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How many times during the day do the hands of a clock overlap?

Imagine an analog clock set to 12 o'clock. Note that the hour and minute hands overlap. How many times each day do both the hour and minute hands overlap? How would you determine the exact times of the day that this occurs?


By the time the minute hand has gone all the way round the clock and is back at 12, one hour later (i.e., at 1 o'clock), the hour hand has moved to 1. Five minutes later, the minute hand reaches 1, but they don't quite meet there because the hour hand would have moved 1/12 of the distance between 1 and 2, to indicate 5 minutes past 1. So the next time after 12 that the minute hand is directly over the hour hand is a bit after 1:05 (we'll calculate the exact time a bit later). 


Similarly, the next time it happens is a bit after 2:10. Then a bit after 3:15, and so on. 



Note that each time they meet, the number of minutes past the hour keeps increasing, so the hour hand would have moved closer to the next number. After 11 o'clock, the minute hand has to travel all the way and by the time they meet it is has to be 12 o'clock again, since we know what the clock looks like at that time. So the two hands overlap 11 times in a 12 hour period. So, in a 24 hour period, they would overlap 22 times.

To answer the second part of the question, let's try to figure out the little bit of extra time the minute hand needs to catch up to the hour hand after every 1:05 hours. Well, after 12 o'clock there are eleven occasions when the two hands match up, and since the clock hands move at constant speeds, those 11 events are spread equally apart around the clock face, so they are 1/11th of an hour apart. That's 5.454545 minutes apart. In other words they meet every after every 1 hour and 5.454545 minutes.

The precise times they overlap (in hours) would be 1 + 1/11, 2 + 2/11, 3+ 3/11, all the way up to 11 + 11/11, which is 12 o'clock again.


Questions related to this:

Q. An analog clock is showing 2 o'clock right now. At what time will the hour and minute hands overlap next?
Q. What is the angle between the hour and minute hands at 4:10?
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