In the 24 hours of a day, how many times does the hour, minute, and second hands of the clock overlap? What is the time? How did you calculate it?

In the 24 hours of a day, how many times does the hour, minute, and second hands of the clock overlap? What is the time? How did you calculate it?

Only twice.
Assuming that the hourly rate is ω (ω = 2 π/12 hours = π/6 hours), the sub-needle's angular velocity is 12 ω (12 ω = 2 π/60 minutes ), the second-hand angular velocity is 72 ω (72 ω = 2 π/60 seconds ).
Assume that the time when the sub-needle overlaps with the hour hand is T, then there are 12 ω T-ω t = 2 π, t = 12/11 hours, which is converted into 1 hour, 5 minutes, 27.3 seconds, obviously, the second-hand needle does not overlap with the hour-hand minute-hand. It can also be calculated that the second-hand needle cannot overlap with the other 10 times when the hour-hand is heavy. Only and will overlap.