Problem 2.3. A square $C$ is completely covered with a whole number of unit squares, without overlaps. If one places as many squares as possible of area 2 inside $C$, with sides parallel to the sides of $C$, without overlaps, it is possible to cover eight ninths of the area of the square. Determine all possible dimensions of such squares.
Solution Problem 2.3 Let $l$ be the side of the square and $n$ the maximum number of squares of area 2 that fit on each side of the square. $l$ and $n$ are integers. The conditions of the problem are that

$$
\left\{\begin{array}{c}
2 n^{2}=\frac{8}{9} l^{2} \\
l^{2}<2(n+1)^{2}
\end{array}\right.
$$

Setting $n=2 n^{\prime}$ and $l=3 l^{\prime}$ (which must be integers, by the first equation above), it results by substituting that

$$
\left\{\begin{array}{c}
n^{\prime}=l^{\prime} \\
n^{\prime 2}-8 n^{\prime}-2<0
\end{array}\right.
$$

so that $1 \leq n^{\prime} \leq 8$. The possible solutions are, therefore,

$$
\left\{\begin{array}{ccc}
n^{\prime}=l^{\prime}=1 & n=2 & l=3 \\
n^{\prime}=l^{\prime}=2 & n=4 & l=6 \\
n^{\prime}=l^{\prime}=3 & n=6 & l=9 \\
n^{\prime}=l^{\prime}=4 & n=8 & l=12 \\
n^{\prime}=l^{\prime}=5 & n=10 & l=15 \\
n^{\prime}=l^{\prime}=6 & n=12 & l=18 \\
n^{\prime}=l^{\prime}=7 & n=14 & l=21 \\
n^{\prime}=l^{\prime}=8 & n=16 & l=24
\end{array}\right.
$$