Puzzle | Program to find number of squares in a chessboard

Puzzle: You are provided with a chessboard and are asked to find the number of squares in it. A chessboard is a board with 8 x 8 grids in it as represented below.

Solution: Looking closely at the chessboard we can see that in addition to the 1 x 1 squares, there can be a combination of 2 x 2, 3 x 3, 4 x 4, 5 x 5, 6 x 6, 7 x 7, and 8 x 8 squares too. To get the total number of squares we need to find all the squares formed.

1 x 1: 8 * 8 = 64 squares.
2 x 2: 7 * 7 = 49 squares.
3 x 3: 6 * 6 = 36 squares.
4 x 4: 5 * 5 = 25 squares.
5 x 5: 4 * 4 = 16 squares.
6 x 6: 3 * 3 = 9 squares.
7 x 7: 2 * 2 = 4 squares.
8 x 8: 1 * 1 = 1 square.

Therefore, we have in all = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares in a chessboard.

General Process

Given an n x n grid, count squares in it.

Examples :



Input: n = 2
Output: 5 (4 squares of 1 unit + 1 square of 2 units)

Input: n = 3
Output: 14 (9 squares of 1 unit + 4 square of 2 units 
                                + 1 square of 1 unit)

For a grid of size n*n the total number of squares formed are:

1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1) / 6 

Below is the implementation of the above formula. Since the value of n*(n+1)*(2n+1) can cause overflow for large values of n, below are some interesting tricks used in the program.

  1. long int is used in return.
  2. n * (n + 1) / 2 is evaluated first as the value n*(n+1) will always be a multiple of 2.

Note that overflow may still happen, but above tricks just reduce chances of overflow.

C

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#include <iostream>
using namespace std;
  
// Function to return count of squares;
long int countSquares(int n)
{
    // A better way to write n*(n+1)*(2n+1)/6
    return (n * (n + 1) / 2) * (2*n + 1) / 3;
}
  
int main()
{
  int n = 4;
  cout << "Count of squares is " << countSquares(n);
  return 0;
}

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Java

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// Java  find number of squares in a 
// chessboard
  
  
class GFG 
{
    // Function to return count of squares;
    static int countSquares(int n)
    {
        // A better way to write n*(n+1)*(2n+1)/6
        return (n * (n + 1) / 2) * (2 * n + 1) / 3;
    }
      
    // Driver code
    public static void main (String[] args)
    {
        int n = 3;
        System.out.println("Count of squares is "
                           +countSquares(n));
    }
}
  
// This code is contributed by Anant Agarwal.

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Python3

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# python code to find number 
# of squares in a chessboard
  
# Function to return count
# of squares;
def countSquares(n):
      
    # better way to write
    # n*(n+1)*(2n+1)/6
    return ((n * (n + 1) / 2
           * (2 * n + 1) / 3)
      
# Driver code
n = 4
print("Count of squares is ",
              countSquares(n))
  
# This code is contributed by sam007.

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C#

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// C# find number of squares in a 
// chessboard
using System;
  
public class GFG {
      
    static int countSquares(int n)
    {
        // A better way to write 
        // n*(n+1)*(2n+1)/6
        return (n * (n + 1) / 2) 
                * (2 * n + 1) / 3;
    }
      
    // Driver code
    public static void Main ()
    {
        int n = 4;
        Console.WriteLine("Count of"
                     + "squares is "
                 + countSquares(n));
    }
}
  
// This code is contributed by Sam007.

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PHP

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<?php
// PHP program to find number
// of squares in a chessboard
  
// Function to return
// count of squares;
function countSquares($n)
{
    // A better way to 
    // write n*(n+1)*(2n+1)/6
    return ($n * ($n + 1) / 2) * 
               (2 * $n + 1) / 3;
}
  
// Driver Code
$n = 4;
echo "Count of squares is " ,
            countSquares($n);
  
// This code is contributed 
// by nitin mittal.
?>

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Output :

Count of squares is 30

This article is contributed by Rishabh. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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