Problem 9.3. To 30 palm trees in different parts of an uninhabited island, a sign is nailed.

- On 15 of them, it is written: "Exactly under 15 signs, treasure is buried."
- On 8 of them, it is written: "Exactly under 8 signs, treasure is buried."
- On 4 of them, it is written: "Exactly under 4 signs, treasure is buried."
- On 3 of them, it is written: "Exactly under 3 signs, treasure is buried."

It is known that only the signs under which there is no treasure are true.

Under what minimum number of signs can the treasure be buried?
Answer: 15.

Solution. Suppose the treasure is not buried under at least 16 plaques. Then there are two plaques with different inscriptions under which there is no treasure. According to the condition, the inscriptions on both should be true, but they contradict each other. Contradiction.

Therefore, the treasure is not buried under more than 15 plaques. Thus, there are at least 15 plaques under which there is treasure. It is not difficult to come up with a suitable example: let the treasures be buried only under plaques with one of the following three phrases.

- "The treasure is buried under exactly 8 plaques."
- "The treasure is buried under exactly 4 plaques."
- "The treasure is buried under exactly 3 plaques."