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
Minimum Perimeter of n blocks

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++

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// 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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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PHP

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<?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.
?>

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Output :

14

References :
http://mathforum.org/library/drmath/view/61595.html
intermath.coe.uga.edu/tweb/gcsu-geo-spr06/aheath/aheath_rectperimeter.doc

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