How often do the clock hands overlap?

Question

On an ordinary analog clock, the hour and minute hands occasionally line up perfectly, one exactly on top of the other (like at 12:00).

How many times do the two hands overlap in a 12-hour period? Careful, it's not 12.

Accepted Answer

The hands overlap 11 times in 12 hours (and 22 times in a full 24-hour day). Think in terms of relative speed. The minute hand completes a full lap every 60 minutes; the hour hand completes a lap every 12 hours. In 12 hours, the minute hand goes around 12 times and the hour hand goes around once. An overlap happens each time the faster minute hand "laps" the slower hour hand, i.e., gains a full lap on it. Count the laps gained. Over 12 hours the minute hand makes 12 revolutions while the hour hand makes 1, so the minute hand gains 12 - 1 = \mathbf{11} full laps on the hour hand. Each lap gain…

How often do the clock hands overlap?

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