Mirror Grid
Description
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 $270^{\circ}$.
Input
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.
Output
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 $270^{\circ}$.
Samples
<div class="test-example-line test-example-line-even test-example-line-0">5</div><div class="test-example-line test-example-line-odd test-example-line-1">3</div><div class="test-example-line test-example-line-odd test-example-line-1">010</div><div class="test-example-line test-example-line-odd test-example-line-1">110</div><div class="test-example-line test-example-line-odd test-example-line-1">010</div><div class="test-example-line test-example-line-even test-example-line-2">1</div><div class="test-example-line test-example-line-even test-example-line-2">0</div><div class="test-example-line test-example-line-odd test-example-line-3">5</div><div class="test-example-line test-example-line-odd test-example-line-3">11100</div><div class="test-example-line test-example-line-odd test-example-line-3">11011</div><div class="test-example-line test-example-line-odd test-example-line-3">01011</div><div class="test-example-line test-example-line-odd test-example-line-3">10011</div><div class="test-example-line test-example-line-odd test-example-line-3">11000</div><div class="test-example-line test-example-line-even test-example-line-4">5</div><div class="test-example-line test-example-line-even test-example-line-4">01000</div><div class="test-example-line test-example-line-even test-example-line-4">10101</div><div class="test-example-line test-example-line-even test-example-line-4">01010</div><div class="test-example-line test-example-line-even test-example-line-4">00010</div><div class="test-example-line test-example-line-even test-example-line-4">01001</div><div class="test-example-line test-example-line-odd test-example-line-5">5</div><div class="test-example-line test-example-line-odd test-example-line-5">11001</div><div class="test-example-line test-example-line-odd test-example-line-5">00000</div><div class="test-example-line test-example-line-odd test-example-line-5">11111</div><div class="test-example-line test-example-line-odd test-example-line-5">10110</div><div class="test-example-line test-example-line-odd test-example-line-5">01111</div><div class="test-example-line test-example-line-odd test-example-line-5"></div>
1
0
9
7
6
Note
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.