When the Hands Sit on Top of Each Other
The problem
Sometime between and the hour hand and the minute hand of an ordinary clock lie exactly on top of one another. At what time does that happen, to the nearest second?
- The minute hand sweeps a full in minutes; the hour hand sweeps in hours.
- Both hands move continuously rather than jumping once per minute.
Take a minute
Reveal hint 1
Give both hands a speed in degrees per minute and track the angle between them as a function of .
Reveal hint 2
At the hour hand already leads by . The hands meet when the minute hand has made up exactly that gap.
Solution
AnswerAbout — precisely minutes past three.
Derivation
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Assign speeds.
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Measure the initial gap.
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Set up the catch-up equation.
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Convert the fraction to seconds.
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Sanity-check the pattern.