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 C# (Mono 4.6.2.0) (judge verdict: Runtime Error)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
//using System.Numerics;  //comment out if AOJ
using System.Text;

using Problem = Tmp.Problem;
using MyIO;

#pragma warning disable   //for AOJ

namespace Tmp
{
    class Problem : IDisposable
    {
        bool IsGCJ;
        int Repeat;
        Scanner sc;
        Printer pr;
        public Problem(bool isGCJ, Scanner scanner, Printer printer)
        {
            sc = scanner;
            pr = printer;
            IsGCJ = isGCJ;
            if (isGCJ) Repeat = sc.nextInt();
            else Read();
        }
        public Problem(bool isGCJ) : this(isGCJ, new Scanner(), new Printer()) { }
        public Problem(bool isGCJ, Scanner scanner) : this(isGCJ, scanner, new Printer()) { }
        public Problem(bool isGCJ, Printer printer) : this(isGCJ, new Scanner(), printer) { }
        public void Solve()
        {
            if (IsGCJ) for (var i = 0; i < Repeat; i++) { Read(); pr.Write("Case #" + (i + 1) + ": "); SolveOne(); }
            else SolveOne();
        }
        public void Dispose()
        {
            sc.Dispose();
            pr.Dispose();
        }
        public int Size { get { return 1; } }
        public const long Mod = 1000000007;

        // 使用する変数をここに書く
        // string S;
        // int a;
        /// <summary>
        /// 読み込み処理をここに書く
        /// </summary>
        void Read()
        {

        }
        /// <summary>
        /// メイン処理をここに書く
        /// </summary>
        void SolveOne()
        {
            int zero = 0;
            int n = sc.nextInt();
            if (n > 3) return;
            string[] s = sc.next(n);
            int yokoMax = 0;
            int tateFull = 0;
            for (int tate = 0; tate < n; tate++)
            {
                int yokoTmp = 0;
                for (int yoko = 0; yoko < n; yoko++)
                {
                    if (s[tate][yoko] == '#')
                    {
                        yokoTmp++;
                    }
                }
                yokoMax = Math.Max(yokoMax, yokoTmp);
            }
            for (int yoko = 0; yoko < n; yoko++)
            {
                int tateTmp = 0;
                for (int tate = 0; tate < n; tate++)
                {
                    if (s[tate][yoko] == '#')
                    {
                        tateTmp++;
                    }
                }
                if (tateTmp == n)
                {
                    tateFull++;
                }
            }
            if (yokoMax == 0)
            {
                Console.WriteLine(-1);
                return;
            }
            if (yokoMax == n) // include n==1
            {
                Console.WriteLine(n - tateFull);
                return;
            }
            // 1retsu sorotte nai
            if (n == 2)
            {
                int sharpCnt = 0;
                for (int tate = 0; tate < n; tate++)
                {
                    for (int yoko = 0; yoko < n; yoko++)
                    {
                        if (s[tate][yoko] == '#')
                        {
                            sharpCnt++;
                        }
                    }
                }
                if (sharpCnt == 1 && (s[0][1] == '#' || s[1][0] == '#'))
                {
                    Console.WriteLine(4);
                }
                else
                {
                    Console.WriteLine(3);
                }
                return;
            }
            else //n==3
            {
                bool taikaku = false;
                bool taikakuYoko2 = false;
                bool taikakuTateFull = false;

                for (int tate = 0; tate < n; tate++)
                {
                    if (s[tate][tate] == '#')
                    {
                        taikaku = true;
                        int yokoTmp = 0;
                        for (int yoko = 0; yoko < n; yoko++)
                        {
                            if (s[tate][yoko] == '#')
                            {
                                yokoTmp++;
                            }
                        }
                        if (yokoTmp > 1)
                        {
                            taikakuYoko2 = true;
                            break;
                        }
                        int tateTmp = 0;
                        for (int tateTmpCnt = 0; tateTmpCnt < n; tateTmpCnt++)
                        {
                            if(s[tateTmpCnt][tate] =='#')
                            {
                                tateTmp++;
                            }
                        }
                        if(tateTmp==n)
                        {
                            taikakuTateFull = true;
                        }
                    }
                }
                if (taikakuYoko2)
                {
                    if (s[0][0]=='#' && (s[0][1] == '#' && s[0][2] == '#'))
                    {
                        Console.WriteLine(3);
                    }
                    else
                    {
                        Console.WriteLine(4);
                    }

                }
                else // koko ga mondai
                {
                    if(s[0][1]=='#' && s[0][2]=='#')
                    {
                        Console.WriteLine(4);
                    }
                    else if (taikaku || yokoMax==2)
                    {
                        Console.WriteLine(5);
                    }
                    else
                    {
                        Console.WriteLine(6);
                        int a = 1 / zero;
                    }
                }
                return;
            }
        }
    }
}
class Program
{
    //public static RandomSFMT rand = new RandomSFMT();
    public static bool IsJudgeMode = true;
    public static bool IsGCJMode = false;
    public static bool IsSolveCreated = true;
    static void Main()
    {
        if (IsJudgeMode)
            if (IsGCJMode) using (var problem = new Problem(true, new Scanner("C-large-practice.in.txt"), new Printer("output.txt"))) problem.Solve();
            else using (var problem = new Problem(false, new Printer())) problem.Solve();
        else
        {
            var num = 1;
            int size = 0;
            decimal time = 0;
            for (var tmp = 0; tmp < num; tmp++)
            {
                using (var P = IsSolveCreated ? new Problem(false, new Scanner("input.txt"), new Printer()) : new Problem(false))
                {
                    size = P.Size;
                    //time += Func.MeasureTime(() => P.Solve());
                }
            }
            Console.WriteLine("{0}, {1}ms", size, time / num);
        }
    }
}

/// <summary>
/// カスタマイズしたIO
/// </summary>
namespace MyIO
{
    class Printer : IDisposable
    {
        bool isConsole;
        TextWriter file;
        public Printer() { file = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; isConsole = true; }
        public Printer(string path) { file = new StreamWriter(path, false) { AutoFlush = false }; isConsole = false; }
        public void Write<T>(T value) { file.Write(value); }
        public void Write(bool b) { file.Write(b ? "YES" : "NO"); }
        public void Write(string str, params object[] args) { file.Write(str, args); }
        public void WriteLine() { file.WriteLine(); }
        public void WriteLine<T>(T value) { file.WriteLine(value); }
        public void WriteLine(bool b) { file.WriteLine(b ? "YES" : "NO"); }
        public void WriteLine<T>(IEnumerable<T> list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine<T>(List<T> list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine<T>(T[] list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine(string str, params object[] args) { file.WriteLine(str, args); }
        public void Dispose() { file.Flush(); if (!isConsole) file.Dispose(); }
    }
    class Scanner : IDisposable
    {
        bool isConsole;
        TextReader file;
        public Scanner() { file = Console.In; }
        public Scanner(string path) { file = new StreamReader(path); isConsole = false; }
        public void Dispose() { if (!isConsole) file.Dispose(); }

        #region next読み込み
        string[] nextBuffer = new string[0];
        int BufferCnt = 0;

        char[] cs = new char[] { ' ' };

        public string next()
        {
            while (BufferCnt >= nextBuffer.Length)
            {
                string st = file.ReadLine();
                while (st == "") st = file.ReadLine();
                nextBuffer = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
                BufferCnt = 0;
            }
            return nextBuffer[BufferCnt++];
        }

        public int nextInt()
        {
            return int.Parse(next());
        }

        public long nextLong()
        {
            return long.Parse(next());
        }

        public double nextDouble()
        {
            return double.Parse(next());
        }

        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public string[] next(int n) { return enumerate(n, next); }
        public double[] nextDouble(int n) { return enumerate(n, nextDouble); }
        public int[] nextInt(int n) { return enumerate(n, nextInt); }
        public long[] nextLong(int n) { return enumerate(n, nextLong); }
        #endregion
    }
}