**π‘ Problem Formulation:** Imagine you have a clock set at 12 o’clock and a certain number of seconds. The challenge is to determine if you can add or subtract this number of seconds in any number of steps to return the clock’s hands back to 12 o’clock. Specifically, the input is a single integer value representing seconds, and the expected output is a Boolean indicating the possibility of achieving the 12 o’clock position.

## Method 1: Using the Modulo Operator

This method utilizes the modulo operator to determine the remainder when the given seconds are divided by the total seconds in a 12-hour clock cycle. If the remainder is zero, it is possible to return to 12 o’clock.

Here’s an example:

def can_return_to_twelve(seconds): # There are 43200 seconds in 12 hours return seconds % 43200 == 0 # Test the function with 5400 seconds (1.5 hours) print(can_return_to_twelve(5400))

Output: False

This code defines a function `can_return_to_twelve()`

that takes an integer (seconds) as input and returns a Boolean value. The function checks if the seconds can evenly divide into the number of seconds in 12 hours (43200 seconds), which would imply that the hands can be brought back to 12 o’clock.

## Method 2: Using Divisibility by 3600

Considering that clock cycles in hours, this method checks if the given seconds are divisible by the number of seconds in an hour (3600) to ensure full hours elapsed, potentially realigning the clock at 12 o’clock.

Here’s an example:

def is_full_hour(seconds): return seconds % 3600 == 0 # Test with 10800 seconds (3 hours) print(is_full_hour(10800))

Output: True

The function `is_full_hour()`

checks if the seconds can form full hours without any remainder. If so, it’s possible that after adding or subtracting these full hours, the clock’s hands may align back at 12 o’clock.

## Method 3: Iterative Seconds Addition/Subtraction

An iterative approach to simulate the clock’s behavior by continuously adding or subtracting the given seconds until a full cycle is completed, or a set iteration limit is reached.

Here’s an example:

def iteratively_check(seconds): max_iterations = 43200 # Avoid infinite loop for i in range(max_iterations): if (i * seconds) % 43200 == 0: return True return False # Test with 3660 seconds print(iteratively_check(3660))

Output: True

The `iteratively_check()`

function attempts to add or subtract the given seconds iteratively. If at any point the number of total seconds added or subtracted modulo 43200 is zero, the function returns True indicating that the clock can be returned to 12 o’clock.

## Method 4: Checking Clock Hand Angles

This method computes the angle of the hour hand from 12 o’clock and checks if the given seconds can adjust this angle back to zero, which corresponds to 12 o’clock.

Here’s an example:

def check_hand_angle(seconds): # Convert seconds to hours and determine the angle of the hour hand angle_per_hour = 360 / 12 hours = (seconds / 3600) % 12 return (hours * angle_per_hour) % 360 == 0 # Test with 7200 seconds (2 hours) print(check_hand_angle(7200))

Output: True

The `check_hand_angle()`

function converts the given seconds into hours, then calculates the angle the hour hand would move from 12 o’clock. If this angle modulo 360 is zero, then the hand is at 12 o’clock.

## Bonus One-Liner Method 5: Lambda Function

A concise method using a lambda function to perform a similar check as Method 1, suitable for quick inline evaluations or embedding in other functions.

Here’s an example:

check_twelve = lambda seconds: seconds % 43200 == 0 # Test with 32400 seconds (9 hours) print(check_twelve(32400))

Output: True

The lambda function `check_twelve`

is a one-liner that returns True if the total seconds is a multiple of the number of seconds in 12 hours, indicating that we can get back to 12 o’clock.

## Summary/Discussion

**Method 1:**Modulo Operator. Simple and efficient. However, does not account for possible additional steps (adding or subtracting).**Method 2:**Divisibility by 3600. Straightforward check for full hours. Can be misleading for non-integer hour increments.**Method 3:**Iterative Addition/Subtraction. Thorough and simulates actual time passage. Potentially slow and may be overkill for simple checks.**Method 4:**Checking Clock Hand Angles. Precise in terms of clock hand positioning. More complicated and not as intuitive as other methods.**Method 5:**Lambda Function. Quick and compact. Offers no insight into the logic and is less readable for beginners.

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.