Data Modelling & AI Data Structure & Algorithm

Print all Knight’s tour possible from a starting point on NxN chessboard

31 July 2024

0

Given a N x N chessboard with a Knight initially standing on the X^th row and Y^th column, the task is to print all possible paths such that the knight must visit each square exactly once.

Example:

Input: N = 5, X = 1, Y = 1
Output:
1 6 15 10 21
14 9 20 5 16
19 2 7 22 11
8 13 24 17 4
25 18 3 12 23
1 6 11 18 21
12 17 20 5 10
7 2 15 22 19
16 13 24 9 4
25 8 3 14 23

… 302 more
Explanation: Initially, the knight is at (1, 2)^th cell. According to the 1st path, the knight will visit the cells in the following order: (1, 1) -> (3, 2) -> (5, 3) -> (4, 5) -> (2, 4) … and so on.

Input: N = 3, X = 1, Y = 3
Output: -1
Explanation: There exist no valid sequence of path such that the knight visit each square exactly once.

Approach: The problem can be solved with the help of Recursion and Backtracking by generating all the possible tours one by one and checking if it satisfies the given conditions. A more thorough explanation of the similar approach is discussed in the Knight’s Tour Problem. Below are the steps to follow:

Create a Recursive function to iterate over all possible paths that the Knight can follow.
Maintain the number of squares visited by the Knight using a variable visited.
Create a function isSafe() which takes the coordinates of a square as an argument and returns whether the square is valid for the next Knight’s move.
Iterate through all the possible 8 moves of the Knight and check whether they are safe to visit. If they are, recursively call for the next move until the number of visited squares is equal to the total number of squares.

Time Complexity: O(8^N*N)

Auxiliary Space: O(N^2)

Below is the implementation of the above approach:

C++

// C++ program of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Stores the 8 possible combinations of
// moves that the knight can follow
int DirX[] = { 2, 1, -1, -2, -2, -1, 1, 2 };
int DirY[] = { 1, 2, 2, 1, -1, -2, -2, -1 };
 
// Function to find if (i, j) is a valid
// cell for the knight to move and it
// exists within the chessboard
bool isSafe(int i, int j, int n,
            vector<vector<int> >& Board)
{
    return (i >= 0 and j >= 0 and i < n and j < n
            and Board[i][j] == 0);
}
 
// Stores whether there exist any valid path
bool isPossible = false;
 
// Recursive function to iterate through all
// the paths that the knight can follow
void knightTour(vector<vector<int> >& ChessBoard, int N,
                int x, int y, int visited = 1)
{
    // Mark the current square of the chessboard
    ChessBoard[x][y] = visited;
 
    // If the number of visited squares are equal
    // to the total number of squares
    if (visited == N * N) {
        isPossible = true;
 
        // Print the current state of ChessBoard
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < N; j++) {
                cout << ChessBoard[i][j] << " ";
            }
            cout << endl;
        }
        cout << endl;
 
        // Backtrack to the previous move
        ChessBoard[x][y] = 0;
        return;
    }
 
    // Iterate through all the eight possible moves
    // for a knight
    for (int i = 0; i < 8; i++) {
 
        // Stores the new position of the knight
        // after a move
        int newX = x + DirX[i];
        int newY = y + DirY[i];
 
        // If the new position is a valid position
        // recursively call for the next move
        if (isSafe(newX, newY, N, ChessBoard)
            && !ChessBoard[newX][newY]) {
            knightTour(ChessBoard, N, newX, newY,
                       visited + 1);
        }
    }
 
    // Backtrack to the previous move
    ChessBoard[x][y] = 0;
}
 
// Driver Code
int main()
{
    vector<vector<int> > ChessBoard(5, vector<int>(5, 0));
    int N = ChessBoard.size();
    int X = 1;
    int Y = 1;
 
    knightTour(ChessBoard, N, X - 1, Y - 1);
 
    // If no valid sequence of moves exist
    if (!isPossible) {
        cout << -1;
    }
 
    return 0;
}

Java

// Java program of the above approach
class GFG {
 
    // Stores the 8 possible combinations of
    // moves that the knight can follow
    static int[] DirX = { 2, 1, -1, -2, -2, -1, 1, 2 };
    static int[] DirY = { 1, 2, 2, 1, -1, -2, -2, -1 };
 
    // Function to find if (i, j) is a valid
    // cell for the knight to move and it
    // exists within the chessboard
    static boolean isSafe(int i, int j, int n, int[][] Board) {
        return (i >= 0 && j >= 0 && i < n && j < n && Board[i][j] == 0);
    }
 
    // Stores whether there exist any valid path
    static boolean isPossible = false;
 
    // Recursive function to iterate through all
    // the paths that the knight can follow
    static void knightTour(int[][] ChessBoard, int N, int x, int y, int visited)
    {
        // Mark the current square of the chessboard
        ChessBoard[x][y] = visited;
 
        // If the number of visited squares are equal
        // to the total number of squares
        if (visited == N * N) {
            isPossible = true;
 
            // Print the current state of ChessBoard
            for (int i = 0; i < N; i++) {
                for (int j = 0; j < N; j++) {
                    System.out.print(ChessBoard[i][j] + " ");
                }
                System.out.println();
            }
            System.out.println();
 
            // Backtrack to the previous move
            ChessBoard[x][y] = 0;
            return;
        }
 
        // Iterate through all the eight possible moves
        // for a knight
        for (int i = 0; i < 8; i++) {
 
            // Stores the new position of the knight
            // after a move
            int newX = x + DirX[i];
            int newY = y + DirY[i];
 
            // If the new position is a valid position
            // recursively call for the next move
            if (isSafe(newX, newY, N, ChessBoard)
                && ChessBoard[newX][newY] == 0) {
                knightTour(ChessBoard, N, newX, newY,
                           visited + 1);
            }
        }
 
        // Backtrack to the previous move
        ChessBoard[x][y] = 0;
    }
 
    // Driver Code
    public static void main(String args[]) {
        int[][] ChessBoard = new int[5][5];
 
        int N = ChessBoard.length;
        int X = 1;
        int Y = 1;
 
        knightTour(ChessBoard, N, X - 1, Y - 1, 1);
 
        // If no valid sequence of moves exist
        if (isPossible == false) {
            System.out.println(-1);
        }
    }
}
 
// This code is contributed by Saurabh Jaiswal

Python3

# Python 3 program of the above approach
 
# Stores the 8 possible combinations of
# moves that the knight can follow
DirX = [2, 1, -1, -2, -2, -1, 1, 2]
DirY = [1, 2, 2, 1, -1, -2, -2, -1]
 
# Function to find if (i, j) is a valid
# cell for the knight to move and it
# exists within the chessboard
def isSafe(i, j, n, Board):
    return i >= 0 and j >= 0 and i < n and j < n and Board[i][j] == 0
 
# Stores whether there exist any valid path
isPossible = False
 
# Recursive function to iterate through all
# the paths that the knight can follow
def knightTour(ChessBoard, N, x, y, visited=1):
    global isPossible
     
    # Mark the current square of the chessboard
    ChessBoard[x][y] = visited
 
    # If the number of visited squares are equal
    # to the total number of squares
    if visited == N * N:
        isPossible = True
 
        # Print the current state of ChessBoard
        for i in range(N):
            for j in range(N):
                print(ChessBoard[i][j], end=" ")
            print()
        print()
 
        # Backtrack to the previous move
        ChessBoard[x][y] = 0
        return
 
    # Iterate through all the eight possible moves
    # for a knight
    for i in range(8):
 
        # Stores the new position of the knight
        # after a move
        newX = x + DirX[i]
        newY = y + DirY[i]
 
        # If the new position is a valid position
        # recursively call for the next move
        if isSafe(newX, newY, N, ChessBoard) and not ChessBoard[newX][newY]:
            knightTour(ChessBoard, N, newX, newY, visited + 1)
 
    # Backtrack to the previous move
    ChessBoard[x][y] = 0
 
# Driver Code
if __name__ == "__main__":
    ChessBoard = [[0 for j in range(5)] for i in range(5)]
    N = len(ChessBoard)
    X = 1
    Y = 1
 
    knightTour(ChessBoard, N, X - 1, Y - 1)
 
    # If no valid sequence of moves exist
    if not isPossible:
        print(-1)

Javascript

<script>
// Javascript program of the above approach
 
// Stores the 8 possible combinations of
// moves that the knight can follow
let DirX = [2, 1, -1, -2, -2, -1, 1, 2];
let DirY = [1, 2, 2, 1, -1, -2, -2, -1];
 
// Function to find if (i, j) is a valid
// cell for the knight to move and it
// exists within the chessboard
function isSafe(i, j, n, Board) {
  return i >= 0 && j >= 0 && i < n && j < n && Board[i][j] == 0;
}
 
// Stores whether there exist any valid path
let isPossible = false;
 
// Recursive function to iterate through all
// the paths that the knight can follow
function knightTour(ChessBoard, N, x, y, visited = 1) {
  // Mark the current square of the chessboard
  ChessBoard[x][y] = visited;
 
  // If the number of visited squares are equal
  // to the total number of squares
  if (visited == N * N) {
    isPossible = true;
 
    // Print the current state of ChessBoard
    for (let i = 0; i < N; i++) {
      for (let j = 0; j < N; j++) {
        document.write(ChessBoard[i][j] + " ");
      }
      document.write("<br>");
    }
    document.write("<br>");
 
    // Backtrack to the previous move
    ChessBoard[x][y] = 0;
    return;
  }
 
  // Iterate through all the eight possible moves
  // for a knight
  for (let i = 0; i < 8; i++) {
    // Stores the new position of the knight
    // after a move
    let newX = x + DirX[i];
    let newY = y + DirY[i];
 
    // If the new position is a valid position
    // recursively call for the next move
    if (isSafe(newX, newY, N, ChessBoard) && !ChessBoard[newX][newY]) {
      knightTour(ChessBoard, N, newX, newY, visited + 1);
    }
  }
 
  // Backtrack to the previous move
  ChessBoard[x][y] = 0;
}
 
// Driver Code
 
let ChessBoard = new Array(5).fill(0).map(() => new Array(5).fill(0));
let N = ChessBoard.length;
let X = 1;
let Y = 1;
 
knightTour(ChessBoard, N, X - 1, Y - 1);
 
// If no valid sequence of moves exist
if (!isPossible) {
  document.write("-1");
}
 
</script>

C#

// C# program of the above approach
using System;
class GFG {
 
    // Stores the 8 possible combinations of
    // moves that the knight can follow
    static int[] DirX = { 2, 1, -1, -2, -2, -1, 1, 2 };
    static int[] DirY = { 1, 2, 2, 1, -1, -2, -2, -1 };
 
    // Function to find if (i, j) is a valid
    // cell for the knight to move and it
    // exists within the chessboard
    static bool isSafe(int i, int j, int n, int[, ] Board)
    {
        return (i >= 0 && j >= 0 && i < n && j < n
                && Board[i, j] == 0);
    }
 
    // Stores whether there exist any valid path
    static bool isPossible = false;
 
    // Recursive function to iterate through all
    // the paths that the knight can follow
    static void knightTour(int[, ] ChessBoard, int N, int x,
                           int y, int visited = 1)
    {
        // Mark the current square of the chessboard
        ChessBoard[x, y] = visited;
 
        // If the number of visited squares are equal
        // to the total number of squares
        if (visited == N * N) {
            isPossible = true;
 
            // Print the current state of ChessBoard
            for (int i = 0; i < N; i++) {
                for (int j = 0; j < N; j++) {
                    Console.Write(ChessBoard[i, j] + " ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
 
            // Backtrack to the previous move
            ChessBoard[x, y] = 0;
            return;
        }
 
        // Iterate through all the eight possible moves
        // for a knight
        for (int i = 0; i < 8; i++) {
 
            // Stores the new position of the knight
            // after a move
            int newX = x + DirX[i];
            int newY = y + DirY[i];
 
            // If the new position is a valid position
            // recursively call for the next move
            if (isSafe(newX, newY, N, ChessBoard)
                && ChessBoard[newX, newY] == 0) {
                knightTour(ChessBoard, N, newX, newY,
                           visited + 1);
            }
        }
 
        // Backtrack to the previous move
        ChessBoard[x, y] = 0;
    }
 
    // Driver Code
    public static void Main()
    {
        int[, ] ChessBoard = new int[5, 5];
 
        int N = ChessBoard.GetLength(0);
        int X = 1;
        int Y = 1;
 
        knightTour(ChessBoard, N, X - 1, Y - 1);
 
        // If no valid sequence of moves exist
        if (isPossible == false) {
            Console.WriteLine(-1);
        }
    }
}
 
// This code is contributed by ukasp.

Output

1 6 15 10 21 
14 9 20 5 16 
19 2 7 22 11 
8 13 24 17 4 
25 18 3 12 23 

1 6 11 18 21 
12 17 20 5 10 
7 2 15 22 19 
16 13 24 9 4 
25 8 3 14 23 

1 6 11 16 21 
12 15 20 5 10 
7 2 13 22 17 
14 19 24 9 4 
25 8 3 18 23 

1 6 17 12 21 
16 11 20 5 18 
7 2 9 22 13 
10 15 24 19 4 
25 8 3 14 23 

1 12 17 6 21 
18 5 20 11 16 
13 2 9 22 7 
4 19 24 15 10 
25 14 3 8 23 

1 16 11 6 21 
10 5 20 15 12 
17 2 13 22 7 
4 9 24 19 14 
25 18 3 8 23 

1 18 11 6 21 
10 5 20 17 12 
19 2 15 22 7 
4 9 24 13 16 
25 14 3 8 23 

1 10 15 6 21 
16 5 20 9 14 
11 2 7 22 19 
4 17 24 13 8 
25 12 3 18 23 

1 16 5 10 21 
6 11 20 15 4 
19 2 17 22 9 
12 7 24 3 14 
25 18 13 8 23 

1 12 5 18 21 
6 17 20 13 4 
11 2 9 22 19 
16 7 24 3 14 
25 10 15 8 23 

1 10 5 16 21 
6 15 20 11 4 
9 2 7 22 17 
14 19 24 3 12 
25 8 13 18 23 

1 18 7 12 21 
8 13 20 17 6 
19 2 5 22 11 
14 9 24 3 16 
25 4 15 10 23 

1 6 17 12 21 
18 11 20 7 16 
5 2 15 22 13 
10 19 24 3 8 
25 4 9 14 23 

1 6 15 12 21 
16 11 20 7 14 
5 2 13 22 19 
10 17 24 3 8 
25 4 9 18 23 

1 12 17 8 21 
18 7 20 3 16 
13 2 11 22 9 
6 19 24 15 4 
25 14 5 10 23 

1 16 13 8 21 
12 7 20 3 14 
17 2 15 22 9 
6 11 24 19 4 
25 18 5 10 23 

1 18 13 8 21 
12 7 20 3 14 
19 2 17 22 9 
6 11 24 15 4 
25 16 5 10 23 

1 10 15 8 21 
16 7 20 3 14 
11 2 9 22 19 
6 17 24 13 4 
25 12 5 18 23 

1 4 15 10 21 
14 9 20 3 16 
19 2 5 22 11 
8 13 24 17 6 
25 18 7 12 23 

1 4 9 18 21 
10 17 20 3 8 
5 2 13 22 19 
16 11 24 7 14 
25 6 15 12 23 

1 4 9 16 21 
10 15 20 3 8 
5 2 11 22 17 
14 19 24 7 12 
25 6 13 18 23 

1 4 17 12 21 
16 11 20 3 18 
5 2 7 22 13 
10 15 24 19 8 
25 6 9 14 23 

1 16 3 10 21 
6 11 20 15 4 
17 2 5 22 9 
12 7 24 19 14 
25 18 13 8 23 

1 18 3 12 21 
8 13 20 17 4 
19 2 7 22 11 
14 9 24 5 16 
25 6 15 10 23 

1 8 3 14 21 
18 13 20 9 4 
7 2 17 22 15 
12 19 24 5 10 
25 6 11 16 23 

1 8 3 14 21 
16 13 20 9 4 
7 2 15 22 19 
12 17 24 5 10 
25 6 11 18 23 

1 14 3 8 21 
4 9 20 13 16 
19 2 15 22 7 
10 5 24 17 12 
25 18 11 6 23 

1 14 3 8 21 
4 9 20 13 18 
15 2 17 22 7 
10 5 24 19 12 
25 16 11 6 23 

1 12 3 18 21 
4 17 20 13 8 
11 2 7 22 19 
16 5 24 9 14 
25 10 15 6 23 

1 10 3 16 21 
4 15 20 11 6 
9 2 5 22 17 
14 19 24 7 12 
25 8 13 18 23 

1 22 11 16 7 
12 17 8 21 10 
25 2 23 6 15 
18 13 4 9 20 
3 24 19 14 5 

1 22 11 16 7 
12 17 8 21 10 
23 2 25 6 15 
18 13 4 9 20 
3 24 19 14 5 

1 24 11 16 7 
12 17 8 25 10 
23 2 21 6 15 
18 13 4 9 20 
3 22 19 14 5 

1 22 11 16 7 
12 17 8 23 10 
21 2 19 6 15 
18 13 4 9 24 
3 20 25 14 5 

1 20 11 16 7 
12 25 8 21 10 
19 2 17 6 15 
24 13 4 9 22 
3 18 23 14 5 

1 18 11 24 7 
12 23 8 19 10 
17 2 15 6 25 
22 13 4 9 20 
3 16 21 14 5 

1 16 11 22 7 
12 21 8 17 10 
15 2 13 6 23 
20 25 4 9 18 
3 14 19 24 5 

1 14 25 20 7 
24 19 8 15 10 
13 2 11 6 21 
18 23 4 9 16 
3 12 17 22 5 

1 24 13 18 7 
14 19 8 23 12 
25 2 11 6 17 
20 15 4 9 22 
3 10 21 16 5 

1 12 15 20 7 
16 21 8 25 14 
11 2 13 6 19 
22 17 4 9 24 
3 10 23 18 5 

1 12 17 22 7 
18 23 8 13 16 
11 2 15 6 21 
24 19 4 9 14 
3 10 25 20 5 

1 12 19 24 7 
20 25 8 13 18 
11 2 17 6 23 
16 21 4 9 14 
3 10 15 22 5 

1 12 23 18 7 
24 17 8 13 22 
11 2 21 6 19 
16 25 4 9 14 
3 10 15 20 5 

1 12 25 18 7 
22 17 8 13 24 
11 2 23 6 19 
16 21 4 9 14 
3 10 15 20 5 

1 12 23 18 7 
22 17 8 13 24 
11 2 25 6 19 
16 21 4 9 14 
3 10 15 20 5 

1 12 21 18 7 
22 17 8 13 20 
11 2 19 6 25 
16 23 4 9 14 
3 10 15 24 5 

1 16 21 10 7 
20 11 8 15 22 
25 2 17 6 9 
12 19 4 23 14 
3 24 13 18 5 

1 16 21 10 7 
22 11 8 15 20 
17 2 25 6 9 
12 23 4 19 14 
3 18 13 24 5 

1 16 21 10 7 
22 11 8 15 20 
17 2 23 6 9 
12 25 4 19 14 
3 18 13 24 5 

1 16 25 10 7 
24 11 8 15 20 
17 2 21 6 9 
12 23 4 19 14 
3 18 13 22 5 

1 16 23 10 7 
22 11 8 15 24 
17 2 19 6 9 
12 21 4 25 14 
3 18 13 20 5 

1 24 19 10 7 
18 11 8 25 20 
23 2 15 6 9 
12 17 4 21 14 
3 22 13 16 5 

1 22 17 10 7 
16 11 8 23 18 
21 2 13 6 9 
12 15 4 19 24 
3 20 25 14 5 

1 20 15 10 7 
14 25 8 21 16 
19 2 11 6 9 
24 13 4 17 22 
3 18 23 12 5 

1 18 23 12 7 
24 13 8 17 22 
19 2 11 6 9 
14 25 4 21 16 
3 20 15 10 5 

1 20 25 14 7 
12 15 8 19 24 
21 2 13 6 9 
16 11 4 23 18 
3 22 17 10 5 

1 22 13 16 7 
12 17 8 21 14 
23 2 15 6 9 
18 11 4 25 20 
3 24 19 10 5 

1 24 13 18 7 
12 19 8 23 14 
25 2 17 6 9 
20 11 4 15 22 
3 16 21 10 5 

1 18 13 20 7 
12 21 8 25 14 
17 2 19 6 9 
22 11 4 15 24 
3 16 23 10 5 

1 18 13 22 7 
12 23 8 19 14 
17 2 21 6 9 
24 11 4 15 20 
3 16 25 10 5 

1 18 13 24 7 
12 25 8 19 14 
17 2 23 6 9 
22 11 4 15 20 
3 16 21 10 5 

1 18 13 24 7 
12 23 8 19 14 
17 2 25 6 9 
22 11 4 15 20 
3 16 21 10 5 

1 24 13 18 7 
14 19 8 23 12 
9 2 25 6 17 
20 15 4 11 22 
3 10 21 16 5 

1 24 13 18 7 
14 19 8 25 12 
9 2 23 6 17 
20 15 4 11 22 
3 10 21 16 5 

1 22 13 18 7 
14 19 8 23 12 
9 2 21 6 17 
20 15 4 11 24 
3 10 25 16 5 

1 20 13 18 7 
14 25 8 21 12 
9 2 19 6 17 
24 15 4 11 22 
3 10 23 16 5 

1 18 13 24 7 
14 23 8 19 12 
9 2 17 6 25 
22 15 4 11 20 
3 10 21 16 5 

1 16 13 22 7 
14 21 8 17 12 
9 2 15 6 23 
20 25 4 11 18 
3 10 19 24 5 

1 14 25 20 7 
24 19 8 15 12 
9 2 13 6 21 
18 23 4 11 16 
3 10 17 22 5 

1 12 23 18 7 
22 17 8 13 24 
9 2 11 6 19 
16 21 4 25 14 
3 10 15 20 5 

1 10 15 20 7 
16 21 8 25 14 
9 2 11 6 19 
22 17 4 13 24 
3 12 23 18 5 

1 10 17 22 7 
18 23 8 11 16 
9 2 13 6 21 
24 19 4 15 12 
3 14 25 20 5 

1 10 19 24 7 
20 25 8 11 18 
9 2 15 6 23 
14 21 4 17 12 
3 16 13 22 5 

1 10 23 16 7 
24 15 8 11 22 
9 2 19 6 17 
14 25 4 21 12 
3 20 13 18 5 

1 10 25 16 7 
20 15 8 11 24 
9 2 21 6 17 
14 19 4 23 12 
3 22 13 18 5 

1 10 21 16 7 
20 15 8 11 22 
9 2 23 6 17 
14 19 4 25 12 
3 24 13 18 5 

1 10 21 16 7 
20 15 8 11 22 
9 2 25 6 17 
14 19 4 23 12 
3 24 13 18 5 

1 10 21 16 7 
22 15 8 11 20 
9 2 17 6 25 
14 23 4 19 12 
3 18 13 24 5 

1 18 21 12 7 
20 13 8 17 22 
25 2 19 6 11 
14 9 4 23 16 
3 24 15 10 5 

1 18 23 12 7 
24 13 8 17 22 
19 2 25 6 11 
14 9 4 21 16 
3 20 15 10 5 

1 18 25 12 7 
24 13 8 17 22 
19 2 23 6 11 
14 9 4 21 16 
3 20 15 10 5 

1 18 23 12 7 
22 13 8 17 24 
19 2 21 6 11 
14 9 4 25 16 
3 20 15 10 5 

1 24 19 12 7 
18 13 8 25 20 
23 2 17 6 11 
14 9 4 21 16 
3 22 15 10 5 

1 22 17 12 7 
16 13 8 23 18 
21 2 15 6 11 
14 9 4 19 24 
3 20 25 10 5 

1 20 15 12 7 
14 25 8 21 16 
19 2 13 6 11 
24 9 4 17 22 
3 18 23 10 5 

1 18 13 24 7 
12 23 8 19 14 
17 2 11 6 25 
22 9 4 15 20 
3 16 21 10 5 

1 20 25 14 7 
10 15 8 19 24 
21 2 11 6 13 
16 9 4 23 18 
3 22 17 12 5 

1 22 11 16 7 
10 17 8 21 12 
23 2 13 6 15 
18 9 4 25 20 
3 24 19 14 5 

1 24 11 18 7 
10 19 8 23 12 
25 2 15 6 17 
20 9 4 13 22 
3 14 21 16 5 

1 16 11 20 7 
10 21 8 25 12 
15 2 17 6 19 
22 9 4 13 24 
3 14 23 18 5 

1 16 11 22 7 
10 23 8 17 12 
15 2 19 6 21 
24 9 4 13 18 
3 14 25 20 5 

1 16 11 24 7 
10 25 8 17 12 
15 2 21 6 23 
20 9 4 13 18 
3 14 19 22 5 

1 16 11 22 7 
10 21 8 17 12 
15 2 25 6 23 
20 9 4 13 18 
3 14 19 24 5 

1 16 11 22 7 
10 21 8 17 12 
15 2 23 6 25 
20 9 4 13 18 
3 14 19 24 5 

1 20 7 14 25 
10 15 24 19 8 
21 2 9 6 13 
16 11 4 23 18 
3 22 17 12 5 

1 22 7 16 25 
12 17 24 21 8 
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21 8 19 14 23

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