# Square pyramidal number (Sum of Squares)

A Square pyramidal number represents sum of squares of first natural numbers. First few Square pyramidal numbers are 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, …

Geometrically these numbers represent number of spheres to be stacked to form a pyramid with square base. Please see this Wiki image for more clarity.

Given a number s (1 <= s <= 1000000000). If s is sum of the squares of the first n natural numbers then print n, otherwise print -1.**Examples :**

Input : 14 Output : 3 Explanation : 1*1 + 2*2 + 3*3 = 14 Input : 26 Output : -1

A **simple solution **is to run through all numbers starting from 1, compute current sum. If current sum is equal to given sum, then we return true, else false.

## C++

`// C++ program to check if a` `// given number is sum of` `// squares of natural numbers.` `#include <iostream>` `using` `namespace` `std;` `// Function to find if the` `// given number is sum of` `// the squares of first n` `// natural numbers` `int` `findS(` `int` `s)` `{` ` ` `int` `sum = 0;` ` ` `// Start adding squares of` ` ` `// the numbers from 1` ` ` `for` `(` `int` `n = 1; sum < s; n++)` ` ` `{` ` ` `sum += n * n;` ` ` `// If sum becomes equal to s` ` ` `// return n` ` ` `if` `(sum == s)` ` ` `return` `n;` ` ` `}` ` ` `return` `-1;` `}` `// Drivers code` `int` `main()` `{` ` ` `int` `s = 13;` ` ` `int` `n = findS(s);` ` ` `n == -1 ? cout << ` `"-1"` `: cout << n;` ` ` `return` `0;` `}` |

## Java

`// Java program to check if a` `// given number is sum of` `// squares of natural numbers.` `class` `GFG` `{` ` ` `// Function to find if the` ` ` `// given number is sum of` ` ` `// the squares of first` ` ` `// n natural numbers` ` ` `static` `int` `findS(` `int` `s)` ` ` `{` ` ` `int` `sum = ` `0` `;` ` ` `// Start adding squares of` ` ` `// the numbers from 1` ` ` `for` `(` `int` `n = ` `1` `; sum < s; n++)` ` ` `{` ` ` `sum += n * n;` ` ` `// If sum becomes equal to s` ` ` `// return n` ` ` `if` `(sum == s)` ` ` `return` `n;` ` ` `}` ` ` `return` `-` `1` `;` ` ` `}` ` ` `// Drivers code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `int` `s = ` `13` `;` ` ` `int` `n = findS(s);` ` ` `if` `(n == -` `1` `)` ` ` `System.out.println(` `"-1"` `);` ` ` `else` ` ` `System.out.println(n);` ` ` `}` `}` |

## Python3

`# Python3 program to find if` `# the given number is sum of` `# the squares of first` `# n natural numbers` `# Function to find if the given` `# number is sum of the squares` `# of first n natural numbers` `def` `findS (s):` ` ` `_sum ` `=` `0` ` ` `n ` `=` `1` ` ` ` ` `# Start adding squares of` ` ` `# the numbers from 1` ` ` `while` `(_sum < s):` ` ` `_sum ` `+` `=` `n ` `*` `n` ` ` `n` `+` `=` `1` ` ` `n` `-` `=` `1` ` ` ` ` `# If sum becomes equal to s` ` ` `# return n` ` ` `if` `_sum ` `=` `=` `s:` ` ` `return` `n` ` ` `return` `-` `1` `# Driver code` `s ` `=` `13` `n ` `=` `findS (s)` `if` `n ` `=` `=` `-` `1` `:` ` ` `print` `(` `"-1"` `)` `else` `:` ` ` `print` `(n)` |

## C#

`// C# program to check if a given` `// number is sum of squares of` `// natural numbers.` `using` `System;` `class` `GFG` `{` ` ` ` ` `// Function to find if the given` ` ` `// number is sum of the squares` ` ` `// of first n natural numbers` ` ` `static` `int` `findS(` `int` `s)` ` ` `{` ` ` `int` `sum = 0;` ` ` ` ` `// Start adding squares of` ` ` `// the numbers from 1` ` ` `for` `(` `int` `n = 1; sum < s; n++)` ` ` `{` ` ` `sum += n * n;` ` ` ` ` `// If sum becomes equal` ` ` `// to s return n` ` ` `if` `(sum == s)` ` ` `return` `n;` ` ` `}` ` ` ` ` `return` `-1;` ` ` `}` ` ` ` ` `// Drivers code` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` `int` `s = 13;` ` ` ` ` `int` `n = findS(s);` ` ` ` ` `if` `(n == -1)` ` ` `Console.Write(` `"-1"` `) ;` ` ` `else` ` ` `Console.Write(n);` ` ` `}` `}` `// This code is contribute by` `// Smitha Dinesh Semwal` |

## PHP

`<?php` `// PHP program to check if a` `// given number is sum of` `// squares of natural numbers.` `// Function to find if the given number` `// is sum of the squares of first n` `// natural numbers` `function` `findS(` `$s` `)` `{` ` ` `$sum` `= 0;` ` ` `// Start adding squares of` ` ` `// the numbers from 1` ` ` `for` `(` `$n` `= 1; ` `$sum` `< ` `$s` `; ` `$n` `++)` ` ` `{` ` ` `$sum` `+= ` `$n` `* ` `$n` `;` ` ` `// If sum becomes equal to s` ` ` `// return n` ` ` `if` `(` `$sum` `== ` `$s` `)` ` ` `return` `$n` `;` ` ` `}` ` ` `return` `-1;` `}` `// Drivers code` `$s` `= 13;` `$n` `= findS(` `$s` `);` `if` `(` `$n` `== -1)` ` ` `echo` `(` `"-1"` `);` `else` ` ` `echo` `(` `$n` `);` `// This code is contributed by Ajit.` `?>` |

## Javascript

`<script>` `// JavaScript to check if a` `// given number is sum of` `// squares of natural numbers.` ` ` `// Function to find if the` ` ` `// given number is sum of` ` ` `// the squares of first` ` ` `// n natural numbers` ` ` `function` `findS(s)` ` ` `{` ` ` `let sum = 0;` ` ` ` ` `// Start adding squares of` ` ` `// the numbers from 1` ` ` `for` `(let n = 1; sum < s; n++)` ` ` `{` ` ` `sum += n * n;` ` ` ` ` `// If sum becomes equal to s` ` ` `// return n` ` ` `if` `(sum == s)` ` ` `return` `n;` ` ` `}` ` ` ` ` `return` `-1;` ` ` `}` `// Driver code` ` ` `let s = 13;` ` ` `let n = findS(s);` ` ` `if` `(n == -1)` ` ` `document.write(` `"-1"` `);` ` ` `else` ` ` `document.write(n);` `// This code is contributed by souravghosh0416.` `</script>` |

**OUTPUT :**

-1

An **alternate solution** is to use Newton Raphson Method.

We know sum of squares of first n natural numbers is **n * (n + 1) * (2*n + 1) / 6**.

We can write solutions as

k * (k + 1) * (2*k + 1) / 6 = s

k * (k + 1) * (2*k + 1) – 6s = 0

We can find roots of above cubic equation using Newton Raphson Method, then check if root is integer or not.

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