Problem:
Score : 500 points

Problem Statement

Snuke has two boards, each divided into a grid with N rows and N columns.
For both of these boards, the square at the i-th row from the top and the j-th column from the left is called Square (i,j).

There is a lowercase English letter written in each square on the first board. The letter written in Square (i,j) is S_{i,j}. On the second board, nothing is written yet.

Snuke will write letters on the second board, as follows:

First, choose two integers A and B ( 0 \leq A, B < N ).

Write one letter in each square on the second board.
Specifically, write the letter written in Square ( i+A, j+B ) on the first board into Square (i,j) on the second board.
Here, the k-th row is also represented as the (N+k)-th row, and the k-th column is also represented as the (N+k)-th column.

After this operation, the second board is called a good board when, for every i and j ( 1 \leq i, j \leq N ), the letter in Square (i,j) and the letter in Square (j,i) are equal.

Find the number of the ways to choose integers A and B ( 0 \leq A, B < N ) such that the second board is a good board.

Constraints

1 \leq N \leq 300

S_{i,j} ( 1 \leq i, j \leq N ) is a lowercase English letter.

Input

Input is given from Standard Input in the following format:

N
S_{1,1}S_{1,2}..S_{1,N}
S_{2,1}S_{2,2}..S_{2,N}
:
S_{N,1}S_{N,2}..S_{N,N}

Output

Print the number of the ways to choose integers A and B ( 0 \leq A, B < N ) such that the second board is a good board.

Sample Input 1

2
ab
ca

Sample Output 1

2

For each pair of A and B, the second board will look as shown below:

The second board is a good board when (A,B) = (0,1) or (A,B) = (1,0), thus the answer is 2.

Sample Input 2

4
aaaa
aaaa
aaaa
aaaa

Sample Output 2

16

Every possible choice of A and B makes the second board good.

Sample Input 3

5
abcde
fghij
klmno
pqrst
uvwxy

Sample Output 3

0

No possible choice of A and B makes the second board good.

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::io;
use std::io::{BufRead, Write};

fn solve<R: BufRead, W: Write>(mut input: R, mut output: W) {
    let mut buf = String::new();
    input.read_line(&mut buf).unwrap();
    let n: usize = buf.trim_right().parse().unwrap();
    let s: Vec<_> = (0..n)
        .map(|_| {
            buf.clear();
            input.read_line(&mut buf).unwrap();
            buf.trim_right().as_bytes().to_owned()
        })
        .collect();
    let x_sum: usize = (0..n)
        .filter(|x_off| {
            (0..n).all(|x| (0..n).all(|y| s[(x + x_off) % n][y] == s[(y + x_off) % n][x]))
        })
        .map(|i| n - i)
        .sum();
    let y_sum: usize = (1..n)
        .filter(|y_off| {
            (0..n).all(|y| (0..n).all(|x| s[x][(y + y_off) % n] == s[y][(x + y_off) % n]))
        })
        .map(|i| n - i)
        .sum();
    writeln!(output, "{}", x_sum + y_sum).unwrap();
}

fn main() {
    let stdin = io::stdin();
    let stdout = io::stdout();
    solve(stdin.lock(), stdout.lock());
}

#[cfg(test)]
mod tests {
    fn test(input: &str, output: &str) {
        let mut v: Vec<u8> = Vec::new();
        ::solve(input.as_bytes(), &mut v);
        assert_eq!(String::from_utf8(v).unwrap(), output);
    }

    #[test]
    fn case1() {
        test(
            "2
ab
ca
", "2\n",
        );
    }

    #[test]
    fn case2() {
        test(
            "4
aaaa
aaaa
aaaa
aaaa
",
            "16\n",
        );
    }

    #[test]
    fn case3() {
        test(
            "5
abcde
fghij
klmno
pqrst
uvwxy
",
            "0\n",
        );
    }
}