Count of uncolored cells in a Pyramid of length H formed using cubes

Given a pyramid, formed using unit area cubes, of given height H. The pyramid is then placed on the ground and painted from outside. The task is to find the count of cubes that are left uncoloured.

Examples:

Input: H = 3
Output: 1
Explanation:
Input: H = 1
Output: 0

Naive Approach: The number of cubes that will remain uncoloured is the number of cubes that are not present in the boundary of the pyramid. So, for every layer of cubes at height h, the number of uncoloured cubes is $(h-2)^2$ where h>1. So, add the number of uncoloured cubes at every level to get the answer.

Below is the implementation of the above approach:

C++

// C++ program for the above problem
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the number of
// uncoloured cubes
int uncolouredCubes(int H)
{
 
    // To store the number of uncoloured
    // cubes
    int res = 0;
 
    for (int h = 2; h <= H; h++) {
        res += (h - 2) * (h - 2);
    }
 
    return res;
}
 
// Driver Code
int main()
{
    int H = 3;
    cout << uncolouredCubes(H);
}

Java

// Java program for the above problem
 
 
class GFG{
 
  // Function to return the number of
  // uncoloured cubes
  static int uncolouredCubes(int H)
  {
 
      // To store the number of uncoloured
      // cubes
      int res = 0;
 
      for (int h = 2; h <= H; h++) {
          res += (h - 2) * (h - 2);
      }
 
      return res;
  }
 
  // Driver Code
  public static void main(String [] args)
  {
      int H = 3;
      System.out.println(uncolouredCubes(H));
  }
   
}
 
// This code is contributed by ihritik

Python3

# python program for the above problem
 
# Function to return the number of
# uncoloured cubes
def uncolouredCubes(H):
 
    # To store the number of uncoloured
    # cubes
    res = 0
 
    for h in range(2, H + 1):
        res = res + (h - 2) * (h - 2)
     
    return res
 
# Driver Code
 
H = 3
print(uncolouredCubes(H))
 
# This code is contributed by ihritik

C#

// C# program for the above problem
 
using System;
class GFG{
 
  // Function to return the number of
  // uncoloured cubes
  static int uncolouredCubes(int H)
  {
 
      // To store the number of uncoloured
      // cubes
      int res = 0;
 
      for (int h = 2; h <= H; h++) {
          res += (h - 2) * (h - 2);
      }
 
      return res;
  }
 
  // Driver Code
  public static void Main()
  {
      int H = 3;
      Console.WriteLine(uncolouredCubes(H));
  }
   
}
 
// This code is contributed by ihritik

Javascript

<script>
// JavaScript program for the above problem
 
// Function to return the number of
// uncoloured cubes
function uncolouredCubes(H){
 
 
    // To store the number of uncoloured
    // cubes
    var res = 0
 
    for(var h = 2 ; h < H + 1; h++)
        res = res + (h - 2) * (h - 2)
     
    return res
}
 
 
// Driver Code
 
var H = 3
document.write(uncolouredCubes(H))
 
// This code is contributed by AnkThon
</script>

Output

Time Complexity: O(N)
Auxiliary Space: O(1)

Efficient Approach: As the number of uncoloured cubes in each layer at height h is $(h-2)^2$ , so this problem can be reduced to find the sum of the squares of the first (H-2) natural numbers. So, the answer is (n * (n + 1) * (2n + 1)) / 6 where n=H-2.

C++

// C++ program for the above problem
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the number of
// uncoloured cubes
int uncolouredCubes(int H)
{
 
    // If H is less than 2, then
    // returning 0
    if (H < 2) {
        return 0;
    }
 
    int n = H - 2;
 
    // Sum of the squares of first
    // n natural numbers
    return (n * (n + 1) * (2 * n + 1)) / 6;
}
 
// Driver Code
int main()
{
    int H = 3;
    cout << uncolouredCubes(H);
}

Java

// Java program for the above problem
import java.util.*;
public class GFG
{
   
// Function to return the number of
// uncoloured cubes
static int uncolouredCubes(int H)
{
   
    // If H is less than 2, then
    // returning 0
    if (H < 2) {
        return 0;
    }
    int n = H - 2;
 
    // Sum of the squares of first
    // n natural numbers
    return (n * (n + 1) * (2 * n + 1)) / 6;
}
 
// Driver Code
public static void main(String args[])
{
    int H = 3;
    System.out.println(uncolouredCubes(H));
}
}
 
// This code is contributed by Samim Hossain Mondal

Python3

# Python program for the above problem
 
# Function to return the number of
# uncoloured cubes
def uncolouredCubes(H):
 
  # If H is less than 2, then
  # returning 0
    if (H < 2):
        return 0;
 
    n = H - 2;
 
    # Sum of the squares of first
    # n natural numbers
    return (n * (n + 1) * (2 * n + 1)) / 6;
 
# Driver Code
H = 3;
print((int)(uncolouredCubes(H)));
 
# This code is contributed by Saurabh Jaiswal.

C#

// C# program for the above problem
using System;
class GFG
{
// Function to return the number of
// uncoloured cubes
static int uncolouredCubes(int H)
{
    // If H is less than 2, then
    // returning 0
    if (H < 2) {
        return 0;
    }
    int n = H - 2;
 
    // Sum of the squares of first
    // n natural numbers
    return (n * (n + 1) * (2 * n + 1)) / 6;
}
 
// Driver Code
public static void Main()
{
    int H = 3;
    Console.Write(uncolouredCubes(H));
}
}
// This code is contributed by Samim Hossain Mondal

Javascript

// Javascript program for the above problem
 
// Function to return the number of
// uncoloured cubes
function uncolouredCubes(H)
{
 
  // If H is less than 2, then
  // returning 0
  if (H < 2) {
    return 0;
  }
 
  let n = H - 2;
 
  // Sum of the squares of first
  // n natural numbers
  return (n * (n + 1) * (2 * n + 1)) / 6;
}
 
// Driver Code
 
let H = 3;
document.write(uncolouredCubes(H));
 
// This code is contributed by gfgking.