Problem 2. A group of adventurers is showing off their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. In addition, it is known that

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

What is the minimum number of adventurers that can be in such a group?
Answer: 16.

Solution. Note that the number of adventurers who have emeralds is equal to the total number of adventurers who have rubies or diamonds. Then, from the second condition, it follows that 5 adventurers have rubies and emeralds, and 6 have emeralds and diamonds. That is, every adventurer who has diamonds must also have emeralds. Then, from the first condition, there cannot be an adventurer who has both sapphires and diamonds. Therefore, there are at least $10+6=16$ adventurers.

Indeed, there can be this many adventurers: let's say we have 6 adventurers who have emeralds and diamonds, 5 adventurers who have rubies, emeralds, and sapphires, and 5 adventurers who have only sapphires. One can verify that this example meets all the conditions.