the sum of all six remainders is no more than $6 \cdot 2=12$, and $100-12=88$). This number is divisible by 3, so it is equal to one of the values $90, 93, 96, 99$ (and all corresponding people can be accommodated in rooms of three). Let's consider several cases.

- If 90 people remain, they need $90: 3=30$ rooms, and the remaining 10 people from the 6 groups will definitely fit into 6 rooms. In total, no more than 36 rooms are needed.
- If 93 people remain, they need $93: 3=31$ rooms, and the remaining 7 people from the 6 groups will definitely fit into 6 rooms. In total, no more than 37 rooms are needed.
- If 96 people remain, they need $96: 3=32$ rooms, and the remaining 4 people will definitely fit into 4 rooms. In total, no more than 36 rooms are needed.
- If 99 people remain, they need $99: 3=33$ rooms, and the remaining one person will definitely fit into 1 room. In total, no more than 34 rooms are needed.

Thus, 37 rooms are definitely enough.

We will also provide an example where at least 37 rooms are necessary. Suppose the distribution of people among the six groups is such: $20, 16, 16, 16, 16, 16$. For the group of 20 people, at least 7 rooms are needed, and for each of the others, at least 6 rooms are needed. In total, at least $7+6 \cdot 5=37$ rooms are needed.