On an analog watch, how many times a day do the hour hand and minute hand matches exactly? Count each match only once.
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The clock and its hands
The best thing about The Clock and Its Hands is that it seems to go one way and resolves another. That little twist is what turns a correct problem into a memorable puzzle.
Hints
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- It is not 24 times: between one overlap and the next, a bit more than one hour passes.
- In 12 hours, the hands overlap 11 times, not 12.
- Think about how much the minute hand gains on the hour hand in one hour.
Solution
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Answer: The hands coincide exactly 22 times per day.
Explanation: What matters is how quickly the minute hand gains on the hour hand. In one hour, the minute hand completes one full turn while the hour hand advances \(1/12\) of a turn, so the gain is:
$$ 1-\frac{1}{12}=\frac{11}{12}. $$
To overlap again, the minute hand must gain one full turn, which takes:
$$ \frac{1}{11/12}=\frac{12}{11} $$
hours. Therefore, in 12 hours there are:
$$ \frac{12}{12/11}=11 $$
overlaps, and in 24 hours:
$$ 11\times2=22. $$
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