How often the hands meet
In one full day, from midnight up to but not including the next midnight, how many times do the hour hand and the minute hand of a clock point in exactly the same direction?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The minute hand turns once an hour, the hour hand once every 12 hours. How fast does the minute hand gain on the hour hand?
Each time the minute hand gains a full turn, the hands meet.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The hands meet each time the minute hand laps the hour hand. Count the laps.
Step 1. In 12 hours the minute hand makes 12 turns and the hour hand 1, so the minute hand gains 11 turns.
Step 2. The hands meet once per turn gained: 11 times in 12 hours, every hours, starting at midnight.
Step 3. In 24 hours that is times.
The trap. “Once an hour” gives 24. But between 11:00 and 1:00 the hands meet only once, at 12:00: the meeting that would fall “around 11:59” is the noon meeting. The same reasoning gives 44 right angles per day, not 48.