Find number of squares inside a given square grid in C++

In this tutorial, we will check out how to find the number of squares inside a given square grid in C++ with the corresponding code.
If the square grid contains a side of N*N, then we are going to find the total number of squares present in it.

calculate the number of squares inside a square grid in C++

If we observe the number of squares that exist in smaller square grids then we can draw a pattern that will guide us to construct a simple formula to find the number of squares in a square grid of any length.

• The total number of squares in a square grid of side 1 =1.
• Total number of squares in a square grid of side 2 =5.  ⇒ 4 small squares, and 1  2×2 square.
• The total number of squares in a square grid of side 3 =14.  ⇒ 9 small squares, 4  2×2 squares and 1  3×3 square.
• The total number of squares in a square grid of side 4 =30. ⇒ 16 small squares, 9  3×3 squares,  4  2×2 squares and 1  4×4 square.
  By observing above sentences we can draw a pattern
• 1×1 :-1^2 = 1.
• 2×2 :- 2^2 + 1^2 = 5.
• 3×3 :- 3^2 + 2^2 + 1^2 = 14.
• 4×4 :- 4^2 + 3^2 + 2^2 + 1^2 = 30.
• Likewise, for nxn is n^2 + (n-1)^2 + (n-2)^2…………….(n-n+1)^2.

Then N(s)n = n^2 + (n-1)^2 + (n-2)^2................(n-n+1)^2 = n(n + 1)(2n + 1)/6

1.By using the formula N(s)n = n^2 + (n-1)^2 + (n-2)^2…………….(n-n+1)^2:

#include <bits/stdc++.h> 
using namespace std; 
int main() 
{ 
    int n,squares=0; 
    cout<<"enter the length of a square:";
    cin>>n;
    while(n>0)
    {
        squares += pow(n, 2); 
        n--;
    }
    cout<<"no of squares in a square grid:"<<squares;
    return 0; 
}

Output:

enter the length of a square:5
no of squares in a square grid:55

2.By using the formula N(s)n =n(n + 1)(2n + 1)/6:

#include <iostream> 
using namespace std; 
  
int squares(int n) 
{
    
    return n * (n + 1) * (2 * n + 1) / 6;
} 
int main() 
{ 
    int m;
    cout<<"enter length of a square:";
    cin>>m;
    cout <<"no of squares in a square grid:"<< squares(m); 
  
    return 0; 
}

Output:

enter length of a square:10
no of squares in a square grid:385

Similarly, you can find the number of squares inside a given square grid of any length by using these programs.

Thus, I hope this tutorial will help you.