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 sys                                                                      
                                                                                
grid = []                                                                       
                                                                                
N = int(raw_input())                                                            
                                                                                
for i in range(0,N):                                                            
    rowChars = list(raw_input())                                                
    rowNums = []                                                                
    for square in rowChars:                                                     
        if square == '.':                                                       
            rowNums.append(0)                                                   
        else:                                                                   
            rowNums.append(1)                                                   
    grid.append(rowNums)                                                        
                                                                                
                                                                                
# Check if totally blank (otherwise there exists a solution)                    
blackSquares = 0                                                                
for i in range(0,N):                                                            
    for j in range(0,N):                                                        
        blackSquares += grid[i][j]                                              
                                                                                
if blackSquares == 0:                                                           
    print "-1"                                                                  
    sys.exit()                                                                  
                                                                                
# Check for all black columns                                                   
colCounts = [0] * N                                                             
for i in range(0,N):                                                            
    for j in range(0,N):                                                        
        colCounts[j] += grid[i][j]                                              
                                                                                
allBlackColCount = 0                                                            
for i in range(0,N):                                                            
    if colCounts[i] == N:                                                       
        allBlackColCount += 1

# Check for any all black rows                                                  
allBlackRowCount = 0                                                            
for i in range(0,N):                                                            
    rowCount = 0                                                                
    for j in range(0,N):                                                        
        rowCount += grid[i][j]                                                  
                                                                                
    if rowCount == N:                                                           
        # Found all black row... we're done                                     
        print "%d" % (N - allBlackColCount)                                     
                                                                                
# No entirely black rows... What we'll do is for each row we will calculate the 
# cost of completing it and take the minimum.                                   
minCost = N                                                                     
for i in range(0,N):                                                            
    blackSquares = 0                                                            
    for j in range(0,N):                                                        
        blackSquares += grid[i][j]                                              
                                                                                
    whiteSquares = N - blackSquares                                             
                                                                                
    completionCost = 0                                                          
    # If the ith column has any black squares, we can use the corresponding row 
    # to fill in the holes in this row for a total number of moves equal to the 
    # number of white squares in the row                                        
    if colCounts[i] > 0:                                                        
        completionCost = whiteSquares                                           
    else:                                                                       
        completionCost = whiteSquares + 1                                       
                                                                                
    minCost = min(minCost, completionCost)                                      
                                                                                
print "%d" % (N - allBlackColCount + minCost)