Warm-up Solution:
59 minutes would require 11 clicks of MM and 4 clicks of the minute button (M), or 15 clicks. This is a big improvement over 59 clicks.
Dennis Shasha is at the Courant Institute of Mathematical Sciences, New York University. His most recent puzzle book, Puzzles for Programmers and Pros, was published in May 2007 by John Wiley and Sons/Wrox.