The shuttle puzzle - A lesson in problem solving

Beginning computer science students often have great difficulty with the analysis and design of solutions to programming related problems. Using the Shuttle Puzzle problem as a vehicle the process of analyzing a problem and synthesizing observations into an algorithm is presented in a detailed, step-by-step manner. Introduction When I was child and went to visit my grandparents, I would often entertain myself with one of a variety of puzzles that magically populated their bookcase. One of the puzzles that I frequently tried, but could never master, was a homemade puzzle consisting of a strip of wood with nine holes. Placed on opposite ends of the strip were marbles of the same color, the center hole was left empty. The object of the puzzle was to reverse the marbles using one of two permissible moves. A marble could slide forward into an empty space, or a marble could jump another marble of the opposite color providing the new location was empty and the jump was in a forward direction. Invariably after making four or five moves I would arrive at a dead end, most often with a marble trapped on one end surrounded by a sea of marbles of the opposite color. With practiced hands my grandfather would rearrange the marbles to their starting locations and swiftly execute a sequence of moves to solve the puzzle. Much later in life, while teaching Computer Science classes at the University of Hartford, I ran into the same puzzle again both as a programming exercise and as a qualifying problem in a computing contest competition (USA Computing Olympiad, 1993). The puzzle, called a Shuttle Puzzle, can be formally described as: A Shuttle Puzzle of size 4 consists of 4 white marbles, 4 black marbles, and a strip of wood with 9 holes (as shown below). The marbles of the same color are placed in the holes at the opposite ends of the strip leaving the center hole empty. The object of the puzzle is to completely reverse the marbles on the strip. In general, a shuttle puzzle of size N consists of N white marbles, N black marbles and 2N+1 holes. There are only two permissible types of moves. You may slide 1 marble 1 space (into an empty hole) or jump 1 marble over 1 marble of the opposite color (into an empty hole). You may not backup, and you may not jump 2 (or more) marbles at once. Write a program that solves the Shuttle Puzzle for sizes 1 <= N <= 10. Remembering my earlier frustrations at attempting a solution, I approached the problem with renewed interest. In generating a solution I found that the Shuttle Puzzle offers an excellent opportunity to teach beginning Computer Science students the rudiments of how