** 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.

- long int is used in return.
- 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

`#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; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// 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. ` |

*chevron_right*

*filter_none*

## Python3

`# 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. ` |

*chevron_right*

*filter_none*

## C#

`// 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. ` |

*chevron_right*

*filter_none*

## PHP

`<?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. ` `?> ` |

*chevron_right*

*filter_none*

**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

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready.

## Recommended Posts:

- Count squares with odd side length in Chessboard
- Puzzle | Dividing a Square into N smaller squares
- Minimum Cuts can be made in the Chessboard such that it is not divided into 2 parts
- Total position where king can reach on a chessboard in exactly M moves
- Total position where king can reach on a chessboard in exactly M moves | Set 2
- Count Distinct Rectangles in N*N Chessboard
- Maximum non-attacking Rooks that can be placed on an N*N Chessboard
- Count positions in a chessboard that can be visited by the Queen which are not visited by the King
- Program to count number of distinct Squares and Cubes upto N
- Find the number of squares inside the given square grid
- Minimum number of squares whose sum equals to given number N | set 2
- Minimum number of squares whose sum equals to a given number N | Set-3
- Minimum number of squares whose sum equals to given number n
- Find a point that lies inside exactly K given squares
- Find the side of the squares which are inclined diagonally and lined in a row
- Find the side of the squares which are lined in a row, and distance between the centers of first and last square is given
- Find K numbers with sum equal to N and sum of their squares maximized
- Find two numbers such that difference of their squares equal to N
- Count number of squares in a rectangle
- Number of perfect squares between two given numbers