Problem Statement

Given two integers hour and minutes, return the smaller angle (in degrees) between the hour hand and the minute hand of a clock.

Example 1:

Input: hour = 12, minutes = 30
Output: 165

Example 2:

Input: hour = 3, minutes = 30
Output: 75

Example 3:

Input: hour = 3, minutes = 15
Output: 7.5

Constraints:

• 1 ≤ hour ≤ 12
• 0 ≤ minutes ≤ 59
• Answers must be within [0, 180].

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Intuition

A clock has two hands — hour and minute — each rotating at different speeds.

We must find the smaller angle between them at a given time.

Key Observations:

1. The clock is a 360° circle.
2. The minute hand moves:
   • 360° in 60 minutes → 6° per minute.
3. The hour hand moves:
   • 360° in 12 hours → 30° per hour.
   • It also moves slightly every minute → 0.5° per minute.

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Formula Derivation

1. Minute hand angle = minutes * 6
2. Hour hand angle = (hour % 12) * 30 + (minutes * 0.5)
   • (hour % 12) ensures that 12 o’clock is treated as 0 hour.
3. Difference = |hour_angle - minute_angle|
4. Smaller angle = Math.min(diff, 360 - diff)

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Java Solution

public class Solution {
    public double angleClock(int hour, int minutes) {
        double minuteAngle = minutes * 6;
        double hourAngle = (hour % 12) * 30 + (minutes * 0.5);
        double diff = Math.abs(hourAngle - minuteAngle);
        return Math.min(diff, 360 - diff);
    }
}

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Complexity Analysis

• Time Complexity: O(1)
  All operations are constant-time arithmetic.
• Space Complexity: O(1)
  No extra space used.

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Dry Run

Let’s dry run with hour = 3, minutes = 15.

Step	Formula	Calculation	Result
Minute hand angle	minutes * 6	15 * 6	90°
Hour hand angle	(3 % 12) * 30 + (15 * 0.5)	90 + 7.5	97.5°
Absolute difference		97.5 – 90
Smaller angle	min(7.5, 360 – 7.5)	min(7.5, 352.5)	7.5°

Output: 7.5

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Example 2: hour = 12, minutes = 30

Step	Formula	Calculation	Result
Minute hand angle	30 * 6	180°
Hour hand angle	(12 % 12) * 30 + (30 * 0.5)	0 + 15	15°
Absolute difference		180 – 15
Smaller angle	min(165, 195)	165°

Output: 165

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Example 3: hour = 9, minutes = 0

Step	Formula	Calculation	Result
Minute hand angle	0 * 6	0°
Hour hand angle	(9 % 12) * 30 + (0 * 0.5)	270 + 0	270°
Absolute difference		270 – 0
Smaller angle	min(270, 90)	90°

Output: 90

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Key Takeaways

1. The minute hand moves 6° per minute.
2. The hour hand moves 0.5° per minute.
3. Always take the smaller angle between the two hands.
4. Using modulo (hour % 12) ensures the formula works even for hour = 12.