# Count positions in a chessboard that can be visited by the Queen which are not visited by the King

• Last Updated : 06 Sep, 2021

Given two integers N and M denoting the dimensions of a chessboard, and two integers X and Y denoting the King’s position, i.e. the cell (X, Y). The task is to find the number of cells the Queen can visit that are not visited by the King if it gets replaced by the King. The King visits all his adjacent cells and the Queen can move diagonally, horizontally, and vertically.

Examples:

Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12.

The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future.

Input: N = 8, M = 8, X = 1, Y = 1
Output: 18
Explanation:
Below is the image to show the movement of King and Queen: Suppose K represents King and * represents cells visited by the king and Q represents Queen and # represent the cells visited by the queen.
So, the total squares can be visited by the queen is 18.

Input: N = 2, M = 1, X = 1, Y = 1
Output: 0

Approach: The idea is to first find the total number of positions that the Queen can visit. Then find the total number of positions that the King can visit. The number of cells that can only be visited by the Queen will be the number of cells the King can visit subtracted from the number of cells the Queen can visit. Follow the below steps to solve the problem:

• Initialize queenMoves by 0 and calculate the total moves of the Queen as follows:

If N – X > 0 and M – Y > 0, queenMoves = queenMoves + min(N – X, M – Y)
If X – 1 > 0 and Y – 1 > 0, queenMoves = queenMoves + min(Y – 1, X – 1)
If X – 1 > 0 and M – Y > 0, queenMoves = queenMoves  + min(X – 1, M – Y)
If N – X > 0 and Y – 1 > 0, queenMoves = queenMoves + min(N – X, Y – 1)
At last, add the answer for horizontal and vertical movements as queenMoves = queenMoves + (N – 1) + (M – 1)

• Initialize kingMoves as 0 and calculate King’s moves as follows:

If X + 1 <= N,  kingMoves = kingMoves + 1
If X – 1 > 0,  kingMoves = kingMoves + 1
If Y + 1 <= M,  kingMoves = kingMoves + 1
If Y – 1 > 0,  kingMoves = kingMoves + 1
If X + 1 <= N and Y + 1 <= M,  kingMoves = kingMoves + 1
If X + 1 <= N and Y – 1 > 0,  kingMoves = kingMoves + 1
If X – 1 > 0 and Y – 1 > 0,  kingMoves = kingMoves + 1
If X – 1 > 0 and Y + 1 <= M,  kingMoves = kingMoves + 1

• Print the absolute difference between the Queen’s and King’s calculated in the above steps as the result.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ` `using` `namespace` `std;`   `// Function to print the number of cells` `// only visited by the queen` `int` `Moves_Calculator(``int` `x, ``int` `y, ` `                     ``int` `row, ``int` `col)` `{` `    `  `    ``// Find all the moves` `    ``int` `total_moves = 0;` `    `  `    ``// Find all moves for x + 1, y + 1` `    ``if` `((row - x) > 0 && (col - y) > 0)` `        ``total_moves += min((row - x),` `                           ``(col - y));`   `    ``// Find all moves for x - 1, y - 1` `    ``if` `((y - 1) > 0 && (x - 1) > 0)` `        ``total_moves += min((y - 1),` `                           ``(x - 1));`   `    ``// Find all moves for x - 1, y + 1` `    ``if` `((x - 1) > 0 && (col - y) > 0)` `        ``total_moves += min((x - 1),` `                         ``(col - y));`   `    ``// Find all moves for x + 1, y - 1` `    ``if` `((row - x) > 0 && (y - 1) > 0)` `        ``total_moves += min((row - x), ` `                             ``(y - 1));`   `    ``total_moves += (row - 1) + (col - 1);`   `    ``// Find all squares visited by King` `    ``// x + 1, in same row` `    ``int` `king_moves = 0;` `    `  `    ``if` `(x + 1 <= row)` `        ``king_moves += 1;`   `    ``// x - 1, in same row` `    ``if` `(x - 1 > 0)` `        ``king_moves += 1;`   `    ``// y + 1, in same column` `    ``if` `(y + 1 <= col)` `        ``king_moves += 1;`   `    ``// y - 1, in same column` `    ``if` `(y - 1 > 0)` `        ``king_moves += 1;`   `    ``if` `(x + 1 <= row && y + 1 <= col)` `        ``king_moves += 1;`   `    ``if` `(x + 1 <= row && y - 1 > 0)` `        ``king_moves += 1;`   `    ``if` `(x - 1 > 0 && y - 1 > 0)` `        ``king_moves += 1;`   `    ``if` `(x - 1 > 0 && y + 1 <= col)` `        ``king_moves += 1;`   `    ``// Return answer` `    ``return` `total_moves - king_moves;` `}`   `// Driver Code` `int` `main()` `{` `    `  `    ``// Dimension of Board` `    ``int` `n = 8, m = 8;` `    `  `    ``// Position of Cell` `    ``int` `x = 1, y = 1;` `    `  `    ``// Function Call` `    ``cout << (Moves_Calculator(x, y, m, n));` `    ``return` `0;` `}`   `// This code is contributed by akhilsaini`

## Java

 `// Java program for the above approach` `import` `java.io.*;`   `class` `GFG{`   `// Function to print the number of cells` `// only visited by the queen` `static` `int` `Moves_Calculator(``int` `x, ``int` `y, ` `                            ``int` `row, ``int` `col)` `{` `    `  `    ``// Find all the moves` `    ``int` `total_moves = ``0``;`   `    ``// Find all moves for x + 1, y + 1` `    ``if` `((row - x) > ``0` `&& (col - y) > ``0``)` `        ``total_moves += Math.min((row - x),` `                                ``(col - y));`   `    ``// Find all moves for x - 1, y - 1` `    ``if` `((y - ``1``) > ``0` `&& (x - ``1``) > ``0``)` `        ``total_moves += Math.min((y - ``1``),` `                                ``(x - ``1``));`   `    ``// Find all moves for x - 1, y + 1` `    ``if` `((x - ``1``) > ``0` `&& (col - y) > ``0``)` `        ``total_moves += Math.min((x - ``1``),` `                              ``(col - y));`   `    ``// Find all moves for x + 1, y - 1` `    ``if` `((row - x) > ``0` `&& (y - ``1``) > ``0``)` `        ``total_moves += Math.min((row - x),` `                                  ``(y - ``1``));`   `    ``total_moves += (row - ``1``) + (col - ``1``);`   `    ``// Find all squares visited by King` `    ``// x + 1, in same row` `    ``int` `king_moves = ``0``;` `    ``if` `(x + ``1` `<= row)` `        ``king_moves += ``1``;`   `    ``// x - 1, in same row` `    ``if` `(x - ``1` `> ``0``)` `        ``king_moves += ``1``;`   `    ``// y + 1, in same column` `    ``if` `(y + ``1` `<= col)` `        ``king_moves += ``1``;`   `    ``// y - 1, in same column` `    ``if` `(y - ``1` `> ``0``)` `        ``king_moves += ``1``;`   `    ``if` `(x + ``1` `<= row && y + ``1` `<= col)` `        ``king_moves += ``1``;`   `    ``if` `(x + ``1` `<= row && y - ``1` `> ``0``)` `        ``king_moves += ``1``;`   `    ``if` `(x - ``1` `> ``0` `&& y - ``1` `> ``0``)` `        ``king_moves += ``1``;`   `    ``if` `(x - ``1` `> ``0` `&& y + ``1` `<= col)` `        ``king_moves += ``1``;`   `    ``// Return answer` `    ``return` `total_moves - king_moves;` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `    `  `    ``// Dimension of Board` `    ``int` `n = ``8``, m = ``8``;`   `    ``// Position of Cell` `    ``int` `x = ``1``, y = ``1``;`   `    ``// Function Call` `    ``System.out.println(Moves_Calculator(x, y, m, n));` `}` `}`   `// This code is contributed by akhilsaini`

## Python3

 `# Python3 program for the above approach`   `# Function to print the number of cells` `# only visited by the queen` `def` `Moves_Calculator(x, y, row, col):`   `    ``# Find all the moves` `    ``total_moves ``=` `0` `    `  `    ``# Find all moves for x + 1, y + 1` `    ``if` `(row ``-` `x) > ``0` `and` `(col ``-` `y) > ``0``:` `        ``total_moves ``+``=` `min``((row ``-` `x), (col ``-` `y))`   `    ``# Find all moves for x - 1, y - 1` `    ``if` `(y ``-` `1``) > ``0` `and` `(x ``-` `1``) > ``0``:` `        ``total_moves ``+``=` `min``((y ``-` `1``), (x ``-` `1``))`   `    ``# Find all moves for x - 1, y + 1` `    ``if` `(x ``-` `1``) > ``0` `and` `(col ``-` `y) > ``0``:` `        ``total_moves ``+``=` `min``((x ``-` `1``), (col ``-` `y))`   `    ``# Find all moves for x + 1, y - 1` `    ``if` `(row ``-` `x) > ``0` `and` `(y ``-` `1``) > ``0``:` `        ``total_moves ``+``=` `min``((row ``-` `x), (y ``-` `1``))`   `    ``total_moves ``+``=` `(row ``-` `1``) ``+` `(col ``-` `1``)`   `    ``# Find all squares visited by King` `    ``# x + 1, in same row` `    ``king_moves ``=` `0` `    ``if` `x ``+` `1` `<``=` `m:` `        ``king_moves ``+``=` `1`   `    ``# x - 1, in same row` `    ``if` `x ``-` `1` `> ``0``:` `        ``king_moves ``+``=` `1`   `    ``# y + 1, in same column` `    ``if` `y ``+` `1` `<``=` `n:` `        ``king_moves ``+``=` `1`   `    ``# y - 1, in same column` `    ``if` `y ``-` `1` `> ``0``:` `        ``king_moves ``+``=` `1`   `    ``if` `x ``+` `1` `<``=` `m ``and` `y ``+` `1` `<``=` `n:` `        ``king_moves ``+``=` `1`   `    ``if` `x ``+` `1` `<``=` `m ``and` `y ``-` `1` `> ``0``:` `        ``king_moves ``+``=` `1`   `    ``if` `x ``-` `1` `> ``0` `and` `y ``-` `1` `> ``0``:` `        ``king_moves ``+``=` `1`   `    ``if` `x ``-` `1` `> ``0` `and` `y ``+` `1` `<``=` `n:` `        ``king_moves ``+``=` `1`   `    ``# Return answer` `    ``return` `total_moves ``-` `king_moves`     `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:`   `    ``# Dimension of Board` `    ``n, m ``=` `8``, ``8` `    `  `    ``# Position of Cell` `    ``x, y ``=` `1``, ``1` `    `  `    ``# Function Call` `    ``print``(Moves_Calculator(x, y, m, n))`

## C#

 `// C# program for the above approach` `using` `System;`   `class` `GFG{`   `// Function to print the number of cells` `// only visited by the queen` `static` `int` `Moves_Calculator(``int` `x, ``int` `y, ` `                            ``int` `row, ``int` `col)` `{` `    `  `    ``// Find all the moves` `    ``int` `total_moves = 0;`   `    ``// Find all moves for x + 1, y + 1` `    ``if` `((row - x) > 0 && (col - y) > 0)` `        ``total_moves += Math.Min((row - x),` `                                ``(col - y));`   `    ``// Find all moves for x - 1, y - 1` `    ``if` `((y - 1) > 0 && (x - 1) > 0)` `        ``total_moves += Math.Min((y - 1), ` `                                ``(x - 1));`   `    ``// Find all moves for x - 1, y + 1` `    ``if` `((x - 1) > 0 && (col - y) > 0)` `        ``total_moves += Math.Min((x - 1), ` `                              ``(col - y));`   `    ``// Find all moves for x + 1, y - 1` `    ``if` `((row - x) > 0 && (y - 1) > 0)` `        ``total_moves += Math.Min((row - x),` `                                  ``(y - 1));`   `    ``total_moves += (row - 1) + (col - 1);`   `    ``// Find all squares visited by King` `    ``// x + 1, in same row` `    ``int` `king_moves = 0;` `    ``if` `(x + 1 <= row)` `        ``king_moves += 1;`   `    ``// x - 1, in same row` `    ``if` `(x - 1 > 0)` `        ``king_moves += 1;`   `    ``// y + 1, in same column` `    ``if` `(y + 1 <= col)` `        ``king_moves += 1;`   `    ``// y - 1, in same column` `    ``if` `(y - 1 > 0)` `        ``king_moves += 1;`   `    ``if` `(x + 1 <= row && y + 1 <= col)` `        ``king_moves += 1;`   `    ``if` `(x + 1 <= row && y - 1 > 0)` `        ``king_moves += 1;`   `    ``if` `(x - 1 > 0 && y - 1 > 0)` `        ``king_moves += 1;`   `    ``if` `(x - 1 > 0 && y + 1 <= col)` `        ``king_moves += 1;`   `    ``// Return answer` `    ``return` `total_moves - king_moves;` `}`   `// Driver Code` `public` `static` `void` `Main()` `{`   `    ``// Dimension of Board` `    ``int` `n = 8, m = 8;`   `    ``// Position of Cell` `    ``int` `x = 1, y = 1;`   `    ``// Function Call` `    ``Console.WriteLine(Moves_Calculator(x, y, m, n));` `}` `}`   `// This code is contributed by akhilsaini`

## Javascript

 ``

Output:

`18`

Time Complexity: O(1)
Auxiliary Space: O(1)

My Personal Notes arrow_drop_up
Recommended Articles
Page :