5. In a room, 15 chairs are arranged in a circle. Three jewelers, when no one is watching, sit on three adjacent chairs, and the one sitting in the middle chair hides a diamond in the chair he is sitting on. The inspector has several detectors that show whether someone has sat on a chair or not. What is the minimum number of detectors he should place on the chairs before the jewelers arrive, so that he can determine from their readings where the diamond is?
Answer: 9 detectors.

Instructions. Evaluation. Consider five chairs such that between any two nearest ones there are two other chairs. If there is no detector in two nearest ones, then by adding to the two chairs between them any of them, we get two possible seating arrangements of jewelers, in which the detector readings are the same. To have a detector in any two nearest ones, it is necessary to place at least three detectors in them. Such different quintets can be formed three times. This means that at least $3 \cdot 3=9$ detectors are needed. Example. They need to be placed in groups of three on adjacent chairs in intervals, between which there are two chairs without detectors. The number of activated detectors easily determines which three chairs were occupied.