Minimum Perimeter of n blocks

We are given n blocks of size 1 x 1, we need to find the minimum perimeter of the grid made by these blocks.
Examples :

Input : n = 4
Output : 8
Minimum possible perimeter with 4 blocks
is 8. See below explanation.

Input : n = 11
Output : 14
The square grid of above examples would be as

Let us take an example to see a pattern. Let us say that we have 4 blocks, following are different possibilities 

  +--+--+--+--+
  |  |  |  |  |  Perimeter = 10
  +--+--+--+--+

  +--+--+--+
  |  |  |  |     Perimeter = 10
  +--+--+--+
        |  |
        +--+

  +--+--+--+
  |  |  |  |     Perimeter = 10
  +--+--+--+
     |  |
     +--+


  +--+--+
  |  |  |        Perimeter = 8
  +--+--+
  |  |  |
  +--+--+

If we do some examples using pen and paper, we can notice that the perimeter becomes minimum when the shape formed is closest to a square. The reason for this is, we want maximum sides of blocks to face inside the shape so that perimeter of the shape becomes minimum.
If the Number of blocks is a perfect square then the perimeter would simply be 4*sqrt(n). 
But, if the Number of blocks is not a perfect square root then we calculate number of rows and columns closest to square root. After arranging the blocks in a rectangular we still have blocks left then we will simply add 2 to the perimeter because only 2 extra side would be left. 
The implementation of the above idea is given below.

C++

// CPP program to find minimum
// perimeter using n blocks.
#include <bits/stdc++.h>
using namespace std;

int minPerimeter(int n)
{
    int l = sqrt(n);
    int sq = l * l;

    // if n is a perfect square
    if (sq == n)
        return l * 4;
    else
    {
        // Number of rows
        long long int row = n / l;

        // perimeter of the
        // rectangular grid
        long long int perimeter
                      = 2 * (l + row);

        // if there are blocks left
        if (n % l != 0)
            perimeter += 2;
        return perimeter;
    }
}

// Driver code
int main()
{
    int n = 10;
    cout << minPerimeter(n);
    return 0;
}

Java

// JAVA Code to find minimum
// perimeter using n blocks
import java.util.*;

class GFG
{
    public static long minPerimeter(int n)
    {
        int l = (int) Math.sqrt(n);
        int sq = l * l;

        // if n is a perfect square
        if (sq == n)
            return l * 4;
        else
        {
            // Number of rows
            long row = n / l;

            // perimeter of the
            // rectangular grid
            long perimeter
                  = 2 * (l + row);

            // if there are blocks left
            if (n % l != 0)
                perimeter += 2;
            return perimeter;
        }
    }

    // Driver code
    public static void main(String[] args)
    {
        int n = 10;
        System.out.println(minPerimeter(n));
    }
}

// This code is contributed by Arnav Kr. Mandal

Python3

# Python3 program to find minimum
# perimeter using n blocks.
import math

def minPerimeter(n):
    l = math.sqrt(n)
    sq = l * l

    # if n is a perfect square
    if (sq == n):
        return l * 4
    else :
        # Number of rows
        row = n / l

        # perimeter of the
        # rectangular grid
        perimeter = 2 * (l + row)

        # if there are blocks left
        if (n % l != 0):
            perimeter += 2
        return perimeter

# Driver code
n = 10
print(int(minPerimeter(n)))

# This code is contributed by
# Prasad Kshirsagar

C#

// C# Code to find minimum
// perimeter using n blocks
using System;

class GFG
{
    public static long minPerimeter(int n)
    {
        int l = (int) Math.Sqrt(n);
        int sq = l * l;

        // if n is a perfect square
        if (sq == n)
            return l * 4;
        else
        {
            // Number of rows
            long row = n / l;

            // perimeter of the
            // rectangular grid
            long perimeter
                  = 2 * (l + row);

            // if there are blocks left
            if (n % l != 0)
                perimeter += 2;
            return perimeter;
        }
    }

    // Driver code
    public static void Main()
    {
        int n = 10;
        Console.Write(minPerimeter(n));
    }
}

// This code is contributed by nitin mittal

PHP

<?php
// PHP program to find minimum
// perimeter using n blocks.

function minPerimeter($n)
{
    $l = floor(sqrt($n));
    $sq = $l * $l;

    // if n is a perfect square
    if ($sq == $n)
        return $l * 4;
    else
    {
        // Number of rows
        $row = floor($n / $l);

        // perimeter of the
        // rectangular grid
        $perimeter = 2 * ($l + $row);

        // if there are blocks left
        if ($n % $l != 0)
            $perimeter += 2;
        return $perimeter;
    }
}

// Driver code
$n = 10;
echo minPerimeter($n);

// This code is contributed
// by nitin mittal.
?>

Javascript

<script>

// JavaScript program for the
// above approach

function minPerimeter(n)
    {
        let l =  Math.sqrt(n);
        let sq = l * l;

        // if n is a perfect square
        if (sq == n)
            return l * 4;
        else
        {
            // Number of rows
            let row = n / l;

            // perimeter of the
            // rectangular grid
            let perimeter
                  = 2 * (l + row);

            // if there are blocks left
            if (n % l != 0)
                perimeter += 2;
            return perimeter;
        }
    }


// Driver Code

    let n = 10;
    document.write(Math.floor(minPerimeter(n)))

</script>

Output :

14

Time complexity : O(logn) 
Auxiliary Space : O(1)

References : 
http://mathforum.org/library/drmath/view/61595.html
intermath.coe.uga.edu/tweb/gcsu-geo-spr06/aheath/aheath_rectperimeter.doc
This article is contributed by Sarthak Kohli. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.