Bulb Switcher

                                            
class Solution {
public:
    int bulbSwitch(int n) {
        return sqrt(n);
    }
};

                                            
class Solution {
    public int bulbSwitch(int n) {
        return (int) Math.sqrt(n);
    }
}

                                            
class Solution(object):
    def bulbSwitch(self, n):
        """
        :type n: int
        :rtype: int
        """
        return int(n ** 0.5)

                                            
int bulbSwitch(int n) {
    return (int)sqrt(n);
}

                                            
public class Solution {
    public int BulbSwitch(int n) {
        return (int)Math.Sqrt(n);
    }
}

                                            
/**
 * @param {number} n
 * @return {number}
 */
var bulbSwitch = function(n) {
    return Math.floor(Math.sqrt(n));
};

                                            
function bulbSwitch(n: number): number {
    return Math.floor(Math.sqrt(n));
};

                                            
class Solution {
    /**
     * @param Integer $n
     * @return Integer
     */
    function bulbSwitch($n) {
        return intval(sqrt($n));
    }
}

                                            
 class Solution {
    func bulbSwitch(_ n: Int) -> Int {
        return Int(sqrt(Double(n)))
    }
}

                                            
class Solution {
    fun bulbSwitch(n: Int): Int {
        return Math.sqrt(n.toDouble()).toInt()
    }
}

                                            
import 'dart:math';

class Solution {
  int bulbSwitch(int n) {
      return sqrt(n).toInt();
  }
}

                                            
package main

func bulbSwitch(n int) int {
    return int(math.Sqrt(float64(n)))
}

                                            
# @return {Integer}
def bulb_switch(n)
    Math.sqrt(n).to_i
end

To solve this coding challenge, it's essential to understand the mechanics of how bulbs are toggled through each round. The challenge can be broken down step-by-step to understand why the final number of bulbs left on is closely related to perfect squares.

Explanation

Initially, every bulb is off. The task requires you to toggle the bulbs in a specific sequence:

In the first round, all bulbs are toggled on.
In the second round, every second bulb is toggled off.
In the third round, every third bulb is toggled (if it was on, it’s turned off; if it was off, it’s turned on).

This pattern continues until the ith round, where every ith bulb is toggled.

Upon careful inspection, it becomes apparent that a bulb ends up being toggled an odd number of times if and only if it has an odd number of divisors. Normally, divisors come in pairs (e.g., the divisors of 6 are 1, 2, 3, 6), but perfect squares have an unpaired middle divisor (e.g., the divisors of 9 are 1, 3, 9).

Thus, the only bulbs that will remain on are those that are at positions of perfect squares (1, 4, 9, 16, ...). Consequently, the number of bulbs that will remain on after n rounds corresponds to the count of perfect squares less than or equal to n.

Therefore, the solution lies in finding the count of such perfect squares. This can be achieved directly by determining the integer part of the square root of n (because the squares of all integers up to ⌊√n⌋ are less than or equal to n).

Step-by-Step Explanation/Detailed Steps in Pseudocode:

Understand the problem mechanics : Bulbs are toggled based on their position and the round.
Identify the pattern : Only bulbs at positions of perfect squares will remain on.
Simplify the problem : We need to count how many perfect squares are <= n.
Calculate the result : This is equivalent to finding the integer part of the square root of n.

Pseudocode

                                            
# Function to calculate the number of bulbs that remain on after n rounds
FUNCTION bulbSwitch(number_of_bulbs):
    # The result is the integer part of the square root of number_of_bulbs
    result = INTEGER_PART(SQUARE_ROOT(number_of_bulbs))
    
    # Return the computed result
    RETURN result

Comments in the Pseudocode:

FUNCTION bulbSwitch(number_of_bulbs) : This denotes the function that we will call with the input number of bulbs.
result = INTEGER_PART(SQUARE_ROOT(number_of_bulbs)) : This line computes the integer part of the square root of the input number of bulbs, which gives the number of perfect squares <= number_of_bulbs.
RETURN result : This returns the computed result as the number of bulbs that remain on after all rounds are completed.

By following the above methodology and pseudocode structure, this coding challenge can be efficiently solved. Every part of the solution is directly tied to a clear mathematical insight, ensuring the correctness and optimality of the approach.