Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

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 JAVA (judge verdict: Accepted)

import java.io.InputStream;
import java.io.IOException;
import java.io.PrintStream;
import java.math.*;
import java.util.*;

public class Main {

  public static void main(String[] args) {
    solve(System.in, System.out);
  }

  private static class Point {

    public Point(int x, int y) {
      this.x = x;
      this.y = y;
    }

    int x;
    int y;

    @Override
    public boolean equals(Object o) {
      if (this == o) {
        return true;
      }
      if (o == null || getClass() != o.getClass()) {
        return false;
      }
      Point point = (Point) o;
      return x == point.x &&
          y == point.y;
    }

    @Override
    public int hashCode() {
      return Objects.hash(x, y);
    }
  }

  static void solve(InputStream is, PrintStream os) {
    Scanner sc = new Scanner(is);

    /* read */

    int k = sc.nextInt();
    Set<Point> existing = new HashSet<>();
    for (int i = 0; i < k; i++) {
      int x = sc.nextInt();
      int y = sc.nextInt();
      existing.add(new Point(x, y));
    }

    Set<Point> ans = travel(existing);

    boolean[][] grid = new boolean[8][8];

    ans.forEach(p -> grid[p.x][p.y] = true);

    for (int y = 0; y < 8; y++) {
      StringBuilder s = new StringBuilder();
      for (int x = 0; x < 8; x++) {
        if (grid[y][x]) s.append('Q'); else s.append('.');
      }
      os.println(s.toString());
    }
  }

  private static Set<Point> travel(Set<Point> existing) {
    if (existing.size() == 8) {
      return existing;
    }
    for (int x = 0; x < 8; x++) {
      for (int y = 0; y < 8; y++) {
        Point p = new Point(x, y);
        if (canPlace(p, existing)) {
          Set<Point> next = new HashSet<>(existing);
          next.add(p);
          Set<Point> result = travel(next);
          if (result != null) {
            return result;
          }
        }
      }
    }
    return null;
  }

  private static boolean canPlace(Point candidate, Set<Point> existing) {
    int x;
    int y;

    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      x++;
      y++;
    }
    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      x++;
      y--;
    }
    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      x--;
      y++;
    }
    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      x--;
      y--;
    }

    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      x++;
    }

    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      x--;
    }

    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      y++;
    }

    x = candidate.x;
    y = candidate.y;
    while (0 <= x && x < 8 && 0 <= y && y < 8) {
      if (existing.contains(new Point(x, y))) {
        return false;
      }
      y--;
    }

    return true;

  }

  private static long gcd(long m, long n) {
    if (m < n) {
      return gcd(n, m);
    }
    if (n == 0) {
      return m;
    }
    return gcd(n, m % n);
  }

  private static long lcm(long m, long n) {
    return m / gcd(m, n) * n;
  }

  private static class Modular {

    private final long mod;

    private Modular(long mod) {
      this.mod = mod;
    }

    private long inverse(long n) {
      return pow(n, mod - 2) % mod;
    }

    private long pow(long x, long n) {
      /*
      long ans = 1;
      while (n > 0) {
        if ((n & 1) == 1) {
          ans = ans * x % mod;
        }
        x = x * x % mod;
        n >>= 1;
      }
      return ans;
      */
      if (n == 0) {
        return 1;
      }
      return n % 2 == 0 ?
          pow(x * x % mod, n / 2) % mod :
          x % mod * pow(x, n - 1) % mod;
    }
  }

  private static class ModularNFixedCombination {

    private final Modular modular;
    private final long[] comb;
    private final long N;

    public ModularNFixedCombination(long N, int size, long mod) {
      this.modular = new Modular(mod);
      this.comb = new long[size + 1];
      this.N = N;

      comb[0] = 1;

      for (int i = 1; i <= size; i++) {
        comb[i] = ((comb[i - 1] * (N - (i - 1))) % mod * modular.inverse(i)) % mod;
      }
    }

    long combination(int n, int k) {
      if (n != N) {
        throw new IllegalStateException();
      }
      return comb[k];
    }

    long repeatedCombination(int n, int k) {
      return combination(n + k - 1, k);
    }
  }

  private static class ModularCombination {

    private final long fact[];
    private final long mod;
    private final Modular modular;

    public ModularCombination(int size, long mod) {
      this.fact = new long[size + 1];
      this.mod = mod;
      this.modular = new Modular(mod);

      this.fact[0] = 1;
      for (int i = 1; i <= size; i++) {
        fact[i] = (fact[i - 1] * i) % mod;
      }
    }

    private long factorial(int n) {
      return fact[n];
    }

    long combination(int n, int k) {
      return
          (
              (
                  (
                      factorial(n) * modular.inverse(factorial(n - k))
                  ) % mod
              ) * modular.inverse(factorial(k))
          ) % mod;
    }

    long repeatedCombination(int n, int k) {
      return combination(n + k - 1, k);
    }
  }

  private static class Scanner {

    private final InputStream is;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    Scanner(InputStream is) {
      this.is = is;
    }

    private boolean hasNextByte() {
      if (ptr < buflen) {
        return true;
      } else {
        ptr = 0;
        try {
          buflen = is.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (buflen <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) {
        return buffer[ptr++];
      } else {
        return -1;
      }
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    public boolean hasNext() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) {
        ptr++;
      }
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) {
        throw new NoSuchElementException();
      }
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    public long nextLong() {
      if (!hasNext()) {
        throw new NoSuchElementException();
      }
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    public int nextInt() {
      long nl = nextLong();
      if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) {
        throw new NumberFormatException();
      }
      return (int) nl;
    }

    public double nextDouble() {
      return Double.parseDouble(next());
    }
  }
}