Maximum bishops that can be placed on N*N chessboard

Given an integer n, the task is to print the maximum number of bishops that can be placed on a n x n chessboard so that no two bishops attack each other. For example, maximum 2 bishops can be placed safely on 2 x 2 chessboard.

Examples:

Input: n = 2
Output: 2
We can place two bishop in a row.



Input: n = 5
Output: 8

Approach: A bishop can travel in any of the four diagonals. Therefore we can place bishops if it is not in any diagonal of another bishop. The maximum bishops that can be placed on an n * n chessboard will be 2 * (n – 1).

  1. Place n bishops in first row
  2. Place n-2 bishops in last row. We only leave two corners of last row

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <iostream>
using namespace std;
  
// Function to return the maximum number of bishops
// that can be placed on an n * n chessboard
int numberOfBishops(int n)
{
    if (n < 1)
        return 0;
    else if (n == 1)
        return 1;
    else
        return 2 * (n - 1);
}
  
// Driver code
int main()
{
    int n = 5;
    cout << numberOfBishops(n);
    return 0;
}

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Java

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// Java implementation of the approach
class gfg 
{
      
// Function to return the maximum 
// number of bishops that can be 
// placed on an n * n chessboard
static int numberOfBishops(int n)
{
    if (n < 1)
        return 0;
    else if (n == 1)
        return 1;
    else
        return 2 * (n - 1);
}
  
// Driver code
public static void main(String[] args)
{
    int n = 5;
    System.out.println(numberOfBishops(n));
}
}
  
// This code is contributed by Mukul Singh.

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Python3

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# Python3 implementation of the 
# approach
import math as mt
  
# Function to return the maximum number 
# of bishops that can be placed on an 
# n * n chessboard
def numberOfBishops(n):
    if (n < 1):
        return 0
    elif (n == 1):
        return 1
    else:
        return 2 * (n - 1)
  
# Driver code
n = 5
print(numberOfBishops(n))
  
# This code is contributed by 
# Mohit kumar 29

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C#

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// C# implementation of the approach
using System;
  
class GFG
{
      
// Function to return the maximum number 
// of bishops that can be placed on an 
// n * n chessboard
static int numberOfBishops(int n)
{
    if (n < 1)
        return 0;
    else if (n == 1)
        return 1;
    else
        return 2 * (n - 1);
}
  
// Driver code
public static void Main()
{
    int n = 5;
    Console.Write(numberOfBishops(n));
}
}
  
// This code is contributed
// by Akanksha Rai

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PHP

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<?php
// PHP implementation of the approach 
  
// Function to return the maximum number 
// of bishops that can be placed on an 
// n * n chessboard 
function numberOfBishops($n
    if ($n < 1) 
        return 0; 
    else if ($n == 1) 
        return 1; 
    else
        return 2 * ($n - 1); 
  
// Driver code 
$n = 5; 
echo numberOfBishops($n);
  
// This code is contributed by Ryuga
?>

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Output:

8

Below is the implementation for bigger values of n:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the difference of 
// two big numbers as string
string substract(string str1, string str2)
{
    string res = "";
    int n1 = str1.length();
    int n2 = str2.length();
  
    // To make substraction easy
    reverse(str1.begin(), str1.end());
    reverse(str2.begin(), str2.end());
  
    int carry = 0;
  
    for (int i = 0; i < n2; i++) {
  
        // Substract digit by bdigit
        int subst = ((str1[i] - '0')
                     - (str2[i] - '0') - carry);
  
        if (subst < 0) {
            subst = subst + 10;
            carry = 1;
        }
        else
            carry = 0;
  
        // Change subst as character and
        // add it to result string
        res.push_back(subst + '0');
    }
  
    for (int i = n2; i < n1; i++) {
        int subst = ((str1[i] - '0') - carry);
  
        if (subst < 0) {
            subst = subst + 10;
            carry = 1;
        }
        else
            carry = 0;
  
        res.push_back(subst + '0');
    }
  
    // Reverse result to make it actual number
    reverse(res.begin(), res.end());
  
    return res;
}
  
string NumberOfBishops(string a)
{
    if (a == "1")
        return a;
    else {
  
        // Substract 1 from number
        a = substract(a, "1");
  
        // Reverse the string to make calculations easier
        reverse(a.begin(), a.end());
  
        int carry = 0;
  
        // Multiply by 2
        for (int i = 0; i < a.size(); i++) {
            int tmp = a[i] - '0';
            tmp *= 2;
            tmp += carry;
            a[i] = '0' + (tmp % 10);
            carry = tmp / 10;
        }
        if (carry > 0)
            a += ('0' + carry);
  
        // Reverse the string to get actual result
        reverse(a.begin(), a.end());
  
        // Return result
        return a;
    }
}
  
// Driver code
int main()
{
    string a = "12345678901234567890";
    cout << NumberOfBishops(a) << endl;
  
    return 0;
}

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Python3

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# Python3 implementation of the approach 
  
# Function to return the difference 
# of two big numbers as string 
def substract(str1, str2): 
  
    res = "" 
    n1 = len(str1) 
    n2 = len(str2) 
  
    # To make substraction easy,
    # reverse the strings
    str1 = str1[::-1
    str2 = str2[::-1]
    carry = 0
  
    for i in range(0, n2): 
  
        # Substract digit by bdigit 
        subst = int(str1[i]) - int(str2[i]) - carry 
  
        if subst < 0
            subst = subst + 10
            carry = 1
          
        else:
            carry = 0
  
        # Change subst as character and 
        # add it to result string 
        res += str(subst) 
  
    for i in range(n2, n1): 
        subst = int(str1[i]) - carry 
  
        if subst < 0:
            subst = subst + 10
            carry = 1
          
        else:
            carry = 0
  
        res += str(subst) 
  
    # Reverse result to make it
    # actual number 
    return res[::-1]
  
def NumberOfBishops(a): 
  
    if a == "1":
        return
          
    else
          
        # Substract 1 from number 
        a = substract(a, "1"
        carry = 0
  
        # Reverse the string to make 
        # calculations easier. Convert the 
        # string to list to manipulate it
        # as strings are immutable in python
        a = list(a[::-1])
          
        # Multiply by 2 
        for i in range(0, len(a)): 
            tmp = (int(a[i]) * 2) + carry 
            a[i] = str(tmp % 10
            carry = tmp // 10
          
        # Convert the list back to string
        a = ''.join(a)
        if carry > 0:
            a += str(carry) 
  
        # Reverse the string to get 
        # actual result 
        return a[::-1
  
# Driver code 
if __name__ == "__main__":
  
    a = "12345678901234567890"
    print(NumberOfBishops(a)) 
  
# This code is contributed 
# by Rituraj Jain

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Output:

24691357802469135778


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