Problem 2: Calculate exactly when the two hands coincide after 11:00. (Yes, I know, the answer is obvious. But calculate it anyway, as it shows that these methods work for extreme cases.) (Answer: 12:00)       It has been called to my attention by my cousin Dan (who also provided the clock diagrams) that the results of coincidence problems can be expressed much more simply and elegantly in terms of hours alone. Thus, e.g., the answer to the problem, "When are the hands coincident after 4:00?" can be rewritten thus: 4:21 9/11 4 + 21/60 + (9/11)(1/60)          4 + 231/660 + 9/660          4 + 240/660          4 + 4/11 Hours This tempts us to offer the following: THEOREM 1: Let n be a number on the clock. Then the time after n-o'clock at which the hands are coincident is n + n/11 hours. Proof: As we have seen above, such solutions begin with an equation of the form 12x = 5n + x (where everything is in minutes.) So x = 5n/11, and thus 12x = 60n/11. To convert this to hours, divide by 60. Thus the time of coincidence is n + n/11 hrs.  QED   Problem 3: At what time after 9:00 are the hands coincident? Express your answer in hours, and then convert to minutes. (Ans: 9 + 9/11 hrs;  9:49 1/11)