Analog Clock Puzzle

How many times do the hour and minute hands meet on an analog clock every day? What are those times?


22 times.

Starting at midnight, it would be roughly 12:00, 1:05:27, 2:10:55, 3:16:22, 4:21:49, 5:27:16, 6:32:44, 7:38:11, 8:43:38, 9:49:05, 10:54:33, and then the same 11 times starting at noon.

The easiest way to solve this:

  1. Every hour, the hour hand moves 1/12 of the way around the clock. Every hour, the minute hand moves the entire way around the clock.
  2. If we put this in terms of how much the minute hand moves relative to the hour hand every hour, this means minute hand effectively moves 1 – 1/12 = 11/12 of the way towards the hour hand each hour.
    • In other words, it would take 12/11 of an hour (roughly 65.455 minutes or roughly 1 hour, 5 minutes, and 27 seconds) for the minute hand to catch up to the hour hand again.
  3. Since there are 24 hours in a day, that would be 24 / (12/11) = 22 times the hands meet each day.

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