Problem Description

This problem asks you to calculate the angle between the hour and minute hands on an analog clock given a specific time.

Given two inputs:

• hour: an integer representing the hour (0-12 format)
• minutes: an integer representing the minutes (0-59)

You need to find the smaller angle formed between the hour hand and the minute hand on a clock face and return it in degrees.

Key points to understand:

• A clock is a circle with 360 degrees
• The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute
• The hour hand moves 360 degrees in 12 hours (720 minutes), which means it moves 30 degrees per hour or 0.5 degrees per minute
• The hour hand doesn't jump from hour to hour - it moves continuously. For example, at 3:30, the hour hand is halfway between 3 and 4
• Since there are two possible angles between any two clock hands (one going clockwise and one going counterclockwise), you need to return the smaller one
• The answer should be between 0 and 180 degrees

For example:

• At 3:00, the hour hand points at 3 and the minute hand points at 12, forming a 90-degree angle
• At 3:15, the minute hand is at 3 (90 degrees from 12), and the hour hand has moved slightly past 3 (at 97.5 degrees from 12), forming a 7.5-degree angle

Intuition

To find the angle between the clock hands, we need to determine where each hand is positioned on the clock face in terms of degrees from the 12 o'clock position.

Let's think about how each hand moves:

For the minute hand: It completes a full 360-degree rotation in 60 minutes. So in 1 minute, it moves 360/60 = 6 degrees. Therefore, at minutes minutes past the hour, the minute hand is at 6 * minutes degrees from 12 o'clock.

For the hour hand: This is trickier because the hour hand moves continuously, not in discrete jumps. It completes a full 360-degree rotation in 12 hours (720 minutes). This means:

• In 1 hour, it moves 360/12 = 30 degrees
• In 1 minute, it moves 30/60 = 0.5 degrees

At any given time, the hour hand's position depends on both the current hour and the minutes that have passed. For example, at 3:30, the hour hand isn't pointing directly at 3; it's halfway between 3 and 4.

So the hour hand's position is: 30 * hour + 0.5 * minutes degrees from 12 o'clock.

Once we have both positions, we calculate the absolute difference between them. However, this difference might be greater than 180 degrees. Since we want the smaller angle, if the difference is more than 180 degrees, we should take 360 - diff instead. This gives us the angle going the "shorter way" around the clock.

The final answer is: min(diff, 360 - diff), ensuring we always return the smaller of the two possible angles between the hands.

Learn more about Math patterns.

Solution Approach

The implementation follows directly from our understanding of how clock hands move:

1. Calculate the hour hand position:

   h = 30 * hour + 0.5 * minutes

   • The base position is 30 * hour degrees (since each hour represents 30 degrees)
   • We add 0.5 * minutes to account for the continuous movement of the hour hand as minutes pass

2. Calculate the minute hand position:

   m = 6 * minutes

   • The minute hand moves 6 degrees per minute from the 12 o'clock position

3. Find the absolute difference between the two positions:

   diff = abs(h - m)

   • This gives us one of the two possible angles between the hands

4. Return the smaller angle:

   return min(diff, 360 - diff)

   • If diff is greater than 180 degrees, then 360 - diff gives us the smaller angle going the other way around the clock
   • We use min() to automatically select the smaller of the two angles

For example, if the time is 3:15:

• Hour hand position: h = 30 * 3 + 0.5 * 15 = 90 + 7.5 = 97.5 degrees
• Minute hand position: m = 6 * 15 = 90 degrees
• Difference: diff = abs(97.5 - 90) = 7.5 degrees
• Final answer: min(7.5, 360 - 7.5) = min(7.5, 352.5) = 7.5 degrees

The algorithm runs in O(1) time and space complexity since we're only performing a fixed number of arithmetic operations.

Example Walkthrough

Let's walk through the solution with the time 9:30.

Step 1: Calculate the hour hand position

• The hour hand moves 30 degrees per hour and 0.5 degrees per minute
• At 9:30: h = 30 * 9 + 0.5 * 30 = 270 + 15 = 285 degrees from 12 o'clock

Step 2: Calculate the minute hand position

• The minute hand moves 6 degrees per minute
• At 30 minutes: m = 6 * 30 = 180 degrees from 12 o'clock

Step 3: Find the absolute difference

• diff = abs(285 - 180) = 105 degrees

Step 4: Return the smaller angle

• We have two possible angles: 105 degrees or 360 - 105 = 255 degrees
• min(105, 255) = 105 degrees

Therefore, at 9:30, the angle between the hour and minute hands is 105 degrees.

Let's verify with another example: 3:15

• Hour hand: h = 30 * 3 + 0.5 * 15 = 90 + 7.5 = 97.5 degrees
• Minute hand: m = 6 * 15 = 90 degrees
• Difference: diff = abs(97.5 - 90) = 7.5 degrees
• Final answer: min(7.5, 360 - 7.5) = 7.5 degrees

The smaller angle at 3:15 is 7.5 degrees.

Solution Implementation

Time and Space Complexity

Time Complexity: O(1)

• The algorithm performs a fixed number of arithmetic operations regardless of input values
• Calculating hour hand position: 30 * hour + 0.5 * minutes - constant time
• Calculating minute hand position: 6 * minutes - constant time
• Computing absolute difference: abs(h - m) - constant time
• Finding minimum between two values: min(diff, 360 - diff) - constant time
• All operations are basic arithmetic that execute in constant time

Space Complexity: O(1)

• The algorithm uses only a fixed amount of extra space
• Variables h, m, and diff store single numeric values
• No data structures that grow with input size are used
• The space requirement remains constant regardless of the input values

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Forgetting to Account for Hour Hand's Continuous Movement

The most common mistake is treating the hour hand as if it jumps discretely from hour to hour, rather than moving continuously as minutes pass.

Incorrect approach:

hour_angle = 30 * hour  # Wrong! This assumes hour hand stays at exact hour position

Correct approach:

hour_angle = 30 * hour + 0.5 * minutes  # Accounts for gradual movement

For example, at 3:30, the incorrect approach would place the hour hand at exactly 90 degrees (pointing at 3), when it should actually be at 105 degrees (halfway between 3 and 4).

2. Not Handling the 12 O'Clock Position Correctly

When the hour is 12, some might incorrectly calculate the hour hand position as 360 degrees instead of 0 degrees.

Potential issue:

# If hour = 12 and minutes = 30
hour_angle = 30 * 12 + 0.5 * 30 = 360 + 15 = 375 degrees  # Wrong!

Solution:

def angleClock(self, hour: int, minutes: int) -> float:
    # Normalize hour to 0-11 range
    hour = hour % 12
    hour_angle = 30 * hour + 0.5 * minutes
    minute_angle = 6 * minutes
    angle_difference = abs(hour_angle - minute_angle)
    return min(angle_difference, 360 - angle_difference)

3. Returning the Larger Angle Instead of the Smaller One

The problem specifically asks for the smaller angle between the two hands. Some solutions might forget to compare with the reflex angle.

Incorrect approach:

return abs(hour_angle - minute_angle)  # Could return angles > 180 degrees

Correct approach:

diff = abs(hour_angle - minute_angle)
return min(diff, 360 - diff)  # Always returns angle ≤ 180 degrees

For instance, if the calculated difference is 270 degrees, the correct answer should be 90 degrees (360 - 270), not 270 degrees.