A problem is a situation in which you have a goal but do not know how to achieve it. Thinking is a process of mental exploration of possible actions and states of the world. Problems can be well-defined problems or ill-defined problems. Well-defined problems are problems in which starting conditions, actions available and goals are all completely specified. Ill-defined problems are problems in which starting conditions, or actions available or goals are not completely specified. Problems can be knowledge-rich or knowledge-lean. Knowledge-rich problems are problems that require extensive specialist knowledge. Knowledge-lean problems are problems that do not require specialist knowledge. Problems may also be classified as non-adversary or adversary problems. Non-adversary problems are problems in which the solver is dealing with inert problem materials with no rational opponent. Adversary problems are problems in which the solver has to deal with a rational opponent.

There are two main historical approaches that are influential in problem-solving and thinking:

The Gestalt approach. This approach to thinking likens problem-solving to seeing new patterns. It stresses the role of insight and understanding of problem-solving. The key process was restructuring. This changing how one represents a problem. A restructuring that leads to a rapid solution is insight. The path to insight is often characterised by restructuring the overall problem in sub-problems. Set is a tendency to persist with one approach to a problem. Functional fixity is a difficulty in thinking of a novel use for a familiar object.

The information processing approach. The problem space is an abstract representation of possible states of a problem. There are two sub-types: state-action spaces (upside-down tree diagram) and goal-subgoal spaces. State-action space is a representation of how problems can be transformed from starting state through intermediate states to the goal The goal-subgoal space is a representation of how an overall problem goal can be broken down into subgoals and sub-subgoals. Three methods can be used to analyse the  state-action tree:

1. Depth-first search (light load on memory)
   Only one possible move is considered at a time.
2. Breadth-first search (heavy working memory load)
   Each possible move at each level is considered. It is an algorithm, this is a problem-solving method that is guaranteed to solve but may do so only with high mental load.
3. Progressive deepening (compromise method)
   The depth-first method is used to a limited depth. When depth limit is reached, the search backs up to start and repeats, avoiding previously explored branches and so on until the whole space has been searched up to the initial depth limit. If a solution is not found, the depth limit is increased. It is an algorithmic method but can be quicker than the breadth-first search, because it has the possibility of being lucky.

The method of hill-climbing uses intermediate evaluations. This is a heuristic method. A heuristic is a problem-solving method that often finds a low effort solution but is not guaranteed to solve the problem. Problems with a large possible alternative actions seem to lend themselves to a hierarchical form. This is subgoal analysis which is often referred to as problem reduction or as means-end analysis. Detour problems are problems in which the hill-climbing method does not work well, as the solver has to move away from the goal at some stage. An example of a detour problem is the hobbits and orcs problem.

Weisberg argues that insight problem-solving arises from ordinary processes of search and problem analysis without any need for special or unusual processes. Experiments using feelings as indicators of closeness to the solution and confidence in solving the solution correlate better with the solution in non-insight tasks than insight-tasks. This provides evidence for a sudden restructuring. FMRI scans showed increased brain activity in the right anterior superior temporal gyrus for insight solutions than for non-insight solutions. This finding supports the view that there are differences in neural processes between insight versus non-insight problem-solving. Thinking out loud impairs the solving of insight tasks, which provides evidence for the idea that insight problem solving uses different unconscious processes that cannot be verbally reported.

There are two main approaches to how to explain insight problem solving:

1. Representational change
   This approach states that we first try an ordinary problem-solving technique in order to solve an insight problem. This leads to an impasse and then restructuring occurs. After the restructuring, there is partial insight and this leads to the correct sequence of steps that achieve the solution or full insight.
2. Progress monitoring
   According to this approach, the main source of difficulty in insight tasks is the use of inappropriate heuristics. When people are looking for a solution, they monitor their progress against some criterion and failure to meet a progress criterion triggers restructuring, rather than impasses, which then lead to the insight solution. This approach uses the maximization heuristic, each move or decision is an attempt to gain as much headway as possible. Insight is most likely to occur when constraint relaxation follows criterion failure.

Expertise is the accumulated high-level knowledge that allows outstanding performance in complex problem areas. The acquisition of domain expertise seems to take at least ten years of extensive studying of the domain. Expertise typically seems to involve extensive memory for familiar patterns which cue appropriate actions. Experts represent problem situations differently from novices as they draw on a more elaborate set of schemata.

Problems that only have one correct solution are called convergent problems. Problems that have many possible answers are called divergent problems. Creative problems are problems to which the solution is novel (for the problem solver) and not coincidental. Boden distinguishes between historically creative and personally creative. Historically creative is novel in the history of the world. Wallas proposed an influential four-stage analysis of creative problem-solving:

1. Preparation
   This rarely leads to a solution and involves conscious work. This stage is essential.
2. Incubation
   The problem is set aside and there is no conscious work.
3. Illumination (inspiration or insight)
   Doesn’t always lead to a solution to the problem. A great idea comes to mind, but must be developed and verified.
4. Verification
   Conscious work must be done on ideas generated through illumination. Solutions can be tested and developed.

In the incubation period, a problem is set aside. It can be set aside immediately after the problem is presented or it can be set aside after conscious work, this is called delayed incubation. There are four main hypotheses regarding incubation effects:

1. Conscious work
   There is a possibility that although people stop pondering over the problem, during other activities there is some conscious work that is carried out.
2. Unconscious work
   This approach argues that incubation effects occur through active but unconscious processing of the problem material.
3. Fatigue dissipation
   This approach suggests that a break is simply an opportunity to rest and return to the problem with more energy.
4. Selective forgetting
   This approach states that misleading strategies, mistaken assumptions and related ‘mental sets’ weaken through forgetting and thus a fresh start or ‘set-shifting’ is facilitated.

In the Simon model to creative problem solving, the emphasis is on the rarity of creative outbursts and states that an individual should use information processing approaches. Incubation is analysed as familiarization with repeated attempts and selective forgetting in between attempts, allowing fresh approaches to be taken.

The geneplore model is a model for creative thinking with stresses the role of a generative and exploratory phase. Creative work involves an initial stage in which pre-inventive structures are generated and then are interpreted during an exploratory phase.

Small cues can have large unconscious effects. A deviant item (e.g: a pattern-breaking item on a poster) can be seen as a cue for unconventional thinking. Brainstorming is stimulating the production of unusual ideas by stressing quantity as against quality and deferment of evaluation of ideas. During brainstorming, criticism is ruled out and free-wheeling is welcomed. Environmental cues and brainstorming can increase creativity. Electrical groups perform better when brainstorming than real groups.