## Task A-3.4.

The fields of a $300 \times 300$ board are the same size as the 14 squares that make up the figure shown in the image. What is the maximum number of such figures that can be placed on this board without overlapping? The figures can be rotated and flipped.

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-18.jpg?height=254&width=189&top_left_y=1615&top_left_x=1459)
## Solution.

Place one such figure on the board and observe the uncovered square marked in the image. Notice that no matter how we place another given figure on the board without overlapping the previous one, that square will remain uncovered.

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-18.jpg?height=437&width=323&top_left_y=2146&top_left_x=798)

On the other hand, from two given figures, we can form a rectangle of dimensions $6 \times 5$ in which only two previously described squares will remain uncovered.

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-19.jpg?height=445&width=383&top_left_y=337&top_left_x=768)

The entire board can be tiled with a total of $60 \cdot 50=3000$ such rectangles, meaning that at most $2 \cdot 3000=6000$ given figures can be placed on the board.