[ Examples and counterexamples. Constructions ] [ Symmetry helps solve the task ]

What is the minimum number of cells that need to be marked on a chessboard so that each cell of the board (marked or unmarked) shares a side with at least one marked cell?
Let's highlight 20 white cells (on the diagram, they are marked with the sign "x").

![](https://cdn.mathpix.com/cropped/2024_05_06_0e468145bb05e34a17aag-04.jpg?height=314&width=323&top_left_y=0&top_left_x=868)

Any black cell is adjacent to no more than two highlighted white cells. Therefore, to ensure that these white cells have marked neighbors, at least ten black cells need to be marked. Similarly, at least ten white cells need to be marked. Thus, on the board, at least twenty cells need to be marked.

An example of twenty marked cells that satisfy the condition can be seen in problem 103868.

## Answer

20 cells.