5. On a $5 \times 5$ board, a cross consisting of five cells (a cell and all its neighbors) is located. What is the minimum number of detectors needed to place on the board cells to accurately determine the position of the cross? (A detector indicates whether a cell belongs to the cross or not, and the detectors trigger simultaneously.)
Answer: 4.

Solution. There are 9 possible positions for the cross on the board, which is the same as the number of positions for the central cell of the cross. A detector has two states, so the total number of possible states for three detectors is $2^{3}=8$, and thus, they cannot distinguish between 9 positions of the cross. Examples of possible arrangements of four detectors are shown in the figures
![](https://cdn.mathpix.com/cropped/2024_05_06_a4325d569d59da29736fg-3.jpg?height=264&width=930&top_left_y=894&top_left_x=1516)

## Grading Criteria.

Only the answer: 0 points.

Example with 4 detectors: 3 points.

Proof of the insufficiency of three detectors: 4 points.