Find Number of Squares in a Chessboard using C++

In this problem, we are given the size of a chessboard. Our task is to create a program to find number of squares in a chessboard in C++.

Problem Description − To find the number of squares in a chessboard. We will have to calculate all the combinations of the square that are inside the chessboard i.e. we will consider squares of side 1x1, 2x2, 3x3 … nxn.

Let’s take an example to understand the problem,

Input: n = 4.

Output: 30

Squares of size 1x1 -> 16
Squares of size 2x2 -> 9
Squares of size 3x3 -> 4
Squares of size 4x4 -> 1
Total number of squares = 16+9+4+1 = 30

Solution Approach:

A simple approach is by using the sum formula for nxn grid.
Let’s deduct the general formula for the sum,
sum(1) = 1
sum(2) = 1 + 4 = 5
sum(3) = 1 + 4 + 9 = 14
sum(4) = 1 + 4 + 9 + 16 = 30
The sum is can be generalised as
sum = 12 + 22 + 32 + 42 + … n2
sum = ( (n*(n+1)*((2*n) + 1))/6 )

Example

#include <iostream>
using namespace std;
int calcSquaresCount(int n){
   int squareCount = ( (n * (n+1) * (2*n + 1))/6 );
   return squareCount;
}
int main() {
   int n = 6;
   cout<<"The total number of squares of size "<<n<<"X"<<n<<" is
   "<<calcSquaresCount(n);
}

Output:

The total number of squares of size 6X6 is 91