10.2. In an $n \times n$ grid, the $n$ columns from left to right are called the 1st column, 2nd column, $\cdots, n$th column. Fill the grid with the $n$ positive integers $1 \sim n$ such that each row and each column contains $n$ different numbers. If the number in a cell is greater than the column number of that cell, then the cell is called "good". Find all positive integers $n$ such that the numbers can be filled in the grid according to the above requirements and satisfy that the number of good cells in each row is the same.
10.2. First, find the total number of good squares.

In the 1st column, there are $n-1$ (all except the one filled with number 1 are good), in the 2nd column, there are $n-2$ (all except the ones filled with numbers 1 and 2 are good), $\cdots \cdots$ the $n$-th column has no good squares. Therefore, the total number of good squares is
$$
(n-1)+(n-2)+\cdots+1=\frac{n(n-1)}{2} \text {. }
$$

Thus, the number of good squares in each row is $\frac{n-1}{2}$, which implies that $n$ is odd.

When $n=2 k+1$ is odd, fill the square at the intersection of the $i$-th row and the $j$-th column with the number $a_{i j}$,
$$
a_{i j}=\left\{\begin{array}{ll}
n-(j-i), & j \geqslant i ; \\
i-j, & j<i .
\end{array}\right.
$$

It is not difficult to verify that in the $2 i$-th row, the good squares are the first $i-1$ and from the $2 i$-th to the $k+i$-th, while in the $2 i-1$-th row, the good squares are the first $i-1$ and from the $2 i-1$-th to the $k+i-1$-th, with exactly $k$ good squares in each row.