Opposing Hands On A Clock

Each hour, as the minute hand whips around, there will be exactly one time where it is opposite the hour hand. This would lead us to believe that there are 12 occurrences of the condition.

However, note that the occurrence for 5pm-6pm and 6pm-7pm is actually the same time ; specifically, it occurs at 6pm. Thus, we have over-counted by 1, so the minute and hour hand are opposite $12-1=11$ times.

The answer is 11. If you start at noon, the first occurrence happens around 12:32 or so. Then as you follow the time line around, each successive occurrence happens closer to the next hour. (these are approximations) 1:36, 2:42, 3:48. Then as you approach the next time, it is around 4:50. Almost 5:00. Then the next occurrence happens at 6:00. People tend to want to count the occurrence between 4:00 and 5:00 twice, forgetting it is the same time. At this point (6:00) it has happened six times. Then you just follow the time line naturally and you get five more times. 5+6=11

Starting at 6 oclock position, imagine the hour hand stays still and the minute hand rotates 360 (and become 7 oclock). In reality, the hour hand has moved 30 degree and therefore it is not aligned with the minute hand. Imagine now the minute hand moved another 30 degree to the number 1. It is still not aligned with the hour hand because it has moved another (30/12) degree and is not perfectly on the number 7 anymore. Thus, after 6 oclock, the minute hand needs to move another 360 degree plus 30 degree plus (30/12) degree plus so on degree to align itself with the hour hand. This is infinite geometric sequence and the sum can be calculated by 360/(1-(1/12)). Now, in a 12 hour period, the minute hand moved 360x12 degrees. Since the hands aligned every 360/(11/12) degree, the answer could be calculated by using the formula 360x12 divided by (360/(11/12)) which is equal to 360x12 x11/(360x12) which is 11