CF1703E.Mirror Grid

You are given a square grid with $n$ rows and $n$ columns. Each cell contains either $0$ or $1$ .

In an operation, you can select a cell of the grid and flip it (from $0 \to 1$ or $1 \to 0$ ). Find the minimum number of operations you need to obtain a square that remains the same when rotated $0^{\circ}$ , $90^{\circ}$ , $180^{\circ}$ and $270^{\circ}$ .

The picture below shows an example of all rotations of a grid.

The first line contains a single integer $t$ ( $1 \leq t \leq 100$ ) — the number of test cases.

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 100$ ) — the size of the grid.

Then $n$ lines follow, each with $n$ characters $a_{i,j}$ ( $0 \leq a_{i,j} \leq 1$ ) — the number written in each cell.

For each test case output a single integer — the minimum number of operations needed to make the square look the same rotated $0^{\circ}$ , $90^{\circ}$ , $180^{\circ}$ and $270^{\circ}$ .

In the first test case, we can perform one operations to make the grid $\begin{matrix}0 & 1 & 0\\ 1 & 1 & \color{red}{1}\\ 0 & 1 & 0\end{matrix}$ . Now, all rotations of the square are the same.

In the second test case, all rotations of the square are already the same, so we don't need any flips.