Question: How many times a day do the minute and hour hands of a clock overlap?
Answer: Did you think the answer was 24 times? Well if you did, it’s time you think again. Let’s do some math.
In T hours, the minute hand completes T laps. In the same amount of time, the hour hand completes T/12 laps.
The first time the minute and hour hands overlap, the minute hand would have completed 1 lap more than the hour hand. So we have T = T/12 + 1. This implies that the first overlap happens after T = 12/11 hours (~1:05 am). Similarly, the second time they overlap, the minute hand would have completed two more laps than the hour hand. So for N overlaps, we have T = T/12 + N.
Since we have 24 hours in a day, we can solve the above equation for N
24 = 24/12 + N
24 = 2 + N
N = 22
Thus, the hands of a clock overlap 22 times a day. Thus the hands of the clock overlap at 12:00, ~1:05, ~2:10, ~3:15, ~4:20, ~5:25, ~6:30, ~7:35, ~8:40, ~9:45, ~10:50. Note that there is no ~11:55. This becomes 12:00.
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Excellent analysis!
Surely its 21, since the 22nd will actually be midnight of the 2nd day?
If we are counting the midnight as the second day then shouldn’t we count the midnight from previous day as today. In the end we count one of midnight (either the previous night or this night). So it ends up being 22.