Place K-knights such that they do not attack each other

16 Jun, 2022
Given integers M, N and K, the task is to place K knights on an M*N chessboard such that they don’t attack each other. The knights are expected to be placed on different squares on the board. A knight can move two squares vertically and one square horizontally or two squares horizontally and one square vertically. The knights attack each other if one of them can reach the other in single move. There are multiple ways of placing K knights on an M*N board or sometimes, no way of placing them. We are expected to list out all the possible solutions. 
Examples:
Input: M = 3, N = 3, K = 5 Output: K A K A K A K A K A K A K K K A K A Total number of solutions : 2  
Input: M = 5, N = 5, K = 13 Output: K A K A K A K A K A K A K A K A K A K A K A K A K Total number of solutions : 1
Approach: This problem can be solved using backtracking. The idea is to place the knights one by one starting from first row and first column and moving forward to first row and second column such that they don’t attack each other. When one row gets over, we move to the next row. Before placing a knight, we always check if the block is safe i.e. it is not an attacking position of some other knight. If it is safe, we place the knight and mark it’s attacking position on the board else we move forward and check for other blocks. While following this procedure, we make a new board every time we insert a new knight into our board. This is done because if we get one solution and we need other solutions, then we can backtrack to our old board with old configuration of knights which can then be checked for other possible solutions. The process of backtracking is continued till we get all our possible solutions. Below is the implementation of the above approach: 
CPP
Java
// C++ implementation of the above approach#include <iostream>using namespace std; /* m*n is the board dimensionk is the number of knights to be placed on boardcount is the number of possible solutions */int m, n, k;int count = 0; /* This function is used to create an empty m*n board */void makeBoard(char** board){    for (int i = 0; i < m; i++) {        for (int j = 0; j < n; j++) {            board[i][j] = '_';        }    }} /* This function displays our board */void displayBoard(char** board){    for (int i = 0; i < m; i++) {        for (int j = 0; j < n; j++) {            cout << " " << board[i][j] << " ";        }        cout << endl;    }    cout << endl;} /* This function marks all the attackingposition of a knight placed at board[i][j]position */void attack(int i, int j, char a, char** board){     /* conditions to ensure that the    block to be checked is inside the board */    if ((i + 2) < m && (j - 1) >= 0) {        board[i + 2][j - 1] = a;    }    if ((i - 2) >= 0 && (j - 1) >= 0) {        board[i - 2][j - 1] = a;    }    if ((i + 2) < m && (j + 1) < n) {        board[i + 2][j + 1] = a;    }    if ((i - 2) >= 0 && (j + 1) < n) {        board[i - 2][j + 1] = a;    }    if ((i + 1) < m && (j + 2) < n) {        board[i + 1][j + 2] = a;    }    if ((i - 1) >= 0 && (j + 2) < n) {        board[i - 1][j + 2] = a;    }    if ((i + 1) < m && (j - 2) >= 0) {        board[i + 1][j - 2] = a;    }    if ((i - 1) >= 0 && (j - 2) >= 0) {        board[i - 1][j - 2] = a;    }} /* If the position is empty,place the knight */bool canPlace(int i, int j, char** board){    if (board[i][j] == '_')        return true;    else        return false;} /* Place the knight at [i][j] positionon board */void place(int i, int j, char k, char a, char** board,           char** new_board){     /* Copy the configurations of    old board to new board */    for (int y = 0; y < m; y++) {        for (int z = 0; z < n; z++) {            new_board[y][z] = board[y][z];        }    }     /* Place the knight at [i][j]    position on new board */    new_board[i][j] = k;     /* Mark all the attacking positions    of newly placed knight on the new board */    attack(i, j, a, new_board);} /* Function for placing knights on boardsuch that they don't attack each other */void kkn(int k, int sti, int stj, char** board){     /* If there are no knights left to be placed,    display the board and increment the count */    if (k == 0) {        displayBoard(board);        count++;    }    else {         /* Loop for checking all thepositions on m*n board */        for (int i = sti; i < m; i++) {            for (int j = stj; j < n; j++) {                 /* Is it possible to place knight at[i][j] position on board? */                if (canPlace(i, j, board)) {                     /* Create a new board and place the                    new knight on it */                    char** new_board = new char*[m];                    for (int x = 0; x < m; x++) {                        new_board[x] = new char[n];                    }                    place(i, j, 'K', 'A', board, new_board);                     /* Call the function recursively for                    (k-1) leftover knights */                    kkn(k - 1, i, j, new_board);                     /* Delete the new board                    to free up the memory */                    for (int x = 0; x < m; x++) {                        delete[] new_board[x];                    }                    delete[] new_board;                }            }            stj = 0;        }    }} // Driver codeint main(){    m = 4, n = 3, k = 6;     /* Creation of a m*n board */    char** board = new char*[m];    for (int i = 0; i < m; i++) {        board[i] = new char[n];    }     /* Make all the places are empty */    makeBoard(board);     kkn(k, 0, 0, board);     cout << endl << "Total number of solutions : " << count;    return 0;}
// Java implementation of the above approach class GFG {     /* m*n is the board dimension    k is the number of knights to be placed on board    count is the number of possible solutions */    static int m, n, k;    static int count = 0;     /* This function is used to create an empty m*n board */    static void makeBoard(char[][] board)    {        for (int i = 0; i < m; i++) {            for (int j = 0; j < n; j++) {                board[i][j] = '_';            }        }    }     /* This function displays our board */    static void displayBoard(char[][] board)    {        for (int i = 0; i < m; i++) {            for (int j = 0; j < n; j++) {                System.out.print(" " + board[i][j] + " ");            }            System.out.println();        }        System.out.println();    }     /* This function marks all the attacking    position of a knight placed at board[i][j]    position */    static void attack(int i, int j, char a, char[][] board)    {         /* conditions to ensure that the        block to be checked is inside the board */        if ((i + 2) < m && (j - 1) >= 0) {            board[i + 2][j - 1] = a;        }        if ((i - 2) >= 0 && (j - 1) >= 0) {            board[i - 2][j - 1] = a;        }        if ((i + 2) < m && (j + 1) < n) {            board[i + 2][j + 1] = a;        }        if ((i - 2) >= 0 && (j + 1) < n) {            board[i - 2][j + 1] = a;        }        if ((i + 1) < m && (j + 2) < n) {            board[i + 1][j + 2] = a;        }        if ((i - 1) >= 0 && (j + 2) < n) {            board[i - 1][j + 2] = a;        }        if ((i + 1) < m && (j - 2) >= 0) {            board[i + 1][j - 2] = a;        }        if ((i - 1) >= 0 && (j - 2) >= 0) {            board[i - 1][j - 2] = a;        }    }     /* If the position is empty,    place the knight */    static boolean canPlace(int i, int j, char[][] board)    {        if (board[i][j] == '_')            return true;        else            return false;    }     /* Place the knight at [i][j] position    on board */    static void place(int i, int j, char k, char a,                      char[][] board, char[][] new_board)    {         /* Copy the configurations of        old board to new board */        for (int y = 0; y < m; y++) {            for (int z = 0; z < n; z++) {                new_board[y][z] = board[y][z];            }        }         /* Place the knight at [i][j]        position on new board */        new_board[i][j] = k;         /* Mark all the attacking positions        of newly placed knight on the new board */        attack(i, j, a, new_board);    }     /* Function for placing knights on board    such that they don't attack each other */    static void kkn(int k, int sti, int stj, char[][] board)    {         /* If there are no knights left to be placed,        display the board and increment the count */        if (k == 0) {            displayBoard(board);            count++;        }        else {             /* Loop for checking all the    positions on m*n board */            for (int i = sti; i < m; i++) {                for (int j = stj; j < n; j++) {                     /* Is it possible to place knight at    [i][j] position on board? */                    if (canPlace(i, j, board)) {                         /* Create a new board and place                        the new knight on it */                        char[][] new_board = new char[m][];                        for (int x = 0; x < m; x++) {                            new_board[x] = new char[n];                        }                        place(i, j, 'K', 'A', board,                              new_board);                         /* Call the function recursively for                        (k-1) leftover knights */                        kkn(k - 1, i, j, new_board);                    }                }                stj = 0;            }        }    }     // Driver code    public static void main(String[] args)    {        m = 4;        n = 3;        k = 6;        count = 0;         /* Creation of a m*n board */        char[][] board = new char[m][];        for (int i = 0; i < m; i++) {            board[i] = new char[n];        }         /* Make all the places are empty */        makeBoard(board);         kkn(k, 0, 0, board);         System.out.println("\n Total number of solutions : "                           + count);    }} // This code is contributed by jainlovely450
 K  K  K 
 A  A  A 
 A  A  A 
 K  K  K 

 K  A  K 
 A  K  A 
 K  A  K 
 A  K  A 

 A  K  A 
 K  A  K 
 A  K  A 
 K  A  K 


Total number of solutions : 3
jainlovely450
simmytarika5
chessboard-problems
Technical Scripter 2018
Backtracking
Backtracking
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