Wednesday, February 17, 2021

At what times are the hands on a clock aligned?



At what times do the hour and minute hands on a clock face become perfectly aligned? Obviously 12:00 is one time. Now think about that at 6:00 where the hands are opposite each other. By the time the minute hand has moved to the 6, the hour hand has already advanced halfway from 6 to 7, and by the time the minute hand moves halfway from 6 to 7, the hour hand has advanced a little more, and so on. At 11:00, by the time the minute hand reaches the hour hand, the hour hand will already be at 12, and the hands will be aligned at 12:00 again. So the two hands actually only align 11 times during a 12-hour period, or 22 times in 24 hours. So the secret then is simply that the hands align at multiples of 12/11ths of an hour, or about 5 minutes and 27 seconds. This can be described as T=12n/11, where 'T' is the time and 'n' is the integers from 1 to 11. You can see in the table below, the right column has the alignment times, but there isn't one for 11, because it becomes 12.

One of my childhood mysteries is finally solved!




Other than 12:00, there are no times when the hour, minute, and second hands become perfectly aligned. See the link below.
















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