A clock has two hands that do not rest. It seems easy to count how many times they are found, but the number that almost everyone says is not correct.
The clock and its hands
Riddle statement
On an analog watch, how many times a day do the hour hand and minute hand matches exactly?
Count each coincidence only once.
Show solution
Solution
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. $