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: Wrong Answer)

using System;
using System.Linq;
using System.Diagnostics;
using System.Collections.Generic;
using static System.Math;
class P
{
    static void Main()
    {
        int n = int.Parse(Console.ReadLine());
        char[][] map = Enumerable.Repeat(0, n).Select(_ => Console.ReadLine().ToArray()).ToArray();
        //[x][y][深さ]
        bool[][][] isSame = Enumerable.Repeat(0, n).Select(_ => Enumerable.Repeat(0, n).Select(__ => new bool[n]).ToArray()).ToArray();
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                for (int k = 0; k < n; k++)
                {
                    isSame[i][j][k] = map[i][j] == map[(i + k) % n][(j + k) % n];
                }
            }
        }
        bool[][] isAllSame = Enumerable.Repeat(0, n).Select(_ => Enumerable.Repeat(true, n).ToArray()).ToArray();
        //スタート
        for (int i = 0; i < n; i++)
        {
            //串のインデックス
            for (int j = 0; j < n; j++)
            {
                //ズラす
                for (int k = 0; k < n; k++)
                {
                    if (!isSame[(n + i - k) % n][k][j])
                    {
                        isAllSame[i][j] = false;
                        break;
                    }
                }
            }
        }
        int count = 0;
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                if (!isAllSame[(i + j) % n][n - (j + 1)]) goto end;
            }
            count++;
            end:;
        }
        Console.WriteLine(count * n);
    }
}