How often do the hour and and minute hand of an analog clock point in the same direction?

The hands start pointing the same direction at midnight, then the hour hand moves clockwise (!) by 360° in 12 hours. The minute hand moves clockwise at the rate of 360° in one hour. So when does the minute hand, moving at 360°/ hour lap the hour hand moving at 30° per hour? When

360t = 30t mod 360

or equivalently

12t = t mod 12

which means t is a multiple of 12/11.

So the times are at 12n/11 hours for n = 0, 1, 2, …, 21, which means 22 times a day. The first time after midnight is 12/11 hours later, or at 5/11 of a second past 1:05.

If you also consider the second hand, the three hands will only line up at midnight and noon.

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