Question 5
An $n\times n$ board has rows and columns numbered $1$ through $n$; the cell in row $i$, column $j$ is denoted $(i,j)$. A subset $A$ of its cells is called good if, for any two cells $(x_1,y),(x_2,y)$ in $A$, none of the cells $(u,v)$ satisfying
$$x_1\lt u\le x_2,\quad v\lt y,$$
or
$$x_1\le u\lt x_2,\quad v>y,$$
belongs to $A$. Find the smallest number of pairwise disjoint good subsets such that every board cell belongs to exactly one subset.
$$x_1\lt u\le x_2,\quad v\lt y,$$
or
$$x_1\le u\lt x_2,\quad v>y,$$
belongs to $A$. Find the smallest number of pairwise disjoint good subsets such that every board cell belongs to exactly one subset.