Problem 4. Consider a $2018 \times$ 2018 board. We call an LC tile a tile consisting of 9 congruent unit squares as shown in the figure below

![](https://cdn.mathpix.com/cropped/2024_06_07_fbc9a4a5e943c67ef798g-3.jpg?height=143&width=157&top_left_y=2341&top_left_x=961)

What is the maximum number of LC tiles that can be placed without overlapping on the board? (Each of the 9 unit squares of the tile must overlap with one of the unit squares of the board; a tile can be rotated, mirrored, etc.)

Alexandru Gîrban
## Solution:

We number the rows and columns from 1 to 2018. We color black the unit squares that have both coordinates divisible by 3.

![](https://cdn.mathpix.com/cropped/2024_06_07_fbc9a4a5e943c67ef798g-4.jpg?height=577&width=828&top_left_y=771&top_left_x=637)

There are $672^{2}$ black squares. We observe that each LC tile covers exactly one black square, so we can place at most $672^{2}$ tiles on the board. On the other hand, the example below indicates a way to place $672^{2}$ tiles on the board. Therefore, the maximum sought is $672^{2}$.
![](https://cdn.mathpix.com/cropped/2024_06_07_fbc9a4a5e943c67ef798g-4.jpg?height=586&width=804&top_left_y=1548&top_left_x=637)