On Time

On a clock the second, minute and hour hands are on the same axis. How many times in a 24 hour day will the second hand be parallel to either of the other two hands?

This is an extension of the Hands Together problem from November 1996. You might like to look at the solution to that problem.

Ling Xiang Ning has cracked this tough nut, well done Xiang Ning! Here is his answer:

First consider the second and minutes hands. In one hour, the minute hand makes one revolution and the second hand goes round 60 times. This means that, in one hour, the second hand passes over the minute hand 60 - 1 = 59 times and the two are also in line (but with 180 degrees between them) 59 times. In 24 hours the two are parallel (24*59*2 = 2832) times.

Now consider the second and hour hands. In one hour the hour hand makes 1/12 revolution while the second hand makes 60 revolutions. So, in 24 hours, the hour hand makes (24*1/12 = 2) revolutions and the second hand goes round (24*60=1440) times. This means that, in 24 hours, the second hand passes over the minute hand 1438 times and the two are also in line (but with 180 degrees between them) another 1438 times. In 24 hours the two are parallel 2876 times.

All three hands, the second, the minute and the hour hands, are parallel 4 times in every 24 hours, once if they are aligned at 12 midnight and then at 06.00 and 12.00 and at 18.00.

So the second hand is parallel to either the minute hand or the hour hand [(2832 + 2876) - 4 = 5704] times.