The Problem
Imagine an analog clock set to 12 o’clock. Note that the hour and minute hands overlap. How many times each day do both the hour and minute hands overlap? How would you determine the exact times of the day that this occurs?
The Solution
My original solution was this:
The hands overlap 24 times a day. They overlap once each hour. They overlap at 12, 1:05, 2:10, 3:15, 4:20, 5:25, 6:30, 7:35, 8:40, 9:45, 10:50 and 11:55. This cycle happens twice, once each for AM and PM.
Bruce Guenter however corrected me in a comment. According to Bruce:
They will overlap 22 times a day, 11 times each for AM and PM. To understand why, ask where the hour hand will be at each of the times you list above. At 6:30, the hour hand will be half way between :30 and :35, so the real overlap will be around 6:33.
At 11:55, the hour hand will be almost at 12:00. By the time the minute hand catches up to the hour hand, it is 12:00 again, back to the first overlap.
Thanks Bruce 
I've got a masters degree in computer science and over 10 years of experience building web-based systems using Java/J2EE, Ruby, Rails and PHP. I'm a strong believer in the effectiveness of Agile Methods.
3 Comments
Uhhhh, not quite. They will overlap 22 times a day, 11 times each for AM and PM. To understand why, ask where the hour hand will be at each of the times you list above. At 6:30, the hour hand will be half way between :30 and :35, so the real overlap will be around 6:33.
At 11:55, the hour hand will be almost at 12:00. By the time the minute hand catches up to the hour hand, it is 12:00 again, back to the first overlap.
Happy puzzling.
Very subtle Bruce. Nice.
Oh, and to find out the exact times of day when they would overlap, you have to use some (relatively) simple linear algebra:
angle_h = 5h + 5m/60 = 5h + m/12
angle_m = m
(angles are in terms of minute markers = 6 degrees)
So, to find where they are the same angle, solve for m:
angle_h = angle_m
5h + m/12 = m
11m/12 = 5h
m = 60h/11
At 12 (h=0), they overlap at 0 minutes, and at h=11 they overlap at 60 minutes (ie 12:00 again). At h=1, they overlap at m=5.4545… which is 1:05:27.2727… and so on.