** 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;` `}` |

## 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.` |

## 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.` |

## 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.` |

## 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.` `?>` |

**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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