Problem Solving

What is the difference between a well-defined and an ill-defined problem?

A well-defined problem has a clearly stated goal, known constraints, and a clear solution path (e.g. Sudoku, a familiar maths formula). An ill-defined problem lacks a clearly defined goal or clear paths to a solution — either too few constraints (what to watch on Netflix) or too many (fitting events into a packed schedule).

What is a routine problem, and how does it differ from an insight problem?

A routine problem is well-defined AND has a known operator — you've encountered it before and can solve it with little effort or creativity, almost automatically. An insight problem requires a sudden restructuring of your mental representation. Progress is not gradual — the solution arrives as an 'aha' moment, often after an incubation period (stepping away from the problem). Metcalfe & Weibe showed that people solving insight problems do not feel progressively 'warmer', unlike routine problems.

Explain the Hobbits & Orcs problem: what type is it and what makes it hard?

It is a well-defined problem that requires insight. Participants must move hobbits and orcs across a river without orcs ever outnumbering hobbits on either side. It is hard because a correct move sometimes requires moving backwards — taking someone back across — which conflicts with the intuitive hill-climbing heuristic. People resist moves that appear to take them further from the goal, even when such moves are necessary.

Explain the Box & Candle problem: what type is it and what makes it hard?

It is an insight problem demonstrating functional fixedness. Participants are given a candle, tacks, and a box of matches and must fix the candle to the wall. The solution is to tack the box to the wall as a shelf. The difficulty is that people fixate on the box's usual function as a container, ignoring its potential as a platform. Insight is needed to break this fixedness and restructure the mental representation of the problem.

Explain the Water Jar problem: what type is it and what makes it hard?

It is a well-defined problem illustrating entrenchment (Einstellung). Participants use jars of different sizes to measure exact volumes. After solving several problems with the same complex method, they develop a mental set and fail to notice a simpler solution available for later problems. It is an over-reliance on a past successful strategy — a temporary inability to shift away from an established mental set.

Why do we rely on algorithms, and what is their key limitation?

Algorithms guarantee a correct solution by systematically searching the entire problem space — every possible solution is considered. We rely on them when accuracy is essential and errors are costly. The limitation is that they are time-consuming: for large or complex problem spaces, checking every option is computationally impractical.

What is analogical transfer? Explain Gick & Holyoak's radiation and army problems.

Analogical transfer is using the same solution structure for a new problem that shares the same underlying logic. In Gick & Holyoak, the radiation problem asks how to destroy a tumour with rays without harming surrounding tissue. The army problem describes a general splitting forces to converge on a fortress from multiple directions. The structural solution is identical — converge multiple weak forces at one point. Participants who read the army story first were much more likely to solve the radiation problem, showing that surface differences do not block transfer when the deep structure is recognised.

What are the five steps of the IDEAL model, and what are the key ways we can explore the problem space?

Identify → Define and represent → Explore solutions → Act → Look back. During the Explore phase we can: use an algorithm (exhaustive search), apply heuristics such as hill-climbing (move closer to the goal) or means-ends analysis (reduce the gap between current and goal state, set sub-goals), use analogical transfer (apply a past solution), or draw on expertise and prior knowledge. The choice of exploration method depends on problem type and available knowledge.