. Some $1 \times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n \times n$ so that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is 2008 , find the least possible value of $n$.

Answer: 77
Following the pattern from the figure, we have space for

$$
6+18+30+\ldots+150=\frac{156 \cdot 13}{2}=1014
$$

dominoes, giving the area $2028>2008$.

![](https://cdn.mathpix.com/cropped/2024_04_17_64f81c74a8132dd111d8g-07.jpg?height=559&width=569&top_left_y=2059&top_left_x=755)

The square $76 \times 76$ is not enough. If it was, consider the "circumferences" of the 1004 dominoes of size $2 \times 3$, see figure; they should fit inside $77 \times 77$ square without overlapping. But $6 \cdot 1004=6024>5929=77 \cdot 77$.

![](https://cdn.mathpix.com/cropped/2024_04_17_64f81c74a8132dd111d8g-08.jpg?height=185&width=263&top_left_y=393&top_left_x=905)