JavaScript Program to Find Number of Squares in an N*N Grid

Given a grid of side N * N, the task is to find the total number of squares that exist inside it.

Examples:

Input: N = 1 
Output: 1 

Input: N = 2 
Output: 5 

There are several approaches to finding the number of squares in an N*N grid which are as follows:

Using direct formula

We can calculate the total number of squares in an N * N grid by calculating the number of squares without iterating through each possible square size. We use the formula to calculate the total number of squares in an N×N grid because it represents the sum of squares from 1 to n².

Formula:

12 + 22 + ……… + n2 = n(n+1)(2n+1) / 6


We can prove this formula using induction.
We can easily see that the formula is true for
n = 1 and n = 2 as sums are 1 and 5 respectively.

Let it be true for n = k-1. So sum of k-1 numbers
is (k - 1) * k * (2 * k - 1)) / 6

In the following steps, we show that it is true 
for k assuming that it is true for k-1.

Sum of k numbers = Sum of k-1 numbers + k2
           = (k - 1) * k * (2 * k - 1) / 6 + k2
           = ((k2 - k) * (2*k - 1) + 6k2)/6
           = (2k3 - 2k2 - k2 + k + 6k2)/6
           = (2k3 + 3k2 + k)/6
           = k * (k + 1) * (2*k + 1) / 6

Example: Implementation of program to find Number of Squares in an N*N grid using direct formula

JavaScript

// Function to count squares 
// using mathematical formula
function countSquaresMath(N) {
    // Calculate the total number 
    // of squares using the formula
    return (N * (N + 1) * (2 * N + 1)) / 6;
}

// Driver code
const N1 = 1;
const N2 = 2;

console.log(countSquaresMath(N1));
// Output: 1
console.log(countSquaresMath(N2));
// Output: 5

Output

1
5

Time Complexity: O(1)

Auxiliary Space: O(1)

Iterative Counting

In this approach, we count the squares in the grid by iterating through each possible square size from 1 to N and counting the number of squares of that size.

Example: Implementation of program to find Number of Squares in an N*N grid using Iterative Counting

JavaScript

// Function to count squares
//  using iterative counting
function countSquaresIterative(N) {
    let totalSquares = 0;
    // Iterate through each 
    // possible square size from 1 to N
    for (let i = 1; i <= N; i++) {
        // Add the number of squares 
        // of size i to the total count
        totalSquares += i * i;
    }
    return totalSquares;
}

// Driver code
const N1 = 1;
const N2 = 10;

console.log(countSquaresIterative(N1));
// Output: 1
console.log(countSquaresIterative(N2));
// Output: 5

Output

1
5

Time Complexity: O(N)

Auxiliary Space: O(1)