Consider an analog clock. How many times a day do a clock’s hands overlap?
Solution: Nothing could be simpler, right? Watch out! The obvious thought is that clock hands overlap 12 times a day. Then if you’re clever, you realize you have to double that conclusion or 24 times a day. But that’s way too easy, so there must be a catch. If you think about it intuitively you’ll eventually conclude that the minute hand overtakes the hour hand somewhat less than once per hour. The obvious solution would be true only if the hour hand remained stationary, but, of course, the hour hand progresses at about 1⁄12 the rate of the minute hand. So a back-of-the-envelope solution suggests that the clock’s hands overlap every 11⁄12 hours or every 65 minutes.
While not exactly right, this solution is close. If the candidate gets this far, most interviewers will want to move on. But if the interviewer asks the candidate to be sure or prove it, the explanation will go some-thing like this, according to Roger Breisch of Batavia, Illinois:
Let’s think about the number of degrees traveled by each hand of the clock. As we know, a circle has 360 degrees. The minute hand travels 360 degrees per hour. The hour hand travels 360 degrees in 12 hours or 30 degrees per hour. Then the question is how long for the minute hand to “catch” the hour hand if the hour hand has a 360-degree head start?
Let x be the number of hours it takes the minute hand to catch the hour hand plus go an additional 360 degrees. So the degrees traveled by the minute hand (x hours 360 degrees/hour) must equal the degrees traveled by the hour hand PLUS 360 (x hours 30 degrees/hour + 360 degrees).
x hours 360 degrees/hour =
x hours 30 degrees/hour + 360 degrees
360x = 30x + 360
330x = 360
x = 1.090909 or 11⁄11 hours
Then 24 divided by 1.090909 or 11⁄11 = 22 overlaps per day.
A much simpler solution comes from the nimble mind of Katherine Lato of Warrenville, Illinois. A trainer at Lucent Technologies, Lato manages to make the clock-hands problem look trivial. Her solution is ingenious if not exactly rigorous, and I’m not sure it really proves anything, but I’d hire her in a New York minute. Let’s show her logic:
The solution is calculated by recognizing that the minute hand will go around an analog clock face 24 times in 24 hours, and the hour hand will go around 2 times in 24 hours. Since both hands travel in the same direction—clockwise—the minute hand will overtake the hour hand 24 2 times or 22 overtakes per day.
Answer: Every 11⁄11 hours, or 22 times a day.