# 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 x^{i}, 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.