# Closest perfect square and its distance

Given a positive integer . The task is to find the perfect square number closest to N and steps required to reach this number from N.
Note: The closest perfect square to N can be either less than, equal to or greater than N and steps are referred to as the difference between N and the closest perfect square.
Examples:

Input: N = 1500
Output: Perfect square = 1521, Steps = 21
For N = 1500
Closest perfect square greater than N is 1521.
So steps required is 21.
Closest perfect square less than N is 1444.
So steps required is 56.
The minimum of these two is 1521 with steps 21.

Input: N = 2
Output: Perfect Square = 1, Steps = 1
For N = 2
Closest perfect square greater than N is 4.
So steps required is 2.
Closest perfect square less than N is 1.
So steps required is 1.
The minimum of these two is 1.

Approach 1:

• If N is a perfect square then print N and steps as 0.
• Else, find the first perfect square number > N and note its difference with N.
• Then, find the first perfect square number < N and note its difference with N.
• And print the perfect square resulting in the minimum of these two differences obtained and also the difference as the minimum steps.

Below is the implementation of the above approach:

## C++

 `// CPP program to find the closest perfect square` `// taking minimum steps to reach from a number`   `#include ` `using` `namespace` `std;`   `// Function to check if a number is` `// perfect square or not` `bool` `isPerfect(``int` `N)` `{` `    ``if` `((``sqrt``(N) - ``floor``(``sqrt``(N))) != 0)` `        ``return` `false``;` `    ``return` `true``;` `}`   `// Function to find the closest perfect square` `// taking minimum steps to reach from a number` `void` `getClosestPerfectSquare(``int` `N)` `{` `    ``if` `(isPerfect(N)) {` `        ``cout << N << ``" "` `             ``<< ``"0"` `<< endl;` `        ``return``;` `    ``}`   `    ``// Variables to store first perfect` `    ``// square number` `    ``// above and below N` `    ``int` `aboveN = -1, belowN = -1;` `    ``int` `n1;`   `    ``// Finding first perfect square` `    ``// number greater than N` `    ``n1 = N + 1;` `    ``while` `(``true``) {` `        ``if` `(isPerfect(n1)) {` `            ``aboveN = n1;` `            ``break``;` `        ``}` `        ``else` `            ``n1++;` `    ``}`   `    ``// Finding first perfect square` `    ``// number less than N` `    ``n1 = N - 1;` `    ``while` `(``true``) {` `        ``if` `(isPerfect(n1)) {` `            ``belowN = n1;` `            ``break``;` `        ``}` `        ``else` `            ``n1--;` `    ``}`   `    ``// Variables to store the differences` `    ``int` `diff1 = aboveN - N;` `    ``int` `diff2 = N - belowN;`   `    ``if` `(diff1 > diff2)` `        ``cout << belowN << ``" "` `<< diff2;` `    ``else` `        ``cout << aboveN << ``" "` `<< diff1;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `N = 1500;`   `    ``getClosestPerfectSquare(N);` `}` `// This code is contributed by` `// Surendra_Gangwar`

## Java

 `// Java program to find the closest perfect square` `// taking minimum steps to reach from a number`   `class` `GFG {`   `    ``// Function to check if a number is` `    ``// perfect square or not` `    ``static` `boolean` `isPerfect(``int` `N)` `    ``{` `        ``if` `((Math.sqrt(N) - Math.floor(Math.sqrt(N))) != ``0``)` `            ``return` `false``;` `        ``return` `true``;` `    ``}`   `    ``// Function to find the closest perfect square` `    ``// taking minimum steps to reach from a number` `    ``static` `void` `getClosestPerfectSquare(``int` `N)` `    ``{` `        ``if` `(isPerfect(N)) {` `            ``System.out.println(N + ``" "` `                               ``+ ``"0"``);` `            ``return``;` `        ``}`   `        ``// Variables to store first perfect` `        ``// square number` `        ``// above and below N` `        ``int` `aboveN = -``1``, belowN = -``1``;` `        ``int` `n1;`   `        ``// Finding first perfect square` `        ``// number greater than N` `        ``n1 = N + ``1``;` `        ``while` `(``true``) {` `            ``if` `(isPerfect(n1)) {` `                ``aboveN = n1;` `                ``break``;` `            ``}` `            ``else` `                ``n1++;` `        ``}`   `        ``// Finding first perfect square` `        ``// number less than N` `        ``n1 = N - ``1``;` `        ``while` `(``true``) {` `            ``if` `(isPerfect(n1)) {` `                ``belowN = n1;` `                ``break``;` `            ``}` `            ``else` `                ``n1--;` `        ``}`   `        ``// Variables to store the differences` `        ``int` `diff1 = aboveN - N;` `        ``int` `diff2 = N - belowN;`   `        ``if` `(diff1 > diff2)` `            ``System.out.println(belowN + ``" "` `+ diff2);` `        ``else` `            ``System.out.println(aboveN + ``" "` `+ diff1);` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `main(String args[])` `    ``{` `        ``int` `N = ``1500``;`   `        ``getClosestPerfectSquare(N);` `    ``}` `}`

## Python3

 `# Python3 program to find the closest` `# perfect square taking minimum steps` `# to reach from a number`   `# Function to check if a number is` `# perfect square or not` `from` `math ``import` `sqrt, floor`     `def` `isPerfect(N):` `    ``if` `(sqrt(N) ``-` `floor(sqrt(N)) !``=` `0``):` `        ``return` `False` `    ``return` `True`   `# Function to find the closest perfect square` `# taking minimum steps to reach from a number`     `def` `getClosestPerfectSquare(N):` `    ``if` `(isPerfect(N)):` `        ``print``(N, ``"0"``)` `        ``return`   `    ``# Variables to store first perfect` `    ``# square number above and below N` `    ``aboveN ``=` `-``1` `    ``belowN ``=` `-``1` `    ``n1 ``=` `0`   `    ``# Finding first perfect square` `    ``# number greater than N` `    ``n1 ``=` `N ``+` `1` `    ``while` `(``True``):` `        ``if` `(isPerfect(n1)):` `            ``aboveN ``=` `n1` `            ``break` `        ``else``:` `            ``n1 ``+``=` `1`   `    ``# Finding first perfect square` `    ``# number less than N` `    ``n1 ``=` `N ``-` `1` `    ``while` `(``True``):` `        ``if` `(isPerfect(n1)):` `            ``belowN ``=` `n1` `            ``break` `        ``else``:` `            ``n1 ``-``=` `1`   `    ``# Variables to store the differences` `    ``diff1 ``=` `aboveN ``-` `N` `    ``diff2 ``=` `N ``-` `belowN`   `    ``if` `(diff1 > diff2):` `        ``print``(belowN, diff2)` `    ``else``:` `        ``print``(aboveN, diff1)`     `# Driver code` `N ``=` `1500` `getClosestPerfectSquare(N)`   `# This code is contributed` `# by sahishelangia`

## C#

 `// C# program to find the closest perfect square` `// taking minimum steps to reach from a number` `using` `System;`   `class` `GFG {`   `    ``// Function to check if a number is` `    ``// perfect square or not` `    ``static` `bool` `isPerfect(``int` `N)` `    ``{` `        ``if` `((Math.Sqrt(N) - Math.Floor(Math.Sqrt(N))) != 0)` `            ``return` `false``;` `        ``return` `true``;` `    ``}`   `    ``// Function to find the closest perfect square` `    ``// taking minimum steps to reach from a number` `    ``static` `void` `getClosestPerfectSquare(``int` `N)` `    ``{` `        ``if` `(isPerfect(N)) {` `            ``Console.WriteLine(N + ``" "` `                              ``+ ``"0"``);` `            ``return``;` `        ``}`   `        ``// Variables to store first perfect` `        ``// square number` `        ``// above and below N` `        ``int` `aboveN = -1, belowN = -1;` `        ``int` `n1;`   `        ``// Finding first perfect square` `        ``// number greater than N` `        ``n1 = N + 1;` `        ``while` `(``true``) {` `            ``if` `(isPerfect(n1)) {` `                ``aboveN = n1;` `                ``break``;` `            ``}` `            ``else` `                ``n1++;` `        ``}`   `        ``// Finding first perfect square` `        ``// number less than N` `        ``n1 = N - 1;` `        ``while` `(``true``) {` `            ``if` `(isPerfect(n1)) {` `                ``belowN = n1;` `                ``break``;` `            ``}` `            ``else` `                ``n1--;` `        ``}`   `        ``// Variables to store the differences` `        ``int` `diff1 = aboveN - N;` `        ``int` `diff2 = N - belowN;`   `        ``if` `(diff1 > diff2)` `            ``Console.WriteLine(belowN + ``" "` `+ diff2);` `        ``else` `            ``Console.WriteLine(aboveN + ``" "` `+ diff1);` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `Main()` `    ``{` `        ``int` `N = 1500;`   `        ``getClosestPerfectSquare(N);` `    ``}` `}` `// This code is contributed by anuj_67..`

## PHP

 ` ``\$diff2``)` `        ``echo` `\$belowN``, ``" "` `, ``\$diff2``;` `    ``else` `        ``echo` `\$aboveN``, ``" "``, ``\$diff1``;` `}`   `// Driver code` `\$N` `= 1500;` `getClosestPerfectSquare(``\$N``);`   `// This code is contributed by ajit.` `?>`

## Javascript

 ``

Output

`1521 21`

Time Complexity: O(n), where n is the given number since we are using brute force to find the above and below perfect squares.
Auxiliary Space: O(1), since no extra space has been taken.

Approach 2:

The above method might take a lot of time for bigger numbers i.e. greater than 106. We would wish to have a faster way to do that.

• Here, we will use maths to solve the above problem in constant time complexity.
• We will first find the square root of the number n.
• We will check if n was a perfect square, and if it was, we will return 0 there itself.
• Else, we will use its square root to find the just above and below perfect square numbers, and return the one which is at the minimum distance.

Below is the implementation of the above approach:

## C++

 `// CPP program to find the closest perfect square` `// taking minimum steps to reach from a number`   `#include ` `using` `namespace` `std;`     `// Function to find the closest perfect square` `// taking minimum steps to reach from a number` `void` `getClosestPerfectSquare(``int` `N)` `{` `  ``int` `x = ``sqrt``(N);` `  `  `  ``//Checking if N is a perfect square` `  ``if``((x*x)==N){` `    ``cout< diff2)` `        ``cout << belowN << ``" "` `<< diff2;` `    ``else` `        ``cout << aboveN << ``" "` `<< diff1;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `N = 1500;`   `    ``getClosestPerfectSquare(N);` `}` `// This code is contributed by` `// Rohit Kumar`

## Java

 `// Java program for the above approach` ` `  `public` `class` `GFG {` `// Function to find the closest perfect square` `// taking minimum steps to reach from a number` `static` `void` `getClosestPerfectSquare(``int` `N)` `{` `    ``int` `x = (``int``)Math.sqrt(N);` `    `  `    ``//Checking if N is a perfect square` `    ``if``((x*x)==N){` `        ``System.out.println(N);` `        ``return``;` `    ``}` `    `  `    ``// If N is not a perfect square,` `    ``// squaring x and x+1 gives us the` `    ``// just below and above perfect squares` ` `  `    ``// Variables to store perfect` `    ``// square number just` `    ``// above and below N` `    ``int` `aboveN = (x+``1``)*(x+``1``), belowN = x*x;` ` `  `    ``// Variables to store the differences` `    ``int` `diff1 = aboveN - N;` `    ``int` `diff2 = N - belowN;` ` `  `    ``if` `(diff1 > diff2)` `        ``System.out.println(belowN+``" "``+diff2);` `    ``else` `        ``System.out.println(aboveN+``" "``+diff1);` `    `  `}` `    ``// Driver Code` `    ``public` `static` `void` `main (String[] args)` `    ``{` `        ``int` `N = ``1500``;` `        `  `        ``getClosestPerfectSquare(N);` `     `  `    ``}` `}` ` `  `// This code is contributed by aditya942003patil`

## Python3

 `# Python3 program to find the closest` `# perfect square taking minimum steps` `# to reach from a number`   `from` `math ``import` `sqrt, floor`   `# Function to find the closest perfect square` `# taking minimum steps to reach from a number`     `def` `getClosestPerfectSquare(N):` `    ``x ``=` `floor(sqrt(N))` `    `  `    ``# Checking if N is itself a perfect square` `    ``if` `(sqrt(N) ``-` `floor(sqrt(N)) ``=``=` `0``):` `        ``print``(N,``0``)` `        ``return`   `    ``# Variables to store first perfect` `    ``# square number above and below N` `    ``aboveN ``=` `(x``+``1``)``*``(x``+``1``)` `    ``belowN ``=` `x``*``x`   `    ``# Variables to store the differences` `    ``diff1 ``=` `aboveN ``-` `N` `    ``diff2 ``=` `N ``-` `belowN`   `    ``if` `(diff1 > diff2):` `        ``print``(belowN, diff2)` `    ``else``:` `        ``print``(aboveN, diff1)`     `# Driver code` `N ``=` `1500` `getClosestPerfectSquare(N)`   `# This code is contributed` `# by Rohit Kumar`

## C#

 `// C# program for the above approach` `using` `System;` `using` `System.Collections.Generic;`   `public` `class` `GFG ` `{`   `  ``// Function to find the closest perfect square` `  ``// taking minimum steps to reach from a number` `  ``static` `void` `getClosestPerfectSquare(``int` `N)` `  ``{` `    ``int` `x = (``int``)Math.Sqrt(N);`   `    ``//Checking if N is a perfect square` `    ``if``((x*x)==N){` `      ``Console.WriteLine(N);` `      ``return``;` `    ``}`   `    ``// If N is not a perfect square,` `    ``// squaring x and x+1 gives us the` `    ``// just below and above perfect squares`   `    ``// Variables to store perfect` `    ``// square number just` `    ``// above and below N` `    ``int` `aboveN = (x+1)*(x+1), belowN = x*x;`   `    ``// Variables to store the differences` `    ``int` `diff1 = aboveN - N;` `    ``int` `diff2 = N - belowN;`   `    ``if` `(diff1 > diff2)` `      ``Console.WriteLine(belowN+``" "``+diff2);` `    ``else` `      ``Console.WriteLine(aboveN+``" "``+diff1);`   `  ``}` `  `  `  ``// Driver Code` `  ``public` `static` `void` `Main (``string``[] args)` `  ``{` `    ``int` `N = 1500;` `    ``getClosestPerfectSquare(N);`   `  ``}` `}`   `// This code is contributed by phasing17`

## Javascript

 `// JavaScript program to find the closest` `// perfect square taking minimum steps` `// to reach from a number`   `// Function to find the closest perfect square` `// taking minimum steps to reach from a number` `function` `getClosestPerfectSquare(N)` `{` `    ``let x = Math.floor(Math.sqrt(N))` `    `  `    ``// Checking if N is itself a perfect square` `    ``if` `(Math.sqrt(N) - Math.floor(Math.sqrt(N)) == 0)` `    ``{` `        ``console.log(N,0)` `        ``return` `    ``}`   `    ``// Variables to store first perfect` `    ``// square number above and below N` `    ``let aboveN = (x+1)*(x+1)` `    ``let belowN = x*x`   `    ``// Variables to store the differences` `    ``let diff1 = aboveN - N` `    ``let diff2 = N - belowN`   `    ``if` `(diff1 > diff2)` `        ``console.log(belowN, diff2)` `    ``else` `        ``console.log(aboveN, diff1)` `}`   `// Driver code` `let N = 1500` `getClosestPerfectSquare(N)`   `// This code is contributed by phasing17`

Output

`1521 21`

Time Complexity: O(logN), as it is using inbuilt sqrt function
Auxiliary Space: O(1), since no extra space has been taken.

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