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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;

namespace AGC023_B___Find_Symmetries
{
    class Program
    {
        static void Main(string[] args)
        {
            Console.WriteLine(solver());
        }
        static long solver()
        {
            var N = int.Parse(Console.ReadLine());
            var baseBoard = new int[N, N];
            for (int i = 0; i < N; i++)
            {
                var read = Console.ReadLine();
                for (int j = 0; j < N; j++) baseBoard[i, j] = read[j] - 'a';
            }

            var count = 0;
            for (int offset = 0; offset < N; offset++)
            {
                var offsetBoard = new int[N, N];
                for (int h = 0; h < N; h++) for (int w = 0; w < N; w++) offsetBoard[h, (w + offset) % N] = baseBoard[h, w];
                var isSymmetry = true;
                for (int h = 0; h < N; h++)
                {
                    for (int w = h; w < N; w++)
                    {
                        if (offsetBoard[h, w] != offsetBoard[w, h])
                        {
                            isSymmetry = false;
                            break;
                        }
                    }
                    if (!isSymmetry) break;
                }
                if (isSymmetry) count++;
            }
            return count * N;
        }
    }
}