## Task A-4.5.

On a board of dimensions $8 \times 8$, triomino tiles of shape $\exists$ are placed such that each triomino tile covers exactly three squares of the board, and they do not overlap with each other.

How many triomino tiles are needed at a minimum to place on the board so that no additional triomino tile can be placed afterward?
## Solution.

Divide the board into 16 squares of $2 \times 2$ as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_6f525b6526cae5162cdfg-22.jpg?height=497&width=508&top_left_y=1319&top_left_x=774)

In each of these squares, at least two fields must be covered, otherwise a tromino-tile can be placed on the uncovered fields. Therefore, at least 32 fields must be covered, and for that, we need at least 11 tromino-tiles.

To show that 11 tromino-tiles are sufficient, we demonstrate this by construction, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_6f525b6526cae5162cdfg-22.jpg?height=489&width=491&top_left_y=2114&top_left_x=788)