A useful analogy for problem solving is to think of the problem as a maze. To solve the problem, you need to travel from a starting point, called the initial state, to an ending point, called the goal state. The initial state includes all the knowledge and resources you currently have available. The goal state refers to the solution that you want to reach. To get from the initial state to the goal state, you have a number of tools you can use, called operators. In a maze, these operators might be "left turn" and "right turn." The set of possible ways to travel from the initial state to the goal state is called the problem space.

The most obvious way to attempt to solve a problem is to simply mentally test every possible path leading from initial state to goal state. This imaginary process of trial and error, called the generate-and-test strategy, considers all possible operators available at every step. The number of operators available is limited by path constraints; for example, in a game of chess, you cannot leap your king across the board. However, even with these constraints, the number of possible paths is overwhelming. For example, imagine that you wanted to make the best possible opening move in a game of chess. There are twenty moves open to you, and there are twenty possible moves open to your opponent. So, considering only the first cycle of play, you must already consider four hundred possible paths. As the game progresses, there are more operators available to each player, so we can consider forty possible moves for each turn. For two cycles of play, then, there are 640,000 paths; for three cycles, over one billion. As you can see, generate-and-test is not a realistic problem- solving method for most complex problems. Instead, we must rely on strategies that consider only a subset of the possibilities.

One strategy that some people use is the "hill-climbing strategy." The hill- climbing strategy states that you should always move toward the goal state. Imagine that you want to solve a maze using this strategy. At any fork, you would always take the direction that led toward the goal state. However, this strategy is fallible; the correct path often winds around and away from the goal state before returning to it later. This is also true for more complex problems, like the river-crossing type. In the river-crossing problems, for example, where the initial state is a number of predators and prey on one side of the river and a boat with a limited capacity, and the goal state is to have everyone on the opposite side of the river, you often must transport beings back to the original side of the river. It may seem like they're going in the wrong direction, but in fact this "downhill" step is necessary to solve the problem. This type of problem is difficult precisely because people are often inclined use the hill-climbing strategy to move toward the goal state.

Another, more effective strategy is to break each problem down into a few subproblems. Evidence suggests that people actually work problems through this way. When working through complex problems, people tend to work very quickly at some points and pause at others. Analysis of their progress suggests that they work quickly when solving a subproblem, and work slowly when deciding what subproblem to tackle next. Smaller problems are easier to solve in general because there are fewer possible paths to consider. As seen in the section on subgoals, experts often break down problems into subproblems and subgoals, suggesting that this strategy is most useful for familiar problems.

Some problems, especially unfamiliar ones, are best solved by starting at the goal state and working backward toward the initial state. Many people solve mazes this way, because there are generally fewer choices to consider at the end than at the beginning. This strategy, while not always helpful, can reduce the overwhelming number of paths available from the initial state. As an example, try working backward on the Water Lilies Problem:

Water lilies are growing on Blue Lake. The water lilies grow rapidly, so that the amount of water surface covered by lilies doubles every twenty-four hours. On the first day of summer, there was just one water lily. On the ninetieth day of the summer, the lake was entirely covered. On what day was the lake half covered? (Answer)

One method that works for many different types of problems is reasoning by analogy. In reasoning by analogy, the problem-solver uses his knowledge about previous, similar problems to find on the best way to solve the current one. Use of analogies hinges on familiarity, since analogies are not helpful without some previous experience with similar problems. Analogies are simply patterns in the structures of problems. Because experts are more likely to see patterns in problems' structures, as we saw in the section on expertise, they are best at reasoning by analogy. Novices tend to examine the superficial aspects of a problem, such as comparing two problems that are both about flowers, rather than the structural aspects of how the problem is set up, such as comparing two problems that both require subtraction.

Mental imagery is often useful in problem solving. Verbal descriptions of problems can become confusing, and a mental image can clear away excessive detail to bring out important aspects of the problem. Imagery is most useful with problems that hinge on some spatial relationship. However, if a problem requires an unusual solution, mental imagery along can be misleading, since it is difficult to change one's understanding of a mental image. In many cases, it helps to draw a concrete picture, since that picture can be turned around, played with, and reinterpreted, yielding new solutions in the way that a mental image cannot.