Problem:
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8 Queens Problem

The goal of 8 Queens Problem is to put eight queens on a chess-board such that none of them threatens any of others. A queen threatens the squares in the same row, in the same column, or on the same diagonals as shown in the following figure.

For a given chess board where $k$ queens are already placed, find the solution of the 8 queens problem.

Input

In the first line, an integer $k$ is given. In the following $k$ lines, each square where a queen is already placed is given by two integers $r$ and $c$. $r$ and $c$ respectively denotes the row number and the column number. The row/column numbers start with 0.

Output

Print a $8 \times 8$ chess board by strings where a square with a queen is represented by 'Q' and an empty square is represented by '.'.

Constraints

There is exactly one solution

Sample Input 1

2
2 2
5 3

Sample Output 1

......Q.
Q.......
..Q.....
.......Q
.....Q..
...Q....
.Q......
....Q...

Solution in C# (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Math;
 
public class Solve{
    static public int mod = 1000000007;
    public static void Main(){
        // 方針
        //
        var n = rint();
        var map = new char[8,8];
        for(int i=0;i<8;i++){
            for(int j=0;j<8;j++){
                map[i,j] = '.';
            }
        }
        var movet = new int[]{-1,-1,-1,0,0,1,1,1};
        var movey = new int[]{-1,0,1,-1,1,-1,0,1};
        for(int i=0;i<n;i++){
            var a = inta();
            map[a[0],a[1]] = 'Q';
            for(int j=0;j<8;j++){
                var zat = a[0];
                var zay = a[1];
                while(0<=zat+movet[j] && zat+movet[j]<8 && 0<=zay+movey[j] && zay+movey[j]<8){
                    zat += movet[j];
                    zay += movey[j];
                    map[zat,zay] = 'X';
                }
            }
        }
        /*
        for(int i=0;i<8;i++){
            for(int j=0;j<8;j++){
                Write(map[i,j]);
            }
            WriteLine();
        }
        */
        var per = new Permutation(8);
        while(true){
            var p = per.now();
            
            //p[1] =0;p[6] = 1;p[2] = 2;p[5] = 3;p[7] = 4;p[4] = 5;p[0] = 6;p[3] = 7;
            
            var map2 = new char[8,8];
            for(int i=0;i<8;i++){
                for(int j=0;j<8;j++){
                    map2[i,j] = map[i,j];
                }
            }
            /*
            for(int i=0;i<8;i++){
            for(int j=0;j<8;j++){
                Write(map2[i,j]);
            }
            WriteLine();
            }
            */
            
            var checker = true;
            for(int i=0;i<8;i++){
                if(map2[i,p[i]] == 'X'){
                    checker = false;
                    break;
                }
                if(map2[i,p[i]] == 'Q') continue;
                if(map2[i,p[i]] == '.'){
                    map2[i,p[i]] = 'Q';
                    for(int j=0;j<8;j++){
                        var zat = i;
                        var zay = p[i];
                        while(0<=zat+movet[j] && zat+movet[j]<8 && 0<=zay+movey[j] && zay+movey[j]<8){
                            zat += movet[j];
                            zay += movey[j];
                            map2[zat,zay] = 'X';
                        }
                        
                        /*
                        if(p[1] == 0 &&p[6] == 1&&p[2] == 2&&p[5] == 3 &&p[7] == 4&&p[4] == 5&&p[0] == 6&&p[3] == 7){
                            //p[0] == 1 &&p[1] == 6&&p[2] == 2&&p[3] == 5 &&p[4] == 7&&p[5] == 4&&p[6] == 0&&p[7] == 3
                            for(int q=0;q<8;q++){
                                for(int w=0;w<8;w++){
                                    Write(map2[q,w]);
                                }
                            WriteLine();
                            }
                            WriteLine();
                        }
                        */
            
                    }
                }
            }
            /*
            if(p[0] == 1 &&p[1] == 6&&p[2] == 2&&p[3] == 5 &&p[4] == 7&&p[5] == 4&&p[6] == 0&&p[7] == 3){
                for(int i=0;i<8;i++){
                    for(int j=0;j<8;j++){
                        Write(map2[i,j]);
                    }
                    WriteLine();
                }
            }
            */
            if(checker){
                for(int i=0;i<8;i++){
                    for(int j=0;j<8;j++){
                        if(map2[i,j] == 'Q'){
                            Write('Q');
                        }else{
                            Write('.');
                        }
                    }
                    WriteLine();
                }
                break;
            }
            var nextper = per.next();
            if(!nextper) break;
            //break;
        }
        
        
    }
    public static void swap(ref int a,ref int b){int temp = a;a= b;b = temp;}
    static void charswap(ref char a,ref char b){char temp = a;a= b;b = temp;}
    static int ncr(int n,int r){if(n<r)return 0;r = Min(r,n-r);long nn = 1;for(int i=n-r+1;i<=n;i++){nn = nn*i%mod;}long rr = 1;for(int i=1;i<=r;i++){rr = rr*i%mod;}rr = square((int)rr,mod-2);nn = nn * rr %mod;return (int)nn;}
    // a^b mod
    static int square(int a,int b){string binary = Convert.ToString(b,2);int bileng = binary.Length;long a_power = a;long value = 1;for(int i=bileng-1;i>=0;i--){if(binary[i] == '1'){value = value*a_power%mod;}a_power = a_power*a_power%mod;}return (int)value;}
    static int square2(int a,int b){long output = 1;var list = new List<long>();int sh = 1;long n = a;list.Add(a);while(sh < b){sh *= 2;n = n*n%mod;list.Add(n);}for(int i=list.Count-1;i>=0;i--){if(b > sh){b -= sh;sh /= 2;output = output*list[i]%mod;}}return (int)output;}
    //各種読取
    static string rstr(){ return ReadLine(); }
    static int rint(){ return int.Parse(ReadLine()); }
    static long rlong(){ return long.Parse(ReadLine()); }
    static string[] stra(){ return ReadLine().Split(' '); }
    static char[] chara(){ string[] a=stra();string b="";for(int i=0;i<a.Length;i++){b+=a[i];}return b.ToCharArray();}
    static int[] inta(){ string[] read_str_array = ReadLine().Split(' '); int[] return_int_array = new int[read_str_array.Length]; for(int countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_int_array[countup_i] = int.Parse(read_str_array[countup_i]); } return return_int_array; }
    static int[,] inta2(int num_array,int in_array){ int[,] int_array2 = new int[num_array,in_array]; for(int i=0;i<num_array;i++){ int[] temp_array = inta(); for(int j=0;j<in_array;j++){ int_array2[i,j] = temp_array[j]; } } return int_array2; }
    static long[] longa(){ string[] read_str_array = ReadLine().Split(' '); long[] return_long_array = new long[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_long_array[countup_i] = long.Parse(read_str_array[countup_i]); } return return_long_array; }
    static double[] doublea(){ string[] read_str_array = ReadLine().Split(' '); double[] return_double_array = new double[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_double_array[countup_i] = long.Parse(read_str_array[countup_i]); } return return_double_array; }
    // -----------------------------
    static long divideup(long divided,long divid){ long temp_divide = divided/divid; if(divided % divid > 0){ temp_divide++; } return temp_divide; }
    static long GCD(long a,long b){ if(a < b){ long temp = a; a = b; b = temp; } if(a % b == 0){ return b; } else{ long temp = b; b = a%b; a = temp; return GCD(a,b); } }
    static long LCM(long a,long b){ return a * b / GCD(a,b); }
    static void WriteArray(int[,] a,int b,int c){for(int i=0;i<b;i++){for(int j=0;j<c;j++){if(j!=0) Write(" ");Write(a[i,j]);}WriteLine();}}
}

class UnionFind{
    private int[] _unionfind;
    private int[] _unionfind_size;
    
    public UnionFind(int a){
        _unionfind = new int[a];
        _unionfind_size = new int[a];
        for(int i=0;i<a;i++){
            _unionfind[i] = i;
            _unionfind_size[i] = 1;
        }
    }
    
    public int root(int x){
        if(_unionfind[x] == x) return x;
        return _unionfind[x] = root(_unionfind[x]);
    }
    
    public void unite(int a,int b){
        a = root(a);
        b = root(b);
        if(a == b) return;
        if(_unionfind_size[a] < _unionfind_size[b]){
            int temp = a;
            a = b;
            b = temp;
        }
        _unionfind[b] = a;
        _unionfind_size[a] += _unionfind_size[b];
    }
    
    public bool same(int a,int b){
        return root(a) == root(b);
    }
    
    public int size(int x){
        return _unionfind_size[root(x)];
    }
}

class PriorityQueue{
    private List<int> _priorityqueue = new List<int>();
    
    public void Enqueue(int a){
        _priorityqueue.Add(a);
        int num = _priorityqueue.Count-1;
        while(true){
            if(num == 0) break;
            if(_priorityqueue[(num-1)/2] < _priorityqueue[num]){ //大小逆の時弄るとこ
                int temp = _priorityqueue[(num-1)/2];
                _priorityqueue[(num-1)/2] = _priorityqueue[num];
                _priorityqueue[num] = temp;
                num = (num-1)/2;
            }else{
                break;
            }
        }
    }
    
    public int Dequeue(){
        int re = _priorityqueue[0];
        int temp = _priorityqueue[0];
        _priorityqueue[0] = _priorityqueue[_priorityqueue.Count-1];
        _priorityqueue[_priorityqueue.Count-1] = temp;
        _priorityqueue.RemoveAt(_priorityqueue.Count-1);
        int num = 0;
        while(true){
            int swapnum = -1;
            if((num+1)*2 < _priorityqueue.Count){
                if(_priorityqueue[num*2+1] > _priorityqueue[(num+1)*2]){ //大小逆の時弄るとこ
                    swapnum = num*2+1;
                }else{
                    swapnum = (num+1)*2;
                }
            }else if(num*2+1 < _priorityqueue.Count){
                swapnum = num*2+1;
            }
            if(swapnum == -1) break;
            if(_priorityqueue[swapnum] > _priorityqueue[num]){ //大小逆の時弄るとこ
                temp = _priorityqueue[swapnum];
                _priorityqueue[swapnum] = _priorityqueue[num];
                _priorityqueue[num] = temp;
                num = swapnum;
            }else{
                break;
            }
        }
        return re;
    }
    
    public int Peek(){
        return _priorityqueue[0];
    }
    
    public int Count(){
        return _priorityqueue.Count;
    }
}
class PriorityQueue2{
    private List<int> _priorityqueue = new List<int>();
    private List<int> _accompany_info = new List<int>();
    
    public void Enqueue(int a,int b){
        _priorityqueue.Add(a);
        _accompany_info.Add(b);
        int num = _priorityqueue.Count-1;
        while(true){
            if(num == 0) break;
            if(_priorityqueue[(num-1)/2] > _priorityqueue[num]){ //大小逆の時弄るとこ
                int temp = _priorityqueue[(num-1)/2];
                  _priorityqueue[(num-1)/2] = _priorityqueue[num];
                  _priorityqueue[num] = temp;
                temp = _accompany_info[(num-1)/2];
                  _accompany_info[(num-1)/2] = _accompany_info[num];
                  _accompany_info[num] = temp;
                num = (num-1)/2;
            }else{
                break;
            }
        }
    }
    
    public int[] Dequeue(){
        var re = new int[]{_priorityqueue[0],_accompany_info[0]};
        var qcount = _priorityqueue.Count;
        var temp = _priorityqueue[0]; _priorityqueue[0] = _priorityqueue[qcount-1]; _priorityqueue[qcount-1] = temp;
        temp = _accompany_info[0]; _accompany_info[0] = _accompany_info[qcount-1]; _accompany_info[qcount-1] = temp;
        _priorityqueue.RemoveAt(qcount-1);
        _accompany_info.RemoveAt(qcount-1);
        int num = 0;
        while(true){
            int swapnum = -1;
            if((num+1)*2 < _priorityqueue.Count){
                if(_priorityqueue[num*2+1] < _priorityqueue[(num+1)*2]){ //大小逆の時弄るとこ
                    swapnum = num*2+1;
                }else{
                    swapnum = (num+1)*2;
                }
            }else if(num*2+1 < _priorityqueue.Count){
                swapnum = num*2+1;
            }
            if(swapnum == -1) break;
            if(_priorityqueue[swapnum] < _priorityqueue[num]){ //大小逆の時弄るとこ
                temp = _priorityqueue[swapnum];_priorityqueue[swapnum] = _priorityqueue[num];_priorityqueue[num] = temp;
                temp = _accompany_info[swapnum];_accompany_info[swapnum] = _accompany_info[num];_accompany_info[num] = temp;
                num = swapnum;
            }else{
                break;
            }
        }
        return re;
    }
    
    public int[] Peek(){
        var temp = new int[]{_priorityqueue[0],_accompany_info[0]};
        return temp;
    }
    
    public int Count(){
        return _priorityqueue.Count;
    }
}
class PriorityQueue3{
    private List<int> _priorityqueue = new List<int>();
    private List<int> _accompany_info = new List<int>();
    private List<int> _accompany_info2 = new List<int>();
    
    public void Enqueue(int a,int b,int c){
        _priorityqueue.Add(a);
        _accompany_info.Add(b);
        _accompany_info2.Add(c);
        int num = _priorityqueue.Count-1;
        while(true){
            if(num == 0) break;
            if(_priorityqueue[(num-1)/2] > _priorityqueue[num]){ //大小逆の時弄るとこ
                int temp = _priorityqueue[(num-1)/2];
                  _priorityqueue[(num-1)/2] = _priorityqueue[num];
                  _priorityqueue[num] = temp;
                temp = _accompany_info[(num-1)/2];
                  _accompany_info[(num-1)/2] = _accompany_info[num];
                  _accompany_info[num] = temp;
                  temp = _accompany_info2[(num-1)/2];
                  _accompany_info2[(num-1)/2] = _accompany_info2[num];
                  _accompany_info2[num] = temp;
                num = (num-1)/2;
            }else{
                break;
            }
        }
    }
    
    public int[] Dequeue(){
        var re = new int[]{_priorityqueue[0],_accompany_info[0],_accompany_info2[0]};
        var qcount = _priorityqueue.Count;
        var temp = _priorityqueue[0]; _priorityqueue[0] = _priorityqueue[qcount-1]; _priorityqueue[qcount-1] = temp;
        temp = _accompany_info[0]; _accompany_info[0] = _accompany_info[qcount-1]; _accompany_info[qcount-1] = temp;
        temp = _accompany_info2[0]; _accompany_info2[0] = _accompany_info2[qcount-1]; _accompany_info2[qcount-1] = temp;
        _priorityqueue.RemoveAt(qcount-1);
        _accompany_info.RemoveAt(qcount-1);
        _accompany_info2.RemoveAt(qcount-1);
        int num = 0;
        while(true){
            int swapnum = -1;
            if((num+1)*2 < _priorityqueue.Count){
                if(_priorityqueue[num*2+1] < _priorityqueue[(num+1)*2]){ //大小逆の時弄るとこ
                    swapnum = num*2+1;
                }else{
                    swapnum = (num+1)*2;
                }
            }else if(num*2+1 < _priorityqueue.Count){
                swapnum = num*2+1;
            }
            if(swapnum == -1) break;
            if(_priorityqueue[swapnum] < _priorityqueue[num]){ //大小逆の時弄るとこ
                temp = _priorityqueue[swapnum];_priorityqueue[swapnum] = _priorityqueue[num];_priorityqueue[num] = temp;
                temp = _accompany_info[swapnum];_accompany_info[swapnum] = _accompany_info[num];_accompany_info[num] = temp;
                temp = _accompany_info2[swapnum];_accompany_info2[swapnum] = _accompany_info2[num];_accompany_info2[num] = temp;
                num = swapnum;
            }else{
                break;
            }
        }
        return re;
    }
    
    public int[] Peek(){
        var temp = new int[]{_priorityqueue[0],_accompany_info[0],_accompany_info2[0]};
        return temp;
    }
    
    public int Count(){
        return _priorityqueue.Count;
    }
}

class Permutation{
    private int[] _permutation;
    private int _length;
    
    public Permutation(int a){
        _permutation = new int[a];
        for(int i=0;i<a;i++){
            _permutation[i] = i;
        }
        _length = a;
    }
    public int[] now(){
        return _permutation;
    }
    
    public bool next(){
        var upper = -1;
        for(int i=0;i<_length-1;i++){
            if(_permutation[i] < _permutation[i+1]) upper = i;
        }
        if(upper == -1){
            return false;
        }
        var bigger = -1;
        for(int i=upper;i<_length;i++){
            if(_permutation[upper] <= _permutation[i]) bigger = i;
        }
        var temp = _permutation[upper];
        _permutation[upper] = _permutation[bigger];
        _permutation[bigger] = temp;
        var list = new List<int>();
        upper++;
        for(int i=upper;i<_length;i++){
            list.Add(_permutation[i]);
        }
        list.Reverse();
        for(int i=upper;i<_length;i++){
            _permutation[i] = list[i-upper];
        }
        return true;
    }
    
    public void prev0(){
        Array.Reverse(_permutation);
    }
    
    public bool prev(){
        var lower = -1;
        for(int i=0;i<_length-1;i++){
            if(_permutation[i] > _permutation[i+1]) lower = i;
        }
        if(lower == -1){
            return false;
        }
        var smaller = -1;
        for(int i=lower;i<_length;i++){
            if(_permutation[lower] >= _permutation[i]) smaller = i;
        }
        var temp = _permutation[lower];
        _permutation[lower] = _permutation[smaller];
        _permutation[smaller] = temp;
        var list = new List<int>();
        lower++;
        for(int i=lower;i<_length;i++){
            list.Add(_permutation[i]);
        }
        list.Reverse();
        for(int i=lower;i<_length;i++){
            _permutation[i] = list[i-lower];
        }
        return true;
    }
    
}