## Task A-3.3.

On a board of dimensions $20 \times 19$, tiles of dimensions $3 \times 1$ are placed such that they cover exactly three squares of the board, and they do not overlap or touch each other, not even at the corners.

Determine the maximum possible number of $3 \times 1$ tiles on this board.
## Solution.

We consider the vertices of unit squares. Each $3 \times 1$ rectangle covers exactly 8 vertices.

Notice that, no matter how it is oriented, a $3 \times 1$ rectangle covers an even number of vertices in each vertical column.

Since each vertical column has 21 vertices, it is possible to cover at most 20 vertices.

There are 20 vertical columns of vertices, so we can cover a total of at most $20 \cdot 20 = 400$ vertices.

This means that we can place at most $\frac{400}{8} = 50$ $3 \times 1$ rectangles on the board.

An example shows that we can indeed place 50 rectangles:

![](https://cdn.mathpix.com/cropped/2024_05_30_b1b3186b66c6729c9326g-17.jpg?height=808&width=780&top_left_y=284&top_left_x=638)