On planet K-Pax, the dial of a clock has 14 equally spaced markings. The day is divided into 28 hours; each hour is divided into 70 minutes and each minute is divided into 70 seconds. Find that number of times that the hour hand and minute are exactly opposite to each other in a day.

Hi,

In this clock, the hour hand will move 360/14 degrees per hour and the minute hand will move 360 degrees per hour. So, the ratio of their speeds is 1:14

We know that in circular motion, if two person move at ratio a:b where a and b are co-prime, they will meet at b-a unique points.

This means that in this clock, they will meet at 13 points. Now, between each of these meetings, the hour hand and the minute hand will be opposite to each other exactly once. Thus, they will be opposite to each other exactly 13 times.