3 [Chessboards and Chess Pieces]

Avoris A. Anjanis

What is the minimum number of cells that need to be marked on a chessboard so that

1) among the marked cells, there are no adjacent cells (having a common side or a common vertex),
2) adding any one cell to these cells would violate point 1?

#
Evaluation. Let's divide the board into nine parts (see the figure). In each of them, a cell must be marked, otherwise the black cell contained in it can be added.

![](https://cdn.mathpix.com/cropped/2024_05_06_278dceb3f0947a8409cfg-20.jpg?height=431&width=421&top_left_y=1508&top_left_x=790)

Example. The nine black cells in the figure satisfy both conditions.

## Answer

9 cells