Problem:
Score : 1300 points

Problem Statement

There is a square-shaped grid with N vertical rows and N horizontal columns.
We will denote the square at the i-th row from the top and the j-th column from the left as (i,\ j).

Initially, each square is either white or black.
The initial color of the grid is given to you as characters a_{ij}, arranged in a square shape.
If the square (i,\ j) is white, a_{ij} is .. If it is black, a_{ij} is #.

You are developing a robot that repaints the grid.
It can repeatedly perform the following operation:

Select two integers i, j (1 ≤ i,\ j ≤ N). Memorize the colors of the squares (i,\ 1), (i,\ 2), ..., (i,\ N) as c_1, c_2, ..., c_N, respectively. Then, repaint the squares (1,\ j), (2,\ j), ..., (N,\ j) with the colors c_1, c_2, ..., c_N, respectively.

Your objective is to turn all the squares black.
Determine whether it is possible, and find the minimum necessary number of operations to achieve it if the answer is positive.

Constraints

2 ≤ N ≤ 500

a_{ij} is either . or #.

Partial Score

In a test set worth 300 points, N ≤ 3.

Input

The input is given from Standard Input in the following format:

N
a_{11}...a_{1N}
:
a_{N1}...a_{NN}

Output

If it is possible to turn all the squares black, print the minimum necessary number of operations to achieve the objective.
If it is impossible, print -1 instead.

Sample Input 1

2
#.
.#

Sample Output 1

3

For example, perform the operation as follows:

Select i = 1, j = 2.

Select i = 1, j = 1.

Select i = 1, j = 2.

The transition of the colors of the squares is shown in the figure below:

Sample Input 2

2
..
..

Sample Output 2

-1

Sample Input 3

2
##
##

Sample Output 3

0

Sample Input 4

3
.#.
###
.#.

Sample Output 4

2

Sample Input 5

3
...
.#.
...

Sample Output 5

5

Solution in Python (2.7.6) (judge verdict: Wrong Answer)

import numpy as np
import copy
 
def sampling_row(mat, row):
    row_str = copy.deepcopy(mat[row, :])
    return row_str
 
def writing_col(mat, row_str, col):
 
    row = 0
    for s in row_str:
        mat[row, col] = s
        row+=1
 
def check(mat, N):
    max_val = 0
    max_row = 0
    max_col = 0
    is_ok = True
    is_fast = True
    for i in range(N):
        for j in range(N):
            row = mat[i, :]
            col = mat[:, j]
            val = 0
 
            no_change = True
            for k in range(N):
                if row[k] == "#" and col[k] == ".":  
                    val += 1
                elif row[k] == "." and col[k] == "#":
                    val -= 1
                # elif row[k] == "." and col[k] == ".":
                #     val -= 1
                # elif row[k] == "#" and col[k] == "#":
                #     val -= 1

                if row[k] == "#" and col[k] == "#" and is_ok: 
                    pass
                else:
                    is_ok = False

                if row[k] == col[k] and no_change:
                    pass
                else:
                    no_change = False
 
            if (max_val < val and not(no_change)) or is_fast:
                max_val = val
                max_row = i
                max_col = j
                is_fast = False
 
    return (max_val, max_row, max_col, is_ok)
 
 
if __name__ == "__main__":
    mat_str = []
 
    N = raw_input()
    N = int(N)
    mat = np.zeros((N, N), str)
    for i in range(N):
        s = raw_input()
        for j in range(N):
            mat[i][j] = s[j]
 
    # row = sampling(mat, 0)
    # writing(mat, row, 1)
 
    is_ok = -1
 
    for s in mat.reshape(N*N):
        if s == "#":
            is_ok = 0
            break
 
    if is_ok != -1:
        while True:
            max_val, max_row, max_col, c_ok = check(mat, N)
            if c_ok == True:
                break
            else:
                writing_col(mat, sampling_row(mat, max_row), max_col)
                is_ok += 1
                # print mat
                
    print is_ok