Given a N x N Chessboard. The task is to count distinct rectangles from the chessboard. For example, if the input is 8 then the output should be 36.
Examples:
Input: N = 4 Output: 10 Input: N = 6 Output: 21
Approach:
Suppose N = 8 i.e. 8 x 8 chessboard is given, So different rectangles that can be formed are:
1 x 1, 1 x 2, 1 x 3, 1 x 4, 1 x 5, 1 x 6, 1 x 7, 1 x 8 = 8
2 x 2, 2 x 3, 2 x 4, 2 x 5, 2 x 6, 2 x 7, 2 x 8 = 7
3 x 3, 3 x 4, 3 x 5, 3 x 6, 2 x 7, 3 x 8 = 6
4 x 4, 4 x 5, 4 x 6, 4 x 7, 4 x 8 = 5
5 x 5, 5 x 6, 5 x 7, 5 x 8 = 4
6 x 6, 6 x 7, 6 x 8 = 3
7 x 7, 7 x 8 = 2
8 x 8 = 1
So total unique rectangles formed = 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36 which is the sum of the first 8 natural numbers. So in general, distinct rectangles that can be formed in N x N chessboard is:
Sum of the first N natural numbers = N*(N+1)/2
= 8*(8+1)/2
= 36
Below is the implementation of the above approach:
C++
// C++ code to count distinct rectangle in a chessboard#include <bits/stdc++.h> using namespace std; // Function to return the count // of distinct rectangles int count(int N){ int a = 0; a = (N * (N + 1)) / 2; return a;}// Driver Codeint main(){ int N = 4; cout<<count(N); }// This code is contributed by nidhi16bcs2007 |
Java
// Java program to count unique rectangles in a chessboardclass Rectangle { // Function to count distinct rectangles static int count(int N) { int a = 0; a = (N * (N + 1)) / 2; return a; } // Driver Code public static void main(String args[]) { int n = 4; System.out.print(count(n)); }} |
Python3
# Python code to count distinct rectangle in a chessboard# Function to return the count # of distinct rectangles def count(N): a = 0; a = (N * (N + 1)) / 2; return int(a);# Driver CodeN = 4;print(count(N)); # This code has been contributed by 29AjayKumar |
C#
// C# program to count unique rectangles in a chessboard using System;class Rectangle { // Function to count distinct rectangles static int count(int N) { int a = 0; a = (N * (N + 1)) / 2; return a; } // Driver Code public static void Main() { int n = 4; Console.Write(count(n)); } } // This code is contributed by AnkitRai01 |
Javascript
// Javascript program to count unique rectangles in a chessboard // Function to count distinct rectangles function count(N) { var a = 0; a = (N * (N + 1)) / 2; return a; } // Driver Code var n = 4; document.write(count(n)); // This code is contributed by bunnyram19. |
Output:
10
Time Complexity: O(1)
Auxiliary Space: O(1)
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