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 Rust (1.15.1) (judge verdict: Accepted)

use std::collections::{BTreeSet, VecDeque};

fn main() {
    //    let n = 3;
    //    for mask in 0..(1 << (n * n)) {
    //        let mut s = vec![vec!['.'; n]; n];
    //        for i in 0..n {
    //            for j in 0..n {
    //                let m = i * n + j;
    //                if mask & (1 << m) != 0 {
    //                    s[i][j] = '#';
    //                }
    //            }
    //        }
    //
    //        let ans1 = solve(&s);
    //        let ans2 = naive(&s);
    //        assert_eq!(ans1, ans2, "{:?} {:?} {:?}", s, ans1, ans2);
    //    }

    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let s: Vec<Vec<char>> = (0..n).map(|_| sc.chars()).collect();

    match solve(&s) {
        Some(ans) => println!("{}", ans),
        None => println!("-1"),
    }
}

fn solve(s: &Vec<Vec<char>>) -> Option<usize> {
    let n = s.len();

    let min_fill = (0..n)
        .map(|r| {
            let holes = s[r].iter().filter(|&&c| c == '.').count();
            let has_column = s.iter().any(|row| row[r] == '#');
            holes + if has_column { 0 } else { 1 }
        })
        .min()
        .unwrap();
    let filled_columns = (0..n)
        .filter(|&c| s.iter().all(|row| row[c] == '#'))
        .count();
    let ans = n - filled_columns + min_fill;
    if ans <= 2 * n {
        Some(ans)
    } else {
        None
    }
}

//fn naive(s: &Vec<Vec<char>>) -> Option<usize> {
//    if s.iter().all(|row| row.iter().all(|&c| c == '.')) {
//        return None;
//    }
//    let n = s.len();
//    let mut q = VecDeque::new();
//    let mut set = BTreeSet::new();
//    set.insert(s.clone());
//    q.push_back((s.clone(), 0));
//    while let Some((s, c)) = q.pop_front() {
//        if s.iter().all(|row| row.iter().all(|&c| c == '#')) {
//            return Some(c);
//        }
//
//        for i in 0..n {
//            for j in 0..n {
//                let mut s = s.clone();
//                let v = s[i].clone();
//                for i in 0..n {
//                    s[i][j] = v[i];
//                }
//                if !set.contains(&s) {
//                    set.insert(s.clone());
//                    q.push_back((s, c + 1));
//                }
//            }
//        }
//    }
//    unreachable!()
//}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}