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Relative Speeds of Hour and Minute Hands:
•  The minute hand completes a full circle (360 degrees) in 60 minutes. Its speed is 6 degrees per minute.
•  The hour hand completes a full circle (360 degrees) in 12 hours (720 minutes). Its speed is 0.5 degrees per minute.
Equation for Coincidence:
Let t be the number of minutes past a given hour 'H'. We can set the angular positions of the hands equal to find when they coincide.
•  Position of minute hand: θ_m = 6t (degrees from 12 o'clock, clockwise)
•  Position of hour hand: θ_h = 30H + 0.5t (degrees from 12 o'clock, clockwise, where H is the hour number 0-11)
For the hands to coincide, their positions must be equal (modulo 360 degrees):
6t ≡ 30H + 0.5t (mod 360)
Rearranging the equation:
5.5t ≡ 30H (mod 360)
(11/2)t = 30H + 360k, for some integer k
11t = 60H + 720k
t = (60H + 720k) / 11
Coincidences in a 12-hour Cycle:
The hour and minute hands coincide 11 times in any 12-hour period. These 11 distinct coincidences occur approximately at:
1.  12:00:00 (exact)
2.  ~1:05:27 (1 hour, 5 minutes, 27 seconds)
3.  ~2:10:54
4.  ~3:16:21
5.  ~4:21:49
6.  ~5:27:16
7.  ~6:32:43
8.  ~7:38:10
9.  ~8:43:38
10. ~9:49:05
11. ~10:54:32
(The next coincidence would be exactly 12 hours after the first 12:00:00 mark).
Counting within the specified duration [3 PM to 3 AM]:
Let's list the coincidence times, converting them to PM/AM for the specified interval:
•  12:00:00 PM (noon)            -  Not in the interval [3 PM, 3 AM]
•  ~1:05:27 PM                     -  Not in the interval [3 PM, 3 AM]
•  ~2:10:54 PM                     -  Not in the interval [3 PM, 3 AM]
•  ~3:16:21 PM                     -  In the interval (1st occurrence)
•  ~4:21:49 PM                     -  In the interval (2nd occurrence)
•  ~5:27:16 PM                     -  In the interval (3rd occurrence)
•  ~6:32:43 PM                     -  In the interval (4th occurrence)
•  ~7:38:10 PM                     -  In the interval (5th occurrence)
•  ~8:43:38 PM                     -  In the interval (6th occurrence)
•  ~9:49:05 PM                     -  In the interval (7th occurrence)
•  ~10:54:32 PM                    -  In the interval (8th occurrence)
•  12:00:00 AM (midnight)        -  In the interval (9th occurrence)
•  ~1:05:27 AM                     -  In the interval (10th occurrence)
•  ~2:10:54 AM                     -  In the interval (11th occurrence)
•  ~3:16:21 AM                     -  Not in the interval (after 3 AM)
Thus, the hour and minute hands coincide 11 times within the 12-hour duration from 3 PM to 3 AM.