Problem Statement:
Given a N*N board with the Knight placed on the first block of an empty board. Moving according to the rules of chess knight must visit each square exactly once. Print the order of each cell in which they are visited.
Input :
N = 8
Output:
0 59 38 33 30 17 8 63
37 34 31 60 9 62 29 16
58 1 36 39 32 27 18 7
35 48 41 26 61 10 15 28
42 57 2 49 40 23 6 19
47 50 45 54 25 20 11 14
56 43 52 3 22 13 24 5
51 46 55 44 53 4 21 12
Following is the Backtracking algorithm for Knight’s tour problem.
If all squares are visited
print the solution
Else
a) Add one of the next moves to solution vector and recursively
check if this move leads to a solution. (A Knight can make maximum
eight moves. We choose one of the 8 moves in this step).
b) If the move chosen in the above step doesn't lead to a solution
then remove this move from the solution vector and try other
alternative moves.
c) If none of the alternatives work then return false (Returning false
will remove the previously added item in recursion and if false is
returned by the initial call of recursion then "no solution exists" )
Following are implementations for Knight’s tour problem. It prints one of the possible solutions in 2D matrix form. Basically, the output is a 2D 8*8 matrix with numbers from 0 to 63 and these numbers show steps made by Knight
// C++ program for Knight Tour problem
#include <bits/stdc++.h>
using namespace std;
#define N 8
int solveKTUtil(int x, int y, int movei, int sol[N][N],
int xMove[], int yMove[]);
/* A utility function to check if i,j are
valid indexes for N*N chessboard */
int isSafe(int x, int y, int sol[N][N])
{
return (x >= 0 && x < N && y >= 0 && y < N
&& sol[x][y] == -1);
}
/* A utility function to print
solution matrix sol[N][N] */
void printSolution(int sol[N][N])
{
for (int x = 0; x < N; x++) {
for (int y = 0; y < N; y++)
cout << " " << setw(2) << sol[x][y] << " ";
cout << endl;
}
}
/* This function solves the Knight Tour problem using
Backtracking. This function mainly uses solveKTUtil()
to solve the problem. It returns false if no complete
tour is possible, otherwise return true and prints the
tour.
Please note that there may be more than one solutions,
this function prints one of the feasible solutions. */
int solveKT()
{
int sol[N][N];
/* Initialization of solution matrix */
for (int x = 0; x < N; x++)
for (int y = 0; y < N; y++)
sol[x][y] = -1;
/* xMove[] and yMove[] define next move of Knight.
xMove[] is for next value of x coordinate
yMove[] is for next value of y coordinate */
int xMove[8] = { 2, 1, -1, -2, -2, -1, 1, 2 };
int yMove[8] = { 1, 2, 2, 1, -1, -2, -2, -1 };
// Since the Knight is initially at the first block
sol[0][0] = 0;
/* Start from 0,0 and explore all tours using
solveKTUtil() */
if (solveKTUtil(0, 0, 1, sol, xMove, yMove) == 0) {
cout << "Solution does not exist";
return 0;
}
else
printSolution(sol);
return 1;
}
/* A recursive utility function to solve Knight Tour
problem */
int solveKTUtil(int x, int y, int movei, int sol[N][N],
int xMove[8], int yMove[8])
{
int k, next_x, next_y;
if (movei == N * N)
return 1;
/* Try all next moves from
the current coordinate x, y */
for (k = 0; k < 8; k++) {
next_x = x + xMove[k];
next_y = y + yMove[k];
if (isSafe(next_x, next_y, sol)) {
sol[next_x][next_y] = movei;
if (solveKTUtil(next_x, next_y, movei + 1, sol,
xMove, yMove)
== 1)
return 1;
else
// backtracking
sol[next_x][next_y] = -1;
}
}
return 0;
}
// Driver Code
int main()
{
// Function Call
solveKT();
return 0;
}
// C# program for
// Knight Tour problem
using System;
class SOLUTION {
static int N = 8;
/* A utility function to
check if i,j are valid
indexes for N*N chessboard */
static bool isSafe(int x, int y, int[, ] sol)
{
return (x >= 0 && x < N && y >= 0 && y < N
&& sol[x, y] == -1);
}
/* A utility function to
print solution matrix sol[N][N] */
static void printSolution(int[, ] sol)
{
for (int x = 0; x < N; x++) {
for (int y = 0; y < N; y++)
Console.Write(sol[x, y] + " ");
Console.WriteLine();
}
}
/* This function solves the
Knight Tour problem using
Backtracking. This function
mainly uses solveKTUtil() to
solve the problem. It returns
false if no complete tour is
possible, otherwise return true
and prints the tour. Please note
that there may be more than one
solutions, this function prints
one of the feasible solutions. */
static bool solveKT()
{
int[, ] sol = new int[8, 8];
/* Initialization of
solution matrix */
for (int x = 0; x < N; x++)
for (int y = 0; y < N; y++)
sol[x, y] = -1;
/* xMove[] and yMove[] define
next move of Knight.
xMove[] is for next
value of x coordinate
yMove[] is for next
value of y coordinate */
int[] xMove = { 2, 1, -1, -2, -2, -1, 1, 2 };
int[] yMove = { 1, 2, 2, 1, -1, -2, -2, -1 };
// Since the Knight is
// initially at the first block
sol[0, 0] = 0;
/* Start from 0,0 and explore
all tours using solveKTUtil() */
if (!solveKTUtil(0, 0, 1, sol, xMove, yMove)) {
Console.WriteLine("Solution does "
+ "not exist");
return false;
}
else
printSolution(sol);
return true;
}
/* A recursive utility function
to solve Knight Tour problem */
static bool solveKTUtil(int x, int y, int movei,
int[, ] sol, int[] xMove,
int[] yMove)
{
int k, next_x, next_y;
if (movei == N * N)
return true;
/* Try all next moves from
the current coordinate x, y */
for (k = 0; k < 8; k++) {
next_x = x + xMove[k];
next_y = y + yMove[k];
if (isSafe(next_x, next_y, sol)) {
sol[next_x, next_y] = movei;
if (solveKTUtil(next_x, next_y, movei + 1,
sol, xMove, yMove))
return true;
else
// backtracking
sol[next_x, next_y] = -1;
}
}
return false;
}
// Driver Code
public static void Main()
{
// Function Call
solveKT();
}
}// Java program for Knight Tour problem
class KnightTour {
static int N = 8;
/* A utility function to check if i,j are
valid indexes for N*N chessboard */
static boolean isSafe(int x, int y, int sol[][])
{
return (x >= 0 && x < N && y >= 0 && y < N
&& sol[x][y] == -1);
}
/* A utility function to print solution
matrix sol[N][N] */
static void printSolution(int sol[][])
{
for (int x = 0; x < N; x++) {
for (int y = 0; y < N; y++)
System.out.print(sol[x][y] + " ");
System.out.println();
}
}
/* This function solves the Knight Tour problem
using Backtracking. This function mainly
uses solveKTUtil() to solve the problem. It
returns false if no complete tour is possible,
otherwise return true and prints the tour.
Please note that there may be more than one
solutions, this function prints one of the
feasible solutions. */
static boolean solveKT()
{
int sol[][] = new int[8][8];
/* Initialization of solution matrix */
for (int x = 0; x < N; x++)
for (int y = 0; y < N; y++)
sol[x][y] = -1;
/* xMove[] and yMove[] define next move of Knight.
xMove[] is for next value of x coordinate
yMove[] is for next value of y coordinate */
int xMove[] = { 2, 1, -1, -2, -2, -1, 1, 2 };
int yMove[] = { 1, 2, 2, 1, -1, -2, -2, -1 };
// Since the Knight is initially at the first block
sol[0][0] = 0;
/* Start from 0,0 and explore all tours using
solveKTUtil() */
if (!solveKTUtil(0, 0, 1, sol, xMove, yMove)) {
System.out.println("Solution does not exist");
return false;
}
else
printSolution(sol);
return true;
}
/* A recursive utility function to solve Knight
Tour problem */
static boolean solveKTUtil(int x, int y, int movei,
int sol[][], int xMove[],
int yMove[])
{
int k, next_x, next_y;
if (movei == N * N)
return true;
/* Try all next moves from the current coordinate
x, y */
for (k = 0; k < 8; k++) {
next_x = x + xMove[k];
next_y = y + yMove[k];
if (isSafe(next_x, next_y, sol)) {
sol[next_x][next_y] = movei;
if (solveKTUtil(next_x, next_y, movei + 1,
sol, xMove, yMove))
return true;
else
sol[next_x][next_y]
= -1; // backtracking
}
}
return false;
}
/* Driver Code */
public static void main(String args[])
{
// Function Call
solveKT();
}
}# Python3 program to solve Knight Tour problem using Backtracking
# Chessboard Size
n = 8
def isSafe(x, y, board):
'''
A utility function to check if i,j are valid indexes
for N*N chessboard
'''
if(x >= 0 and y >= 0 and x < n and y < n and board[x][y] == -1):
return True
return False
def printSolution(n, board):
'''
A utility function to print Chessboard matrix
'''
for i in range(n):
for j in range(n):
print(board[i][j], end=' ')
print()
def solveKT(n):
'''
This function solves the Knight Tour problem using
Backtracking. This function mainly uses solveKTUtil()
to solve the problem. It returns false if no complete
tour is possible, otherwise return true and prints the
tour.
Please note that there may be more than one solutions,
this function prints one of the feasible solutions.
'''
# Initialization of Board matrix
board = [[-1 for i in range(n)]for i in range(n)]
# move_x and move_y define next move of Knight.
# move_x is for next value of x coordinate
# move_y is for next value of y coordinate
move_x = [2, 1, -1, -2, -2, -1, 1, 2]
move_y = [1, 2, 2, 1, -1, -2, -2, -1]
# Since the Knight is initially at the first block
board[0][0] = 0
# Step counter for knight's position
pos = 1
# Checking if solution exists or not
if(not solveKTUtil(n, board, 0, 0, move_x, move_y, pos)):
print("Solution does not exist")
else:
printSolution(n, board)
def solveKTUtil(n, board, curr_x, curr_y, move_x, move_y, pos):
'''
A recursive utility function to solve Knight Tour
problem
'''
if(pos == n**2):
return True
# Try all next moves from the current coordinate x, y
for i in range(8):
new_x = curr_x + move_x[i]
new_y = curr_y + move_y[i]
if(isSafe(new_x, new_y, board)):
board[new_x][new_y] = pos
if(solveKTUtil(n, board, new_x, new_y, move_x, move_y, pos+1)):
return True
# Backtracking
board[new_x][new_y] = -1
return False
# Driver Code
if __name__ == "__main__":
# Function Call
solveKT(n)<script>
// Javascript program for Knight Tour problem
let N = 8;
// A utility function to check if i,j are
// valid indexes for N*N chessboard
function isSafe(x, y, sol)
{
return(x >= 0 && x < N && y >= 0 &&
y < N && sol[x][y] == -1);
}
// A utility function to print solution
// matrix sol[N][N]
function printSolution(sol)
{
for(let x = 0; x < N; x++)
{
for(let y = 0; y < N; y++)
document.write(sol[x][y] + " ");
document.write("<br/>");
}
}
// This function solves the Knight Tour problem
// using Backtracking. This function mainly
// uses solveKTUtil() to solve the problem. It
// returns false if no complete tour is possible,
// otherwise return true and prints the tour.
// Please note that there may be more than one
// solutions, this function prints one of the
// feasible solutions.
function solveKT()
{
let sol = new Array(8);
for(var i = 0; i < sol.length; i++)
{
sol[i] = new Array(2);
}
// Initialization of solution matrix
for(let x = 0; x < N; x++)
for(let y = 0; y < N; y++)
sol[x][y] = -1;
// xMove[] and yMove[] define next move of Knight.
// xMove[] is for next value of x coordinate
// yMove[] is for next value of y coordinate
let xMove = [ 2, 1, -1, -2, -2, -1, 1, 2 ];
let yMove = [ 1, 2, 2, 1, -1, -2, -2, -1 ];
// Since the Knight is initially at the first block
sol[0][0] = 0;
// Start from 0,0 and explore all tours using
// solveKTUtil()
if (!solveKTUtil(0, 0, 1, sol, xMove, yMove))
{
document.write("Solution does not exist");
return false;
}
else
printSolution(sol);
return true;
}
// A recursive utility function to solve Knight
// Tour problem
function solveKTUtil(x, y, movei, sol, xMove, yMove)
{
let k, next_x, next_y;
if (movei == N * N)
return true;
// Try all next moves from the
// current coordinate x, y
for(k = 0; k < 8; k++)
{
next_x = x + xMove[k];
next_y = y + yMove[k];
if (isSafe(next_x, next_y, sol))
{
sol[next_x][next_y] = movei;
if (solveKTUtil(next_x, next_y, movei + 1,
sol, xMove, yMove))
return true;
else
sol[next_x][next_y] = -1; // backtracking
}
}
return false;
}
// Driver code
// Function Call
solveKT();
</script>0 59 38 33 30 17 8 63
37 34 31 60 9 62 29 16
58 1 36 39 32 27 18 7
35 48 41 26 61 10 15 28
42 57 2 49 40 23 6 19
47 50 45 54 25 20 11 14
56 43 52 3 22 13 24 5
51 46 55 44 53 4 21 12