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Types and Characteristics of Problems in Problem Solving

"Solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim that was not immediately understandable. Solving problems is the specific achievement of intelligence and intelligence is the specific gift of mankind."
— George Polya (1962)

Introduction​

In our day-to-day life we constantly solve problems — in the classroom, family, or workplace. Problem solving is nearly inescapable. We engage in it whenever we want to reach a goal that is not immediately available, and something is blocking our path.

To appreciate the complexity of problem solving, try these warm-up problems:

Warm-Up Problems
  1. What one mathematical symbol placed between 2 and 3 gives a number greater than 2 and less than 3?
  2. Rearrange the letters NEWDOOR to make one word.
  3. How many pets do you have if all of them are birds except two, all are cats except two, and all are dogs except two?

(Answers at the bottom of this page)

Problems span a vast range: from personal decisions (who to marry, career choices) to scientific challenges (curing cancer, proving theorems) to everyday tasks (preparing a meal, planning a study schedule). Understanding the types of problems we face — and why some are harder than others — is the foundation of the psychology of problem solving.

Source

📄 MPC-001 Block-4/Unit-1.pdf - Pages 5-8 📚 MPC-001 Cognitive Psychology, Learning and Memory, IGNOU

🎬 Video Resources​

Types of Problems: A Spectrum​

Problems vary along a spectrum from ill-defined to well-defined:

Well-defined problems (mathematical equations, jigsaw puzzles):

  • The nature of the problem is clear
  • The information needed to solve it is available
  • One can make straightforward judgments about whether a potential solution is appropriate

Ill-defined problems (how to bring peace, how to improve one's relationships):

  • The specific nature of the problem may be unclear
  • The information required to solve it is even less obvious
  • Multiple solutions may be valid; no clear criterion for success

📖 Wikipedia: Problem solving — Comprehensive overview of problem solving types, strategies, and research.

Greeno's Classification of Well-Defined Problems​

Greeno (1978) proposed a classification of well-defined problems based on the psychological skills and knowledge required to solve them. He identified three core types:

1. Arrangement Problems​

The problem solver must rearrange or recombine elements in a way that satisfies certain criteria. Usually, several different arrangements can be made, but only one or a few will produce the correct solution.

Examples:

  • Anagram problems: Rearrange NEWDOOR to spell one word → WOODERN or DOORMEN
  • Jigsaw puzzles: Fit pieces into the correct configuration
  • String arrangement problems: Classic lab tasks where you must tie two strings together

Key skill required: Spatial reasoning and systematic search through possible arrangements.

2. Problems of Inducing Structure​

The person must identify existing relationships among elements and then construct a new relationship among them to solve the problem. The solver must determine not only the relationships but also the structure and size of the elements involved.

Examples:

  • Analogy problems: "Doctor is to patient as teacher is to ___?"
  • Number series: "2, 4, 8, 16, ___ ?"
  • Pattern recognition tasks: Identifying the rule governing a sequence

Key skill required: Relational reasoning and abstract thinking — seeing the underlying pattern.

3. Transformation Problems​

One takes into consideration an initial state and attempts to change it to a goal state through a series of permitted operations.

The Tower of Hanoi is the classic example:

  • Initial state: Three disks stacked on peg A (largest at bottom)
  • Goal state: All three disks on peg C in the same order
  • Operators: Move only one disk at a time; never place a larger disk on a smaller one

According to Greeno (1978), solving transformation problems primarily requires skills in planning based on a method called means-end analysis — identifying differences between the current state and goal state, then selecting operations to reduce those differences.

📖 Wikipedia: Tower of Hanoi — The mathematical puzzle used to study transformation problem solving.

Other transformation problem examples:

  • River-crossing puzzles (farmer, fox, chicken, grain)
  • Rubik's Cube
  • Chess problems (achieving checkmate in n moves)
Comparison Table
Problem TypeCore TaskKey SkillExample
ArrangementRearrange elements to fit criteriaSpatial reasoning, systematic searchAnagrams, jigsaw puzzles
Inducing StructureIdentify and construct relationshipsRelational reasoning, abstract thinkingAnalogies, number series
TransformationMove from initial state to goal statePlanning, means-end analysisTower of Hanoi, chess

Characteristics of Difficult Problems​

Not all problems within these categories are equally hard. Researchers have identified specific characteristics that make problems difficult to solve:

Detailed Explanation of Each Characteristic​

Clarity-related difficulties:

CharacteristicDefinition
IntransparencyLack of clarity about the nature of the situation itself
Commencement opacityConfusion about how to even start stating or approaching the problem
Continuation opacityContinuing confusion throughout the problem — no clarity emerges as work progresses
InexpressivenessInability to express the problem clearly to oneself or others

Goal-related difficulties:

CharacteristicDefinition
PolytelyThe problem has multiple goals; reaching and selecting a particular goal is difficult
OppositionGoals within the problem conflict with each other

Complexity-related difficulties:

CharacteristicDefinition
ComplexityLarge numbers of items, too many interrelationships and decisions required
EnumerabilityIt is not possible to list or quantify all aspects of the problem
ConnectivityA hierarchy of problems exists — solving one opens up others
HeterogeneityThe problem is not homogeneous, making consistent handling difficult

Dynamic difficulties:

CharacteristicDefinition
TransienceThe problem keeps changing while one is trying to solve it
DynamicsTime plays a central role in the problem structure
Temporal constraintsA time limit applies — must be solved within a period
Temporal sensitivityThe problem is influenced and affected by timing
Phase effectsChanges occur in different phases of the problem
Dynamic unpredictabilityThe problem involves high unpredictability and complexity
Real-World Application

Complex, real-world problems like climate change or healthcare system reform exhibit all of these characteristics simultaneously — they are intransparent, polytelic, heterogeneous, transient, and dynamically unpredictable. This is why real-world problem solving is so much harder than laboratory tasks.

📖 Wikipedia: Wicked problem — A category of problems that exhibit many of these difficult characteristics; used in policy and design fields.

Why Lab Problems vs. Real-World Problems Differ​

Early psychological research on problem solving used simple, well-defined laboratory tasks (like the Tower of Hanoi) on the assumption that they captured the essential properties of real-world problems. Researchers could:

  • Trace problem-solving steps precisely
  • Control variables systematically
  • Define clear success criteria

However, real-world problems differ critically from lab problems:

  • They are emotionally loaded (clinical psychology research shows poor emotional control disrupts problem solving — Rath et al., 2004)
  • They lack clear success criteria
  • They involve multiple conflicting stakeholders
  • They change while being solved (transience)
Recent Research

A 2022 meta-analysis in Psychological Bulletin (Sala & Gobet) examined whether training on laboratory problem-solving tasks (like the Tower of Hanoi) transfers to real-world problem solving. Finding: near-zero transfer to novel real-world problems. This supports the view that real-world problem solving requires domain-specific knowledge, not just general problem-solving skill.

Problem Solving as a Cognitive Process​

Problem solving is considered one of the most complex of all intellectual functions — a higher-order cognitive process that requires modulation and control of more routine skills.

It consists of two related processes:

1. Problem Orientation — the motivational/attitudinal/affective approach to problematic situations (how one feels about and frames the problem)

2. Problem-Solving Skills — the actual cognitive-behavioural steps used to reach a solution

When cognitive skills are successfully implemented, they lead to effective problem resolution.

📖 Wikipedia: Cognition — The mental representation and manipulation of information used in problem solving.

Key definitions:

  • Baron (2001): Problem solving involves efforts to develop or choose among various responses in order to attain desired goals
  • Witting and Williams (1984): Problem solving is the use of thought processes to overcome obstacles and work towards goals

🌐 External Resources​

🧠 Memory Aids​

Greeno's Three Types — AIT​

Arrangement (rearrange elements) · Inducing Structure (find/build relationships) · Transformation (move from initial to goal state)

Difficult Problem Characteristics — ICTEP-CEHTTDP​

Group by theme:

  • Clarity: Intransparency, Commencement opacity, Continuation opacity, Expressiveness (lack of)
  • Goals: Polytely, Opposition
  • Complexity: Complexity, Enumerability, Heterogeneity, Connectivity
  • Dynamics: Transience, Temporal constraints, Temporal sensitivity, Dynamics, Phase effects, Dynamic unpredictability

Well-Defined vs. Ill-Defined — the "GPS" analogy​

Well-defined = GPS with a known destination (clear route, clear arrival).
Ill-defined = Wandering without knowing your destination (unclear route, unclear success).

✅ Self-Assessment Questions​

  1. What is a problem? How does problem solving differ from routine task completion?
  2. Distinguish between well-defined and ill-defined problems with two examples of each.
  3. Describe Greeno's three types of well-defined problems with an example of each.
  4. What is the Tower of Hanoi, and which of Greeno's problem types does it represent?
  5. List any five characteristics that make a problem difficult.
Warm-Up Answers
  1. A decimal point → 2.3 (greater than 2, less than 3)
  2. DOORMEN (or WOODNER)
  3. You have 3 pets — one bird, one cat, and one dog (each "except two" = the other two species)

🔗 Related Topics​

Source PDFs: