Please explain how to assess the angle between the hour and minute hands of a clock, based on the current time.

Question Analysis

This question is asking you to calculate the angle between the hour and minute hands of a clock given a specific time. It's a typical problem in coding interviews that tests your understanding of geometry and arithmetic operations. The problem requires you to understand how the positions of the hour and minute hands are determined by the time and how to translate these positions into an angle.

Answer

To solve this problem, you can follow these steps:

1. Understand the movements of the clock hands:

   • The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute (360/60 = 6 degrees).
   • The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees per hour (360/12 = 30 degrees). Additionally, it moves 0.5 degrees per minute because it continuously moves as the minutes pass (30 degrees per hour / 60 minutes = 0.5 degrees per minute).

2. Calculate the positions of the clock hands:

   • For the minute hand at M minutes, the angle from the 12 o'clock position is Minute Angle = M * 6.
   • For the hour hand at H hours and M minutes, the angle from the 12 o'clock position is Hour Angle = H * 30 + M * 0.5.

3. Calculate the angle between the two hands:

   • The difference in their positions gives the angle between them: Angle = |Hour Angle - Minute Angle|.
   • Since the angle between the hands can be greater than 180 degrees, you should return the smaller of the two possible angles: Angle = min(Angle, 360 - Angle).

Here's a concise code implementation in Python:

def clock_angle(H, M):
    # Calculate the angles
    minute_angle = M * 6
    hour_angle = H * 30 + M * 0.5
    
    # Calculate the angle between the two hands
    angle = abs(hour_angle - minute_angle)
    
    # Return the smaller angle
    return min(angle, 360 - angle)

# Example usage
H = 3
M = 30
print(clock_angle(H, M))  # Output should be 75

This function calculates the angle between the hour and minute hands based on given hours H and minutes M and returns the smallest angle.