11.4. What is the minimum number of cells that need to be marked on an $8 \times 9$ board so that among any five consecutive cells in a row, column, or diagonal, there is a marked cell

Fig. 11.4a
Answer: 14 cells.

Solution. Example.

See Fig. 11.4a.

Estimate. Let's highlight 14 rectangles of size $\quad 1 \times 5 \quad$ on the board, touching only two central cells (see Fig. 11.4b). In each of them, there must be at least one marked cell, meaning there are at least 14 marked cells.

![](https://cdn.mathpix.com/cropped/2024_05_06_0d3f1d5c0c0c0a5a42efg-2.jpg?height=460&width=583&top_left_y=1129&top_left_x=702)

Examiners are recommended to check the example very carefully, especially all the diagonals.

Grading Criteria.

«+» A complete and well-reasoned solution is provided

«干» The correct answer and correct estimate are provided, but the example is missing or incorrect «干» The correct answer and correct example are provided, but the estimate is missing or incorrect «-» Only the answer is provided

«-» The problem is not solved or is solved incorrectly