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 C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <vector>
#include <unordered_map>
int mod(int x, int n) {
    return ((x % n) + n) % n;
}
int main() {
	int n;
    std::cin >> n;
    std::vector<std::vector<char>> v(n,std::vector<char>(n, 'a'));
    std::unordered_map<char, std::vector<std::pair<int, int> > > m;
    for(int i = 0; i < n; ++i) {
        for(int j = 0; j < n; ++j) {
            std::cin>> v[i][j];
            if (m.find(v[i][j]) == m.end()) {
                std::vector<std::pair<int, int> > q;
                q.push_back(std::pair<int, int> (i, j));
                m.insert(std::pair<  char, std::vector<std::pair<int, int> > > (v[i][j], q));
            } else {
                m[v[i][j] ].push_back(std::pair<int, int> (i, j));
            }
        }
    }
    std::vector<char> hui;
    for (auto it = m.begin(); it != m.end(); ++it) {
        if (it->second.size() % 2 == 1) {
            hui.push_back(it->first);
        }
    }
    int res = 0;
    for(int a = 0; a < n; ++a) {
        for(int b = 0; b < n; ++b) {
            int fl = 0;
            for(int i = 0; i < n; ++i) {
                for(int j = i; j < n; ++j) {
                    if (v[mod(i - a,n)][mod(j - b,n)] != v[mod(j - a,n)][mod(i - b,n)]) {
                        fl = 1;
                        //std::cout << a << b <<'\n';
                        //std::cout << v[mod(i - a,n)][mod(j - b,n)] << v[mod(i - a,n)][mod(i - a,n)];
                        break;
                    }
                }
                if (fl) break;
            }
            if (fl == 0) res += 1;

        }
    }
    std::cout << res << '\n';
}