4. A table of dimensions $8 \times 8$ is divided into 64 unit squares. We place dominoes in the shape of a + (see image) on the table such that their interiors do not overlap, the edges of the dominoes overlap with the edges of the corresponding unit squares, and the dominoes themselves lie entirely on the table. What is the maximum number of dominoes that can be placed on the table in this way?

![](https://cdn.mathpix.com/cropped/2024_06_07_44798e3c82dc47fe22e9g-4.jpg?height=168&width=165&top_left_y=2052&top_left_x=1645)

Solve the problems independently. Time for solving: $3 \frac{1}{2}$ hours.

The use of notes, literature, or a pocket calculator is not allowed.

45th Mathematical Competition for High School Students in Slovenia

Idrija, May 12, 2001

## PROBLEMS FOR THE FOURTH YEAR
III/4. Obviously, at each edge of the table, at most two unit squares are covered. Therefore, the maximum number of unit squares that can be fully covered is $4 \cdot 2 + 6 \cdot 6 = 44$ unit squares. Since each domino covers 5 unit squares, the maximum number of dominos that can be placed on the table is $\frac{44}{5} < 9$ dominos. The image on the right, however, shows that 8 dominos can indeed be placed.

![](https://cdn.mathpix.com/cropped/2024_06_07_44798e3c82dc47fe22e9g-8.jpg?height=302&width=309&top_left_y=954&top_left_x=1573)