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 1 hour 5 minutes and 27 seconds. This can be described as T=n x 12/11, 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.

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