How often are the clock hands exactly opposite?

Question

An ordinary analogue clock has an hour hand and a minute hand, both moving smoothly.

In 24 hours, how many times are the two hands pointing in exactly opposite directions, that is, 180 degrees apart, as at 6 o'clock?

Be careful: the naive answer "once an hour, so 24" is wrong.

Accepted Answer

22 times. Relative motion. In 12 hours the minute hand completes 12 laps and the hour hand completes 1 lap. So the minute hand gains exactly 11 laps on the hour hand every 12 hours. During each gained lap, the minute hand is exactly half a lap (180 degrees) ahead of the hour hand exactly once. So the hands are opposite 11 times in 12 hours, and 22 times in 24 hours. Why not 24? The events are not spaced one hour apart. They are spaced 12/11 of an hour apart, about 65 minutes 27 seconds. Starting from 6:00, the next opposition is at about 7:05, then 8:11, 9:16, 10:22, 11:27, 12:33, 1:38, 2:44,…

How often are the clock hands exactly opposite?

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