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

• Difficulty Level : Hard
• Last Updated : 21 Nov, 2018

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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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:

 `// C++ implementation of the above approach ` `#include ` `using` `namespace` `std; ` ` `  `/* m*n is the board dimension ` `k is the number of knights to be placed on board  ` `count 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) ` `{ ` `    ``cout << endl ` `         ``<< endl; ` `    ``for` `(``int` `i = 0; i < m; i++) { ` `        ``for` `(``int` `j = 0; j < n; j++) { ` `            ``cout << ``" "` `<< board[i][j] << ``" "``; ` `        ``} ` `        ``cout << endl; ` `    ``} ` `} ` ` `  `/* This function marks all the attacking ` ` ``position 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] position ` `   ``on 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 board ` `   ``such 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 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 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 code ` `int` `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; ` `} `

Output:

``` 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
```

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