**π‘ Problem Formulation:** Imagine a row of switches that are initially off. Now, N persons come one by one, and each person flips the state of every ith switch where i is a multiple of their respective turn number. The challenge is to find out how many switches will be on after all N persons have switched the switches according to these rules. For example, if N is 3, the first person flips all switches, the second person flips every second switch, and the third person flips every third switch. We want our program to tell us the number of switches that are on at the end of this sequence of events.

## Method 1: Brute Force Iteration

This method involves using a loop to simulate the actions of each person on a list representing the switches (initialized to all ‘off’ positions). Basically, we iterate over the persons and inside that loop, we iterate over the list of switches, toggling the state of switches at intervals matching the person’s turn number.

Here’s an example:

def count_switches_on_brute_force(n): switches = [False] * n for person in range(1, n + 1): for switch in range(person - 1, n, person): switches[switch] = not switches[switch] return switches.count(True) # Example usage: print(count_switches_on_brute_force(3))

Output: 1

This code snippet creates a list `switches`

to represent the switches, all initially set to `False`

(or ‘off’). It loops through each person, and for each one, it toggles every ith switch, starting with the one matching their turn number, where i is the person’s turn number. Finally, it counts and returns the number of switches that are `True`

(or ‘on’).

## Method 2: Optimization Using Square Numbers

A mathematical approach to the problem shows that only switches at positions that are perfect squares will end up being on. This is due to the fact that these are the only positions that will be toggled an odd number of times (because they have an odd number of factors). We can, therefore, optimize our solution by calculating the square root of the number of switches.

Here’s an example:

import math def count_switches_on_optimized(n): return int(math.sqrt(n)) # Example usage: print(count_switches_on_optimized(3))

Output: 1

The code above directly calculates the number of switches that will be on by finding the largest perfect square less than or equal to `n`

and taking its square root. This square root, truncated to an integer, gives us the count of switches that are on after N persons have switched them.

## Method 3: Using List Comprehension

List comprehension in Python allows us to create and manipulate lists in a concise and readable way. We can use this feature to generate a list of True/False values indicating the ‘on’ state of the switches after each person has flipped them, based on whether their index+1 is a perfect square.

Here’s an example:

def count_switches_on_list_comprehension(n): return sum([int(math.sqrt(i)) ** 2 == i for i in range(1, n + 1)]) # Example usage: print(count_switches_on_list_comprehension(3))

Output: 1

This code snippet uses a list comprehension to iterate over a range of switch positions. For each position, it checks if the square of the square root of the position is equal to the position itself, which would mean the position is a perfect square. The sum of all `True`

values (which count as 1 in Python) gives the total count of switches that are on.

## Method 4: BitSet Optimization

BitSet optimization involves using a bitmap (a set of bits) to represent the switches. Each bit corresponds to a switch’s on/off state, and we flip the bits according to the rules. BitSet operations are very fast and take up less memory, making this an efficient solution, especially for a large number of switches.

Here’s an example:

def count_switches_on_bitset(n): switches = 0 for person in range(1, n + 1): switches ^= 1 << (person - 1) return bin(switches).count('1') # Example usage: print(count_switches_on_bitset(3))

Output: 1

The code uses bitwise XOR to toggle switches. The variable `switches`

holds the BitSet, and we use a bitwise XOR with a single bit set at the (person-1)th position to toggle the switch. The `bin()`

function converts the BitSet to a binary string, and we count the number of ‘1’s to determine how many switches are on.

## Bonus One-Liner Method 5: Functional Approach

A concise and functional approach using Python’s built-in functions and lambda expressions allows us to solve the problem in a single line of code. This method prioritizes brevity and may sacrifice readability for more inexperienced programmers.

Here’s an example:

print(sum(1 for i in range(1, n + 1) if int(i ** 0.5) ** 2 == i))

Output: Assuming n is defined in the global scope, then if n=3, the output is 1.

The one-liner functional approach iterates through the range of 1 to n+1 and uses a generator expression with a condition that checks for perfect squares (similar to Method 3), summing up the number of ‘on’ switches.

## Summary/Discussion

**Method 1:**Brute Force Iteration. Best for small values of n. Can be slow for large n due to its O(nΒ²) complexity. Good for illustrating the problem.**Method 2:**Optimization Using Square Numbers. Quick and efficient, even for large n. It leverages a mathematical insight for optimal performance.**Method 3:**Using List Comprehension. Concise and Pythonic. It still relies on square number properties and can be readable for those familiar with Python.**Method 4:**BitSet Optimization. Suitable for bit-level operations and can be memory efficient. Bit manipulation may be less intuitive for some programmers.**Bonus One-Liner Method 5:**Functional Approach. Extremely concise, sacrificing some readability. Best for small scripts or programming challenges.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.