Problem Solving Science: How the Mind Finds Solutions

What Makes Something a Problem

In cognitive science, a problem exists whenever there is a gap between where you are (the initial state) and where you want to be (the goal state), and the path between them is not immediately obvious. Allen Newell and Herbert Simon formalized this idea in their theory of problem spaces. A problem space consists of all possible states that can be reached from the initial state through legal operations, and solving the problem means finding a path through this space to the goal.

Problems vary along several dimensions. Well-defined problems have clear initial states, goal states, and legal operations, like a chess puzzle or a mathematics equation. Ill-defined problems have vague or ambiguous goals, like deciding on a career or designing a building. Most real-world problems are ill-defined, which makes them considerably harder to study in the laboratory but more representative of the challenges people actually face.

Problems also differ in whether they require reproductive thinking (applying known methods to familiar problem types) or productive thinking (generating novel approaches to unfamiliar challenges). The Gestalt psychologists, working in the early twentieth century, were particularly interested in productive thinking and the role of insight in problem solving.

Problem-Solving Strategies

Cognitive scientists have identified several general strategies that people use when solving problems. Algorithms are step-by-step procedures guaranteed to produce a correct solution if followed correctly. Long division is an algorithm, as is the method for solving a Rubik cube by following a memorized sequence of moves. Algorithms are reliable but often impractical for complex problems because the number of steps required may be astronomically large.

Heuristics are mental shortcuts that usually lead to good solutions but are not guaranteed to work. Means-ends analysis, identified by Newell and Simon, is one of the most powerful general-purpose heuristics. It involves comparing the current state to the goal state, identifying the largest difference between them, and then selecting an operation that reduces that difference. This process is repeated until the goal is reached. Hill climbing is a simpler heuristic that involves always choosing the option that seems to move closest to the goal, though this can sometimes lead to dead ends when the best path requires temporarily moving away from the goal.

Working backward is a strategy that starts from the goal state and works toward the initial state. This is particularly useful when the goal is clearly specified and there are fewer possible operations leading to it than away from the initial state. Mathematicians frequently use this approach when constructing proofs.

Analogical reasoning involves recognizing that a current problem shares structural features with a previously solved problem and transferring the solution strategy. Dedre Gentner has shown that people are surprisingly poor at noticing useful analogies unless they are explicitly prompted, even when the analogous solution would be highly effective. People tend to focus on surface features (whether two problems look similar) rather than structural features (whether they share the same underlying logic).

Insight and the Aha Moment

Some problems are solved not through gradual, step-by-step progress but through a sudden flash of insight, an aha moment when the solution becomes apparent all at once. The Gestalt psychologists studied insight extensively, using problems like the nine-dot problem (connect nine dots arranged in a square using four straight lines without lifting the pen) and the candle problem (attach a candle to a wall using only a box of thumbtacks and matches).

Karl Duncker described the concept of functional fixedness, the tendency to see objects only in terms of their typical function, which blocks insight into novel uses. In the candle problem, people struggle because they see the box as a container for thumbtacks rather than as a potential shelf for the candle. Overcoming functional fixedness requires restructuring the mental representation of the problem, seeing familiar elements in a new way.

Neuroimaging studies by Mark Jung-Beeman and colleagues have shown that insight solutions are associated with a burst of activity in the right anterior superior temporal gyrus, a brain region involved in making distant semantic associations. This finding suggests that insight involves connecting ideas that are not normally associated with each other, and that the right hemisphere of the brain may play a special role in this process. Interestingly, people are more likely to solve problems through insight when they are in a positive mood, possibly because positive affect broadens the scope of attention and facilitates access to remote associations.

Expertise and Problem Solving

Expert problem solvers do not simply think faster than novices; they think differently. Research on chess expertise by Adriaan de Groot and later by Herbert Simon and William Chase showed that chess masters recognize meaningful patterns of pieces (chunks) that allow them to evaluate positions quickly and identify promising moves. A grandmaster can look at a chess position and immediately recognize it as similar to positions seen in thousands of previous games, while a novice must analyze each piece individually.

This pattern-based approach to expertise generalizes across domains. Expert physicians recognize disease patterns, expert programmers recognize code structures, and expert firefighters recognize situational patterns that indicate danger. In each case, extensive practice has built a large library of stored patterns that allow the expert to bypass the slow, deliberate problem-solving strategies that novices must rely on.

Anders Ericsson identified deliberate practice as the key factor in developing expertise. Deliberate practice involves working on tasks that are just beyond your current ability, receiving immediate feedback, and focusing on correcting weaknesses. Simple repetition is not enough; the practice must be structured, effortful, and targeted at specific aspects of performance that need improvement. Ericsson estimated that achieving true expertise in any complex domain requires roughly 10,000 hours of deliberate practice, though the exact number varies across fields and individuals.

Obstacles to Effective Problem Solving

Several cognitive tendencies can interfere with effective problem solving. Confirmation bias causes people to seek evidence that supports their current hypothesis while ignoring evidence that contradicts it. Mental set, or Einstellung, is the tendency to apply a familiar strategy even when a better approach is available. Abraham Luchins demonstrated this with his water jug problems, where participants who learned a complex solution method continued to use it even when a much simpler solution became available.

Anchoring effects can bias problem solving when an initial estimate or approach disproportionately influences subsequent thinking. Framing effects change how people approach a problem depending on how it is described: the same medical decision can produce different choices depending on whether outcomes are described in terms of survival rates or mortality rates.

Stress and time pressure also impair problem solving by narrowing attention, reducing working memory capacity, and increasing reliance on habitual rather than flexible strategies. Under pressure, people are more likely to fall back on familiar approaches and less likely to notice creative alternatives, which is why high-stakes situations often produce worse decision making than low-stakes ones.

Computational Models of Problem Solving

The General Problem Solver (GPS), created by Newell and Simon in 1957, was one of the first computer programs designed to simulate human problem-solving behavior. GPS used means-ends analysis to work toward goals in a variety of problem domains. While GPS could solve well-defined problems, it struggled with the ill-defined problems that humans face in everyday life.

Modern computational approaches to problem solving include constraint satisfaction models, which represent problems as sets of variables that must satisfy certain conditions simultaneously, and reinforcement learning models, which learn problem-solving strategies through trial and error with feedback. These computational frameworks help cognitive scientists formalize their theories and generate testable predictions about human behavior.