Problem 125 Solution
Let both hands be at 12. It will take the hour hand 1 hour to return to where the minute hand started. However at that time the minute hand will have advanced 1/12 of an hour. When the hour hand gets there the minute hand will have advanced (1/12)2 or an hour. When the hour hand gets there the minute hand will have advanced (1/12)3 or an hour. So the total distance the hour hand will cover until it reaches the minute hand is the sum for i= 0 to infinite of (1/12)i. Everyone should know that the infinite series for xi, where x<1 and i from 0 to infinity is 1/(1-x). In this case the answer is 1/(1-1/12) = 1/(11/12) = 12/11 = 65.45 minutes.
Björn from Germany suggested another solution. Let both hands be at 12. Over the next 12 hours they will cross 11 times. Thus they every 12/11 hours.
Michael Shackleford, A.S.A.