Problem 2. A group of adventurers is showing off their loot. It is known that exactly 13 adventurers have rubies; exactly 9 have emeralds; exactly 15 have sapphires; exactly 6 have diamonds. In addition, it is known that

- if an adventurer has sapphires, then they have either emeralds or diamonds (but not both at the same time);
- if an adventurer has emeralds, then they have either rubies or sapphires (but not both at the same time).

What is the minimum number of adventurers that can be in such a group?
Answer: 22.

Solution. Note that the number of adventurers who have sapphires is equal to the total number of adventurers who have emeralds or diamonds. Then, from the first condition, it follows that 9 adventurers have sapphires and emeralds, and 6 have sapphires and diamonds. That is, every adventurer who has emeralds must also have sapphires. Then, from the second condition, there cannot be an adventurer who has both emeralds and rubies. Therefore, there must be at least $13+9=22$ adventurers.

This number of adventurers is indeed possible: let's say we have 9 adventurers who have sapphires and emeralds, 6 adventurers who have sapphires, diamonds, and rubies, and 7 adventurers who have only rubies. One can verify that this example meets all the conditions.