Chapter 6, Problem 6.25E

(Knight’s Tour: Brute-Force Approaches) In Exercise 6.24 we developed a solution to the Knights Tour problem. The approach used, called the ‘accessibility heuristic, generates many solutions and executes efficiently.

As computers continue increasing in power, we’ll be able to solve many problems with sheer computer power and relatively unsophisticated algorithms. Let’s call this approach brute-force problem solving.

1. Use random number generation to enable the knight to walk around the chess board (in its legitimate L-shaped moves, of course) at random. Your program should run one tour and print the final chessboard. How far did the knight get?
2. Most likely, the preceding program produced a relatively short tour. Now modify your program to attempt 1,000 tours. Use a one-dimensional array to keep track of the number of tours of each length. When your program finishes attempting the 1000 tours, it should print this information in a tabular format. What was the best result?
3. Most likely, the preceding program gave you some “respectable tours but no full tours. Now “pull all the stops out” and simply let your program run until it produces a full tour. [Caution: This version of the program could run for hours on a powerful computer.] Once again, keep a table of the number of tours of each length and print this table when the first full tour is found. How many tours did your program attempt before producing a full tour? How much time did it take?
4. Compare the brute-force version of the Knight’s Tour with the accessibility-heuristic version. Which required a more careful study of the problem? Which algorithm was more difficult to develop? Which required more computer power? Could we be certain (in advance) of obtaining a full tour with the accessibility-heuristic approach? Could we be certain (in advance) of obtaining a full tour with the brute-force approach? Argue the pros and cons of brute-force problem solving in general.