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You can quickly prove to yourself that these six are the only ways that A, B, and C can be arranged. However, you can derive the same number by using a theorem from the branch of mathematics called combinatorics the study of the way things can be combined. According to the theorem, the number of ways that N objects can be arranged is equal to N! (N factorial). The factorial of a number is the product of all whole numbers equal to or less than itself down to 1. Therefore, 3! is 3 2 1, or 6. If you had four objects to arrange, there would be 4!, or 24, permutations. With five objects, the number is 120, and with six it is 720. With 1000 objects the number of possible permutations is huge! The graph in Figure 10-2 gives you a visual feel for what is sometimes referred to as a combinatoric explosion. Once there are more than a handful of possibilities, it very quickly becomes difficult to examine (indeed, even to enumerate) all the arrangements. This same sort of combinatorial explosion can occur in paths through search spaces. Because of this, only the simplest of problems lend themselves to exhaustive searches. An exhaustive search is one that examines all nodes. Thus, it is a brute-force technique. Brute force always works but is not often practical for large problems because it consumes too much time, too many computing resources, or both. For this reason, AI-based search techniques were developed.

Figure 10-2

pivot hole as well as drilling a 1 4-inch hole near the base of the key bit. An engraving chisel with a 1 16-inch-wide cutting edge is used to remove material from one side of the keyhole. A regular cold chisel can then score the line on the opposite side of the keyhole (Fig. 23.6). The waste can then be broken away from the center of the keyhole. Clean up any rough edges with a file. 7. Rivet the front bolt guide in place. The rear bolt guide is fitted to the backplate but not riveted at this time. The bolt will need to be removed later to fit the mainspring and key.

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When TanZen was completed in late July of 2008, I was ready for a much-needed vacation I had been working twenty-hour days for the final month or two of the project During TanZen s first five months on the App Store, I continued at my day job, all the while enjoying all the extra income I kept busy making updates to the game, but I wasn t finding time to create anything new The App Store was not a steady source of income, as sales varied wildly from week to week It was time to make a new game to bring in some more cash and spread out the risk When talking to people about the App Store, I liken the medium to the music industry Applications generally become popular quickly or not at all.

There are several ways to search for a solution. The four most fundamental are Depth first Breadth first Hill climbing Least cost In the course of this chapter, each of these searches is examined.

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Evaluating the performance of an AI-based search technique can be complicated. Fortunately, for our purposes we are concerned only with these two measurements: How quickly a solution is found How good the solution is There are several types of problems for which all that matters is that a solution, any solution, be found with the minimum effort. For these problems, the first measurement is especially important. In other situations, the quality of the solution is more important. The speed of a search is affected both by the size of the search space and by the number of nodes actually traversed in the process of finding the solution. Because backtracking from dead ends is wasted effort, you want a search that seldom retraces its steps. In AI-based searching, there is a difference between finding the best solution and finding a good solution. Finding the best solution can require an exhaustive search because sometimes this is the only way to know that the best solution has been found. Finding a good solution, in contrast, means finding a solution that is within a set of constraints it does not matter if a better solution might exist. As you will see, the search techniques described in this chapter all work better in certain situations than in others, so it is difficult to say whether one search method is always superior to another. But some search techniques have a greater probability of being better for the average case. In addition, the way a problem is defined can sometimes help you choose an appropriate search method.

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