Posted: April 9th, 2015

Problem-Solving

Unit 8 – Problem-Solving

Activity 2: The Tower of Hanoi
Try to solve The Tower of Hanoi Problem:
Figure 1 – The tower of Hanoi Problem

You are presented with three pegs and three rings of differing sizes. Initially all three rings are on peg A and your task is to move all three rings to peg C. However, you may only move one ring at a time, you cannot place any ring on top of a smaller one, and you cannot put the pegs on the floor or anywhere else.

As I suspect you won’t have either rings or pegs to try solving this task, you need to adapt the problem by
drawing three circles on a piece of paper and label them A, B & C. Now place three different sized coins in circle A with the largest at the bottom and the smallest at the top. Now transfer all of the coins to circle C, remembering that you can only move one at a time and that a coin cannot be placed on one smaller than itself.

How many moves did you take?

Try this with other people and observe how people solve the problem.

Describe the initial state of the problem, the goal state and the legal moves (= things that can be done) and restrictions (things that can’t be done)?

What is the minimum number of moves needed to solve the problem? Post your answers to your Discussion Board Group.

A site that allows you to try the Tower of Hanoi problem electronically can be found here: Tower of hanoi DHTML game

If you like, you can try a similar problem: Missionaries & Cannibals

Types of Problems
Several types of problems are commonly distinguished in problem-solving research:

The Tower of Hanoi is a problem that has been used in many laboratory-based studies. Our above list of six example problems contains some other commonly used laboratory problems (How can I draw three lines through a 3 by 3 array of dots without taking my pencil off the paper) but it also gives real world problems (e.g., How can I get a pay increase?) Real world and laboratory problems differ in how well they are defined. Traditionally, problem-solving research has used well defined problems, which means the solver is provided with all the information that is needed to solve the problem. There are four different sorts of information: Information about the initial state of the problem.
•    Information about the initial state of the problem
•    Information about the goal state
•    Information about legal operators (things that can be done)
•    Information about operator restrictions (things that can’t be done)

When this mean-ends analysis is complete, a mental plan has been produced consisting of a sequence of
operators that can be applied (in the reverse order in which they were generated) to enable the main goal to be attained.
Activity 4
Describe a concrete example of means-ends analysis in a real-world situation.

Post your answers to your Discussion Board Group

The examples above (and your own example) indicate that heuristics such as means-ends analysis are
powerful strategies even when dealing with everyday-type problems and tasks. Some everyday problems are well-defined and have (relatively) small problem spaces to search through. Smyth et al. (1994) argue that general-purpose heuristics such as means-ends analysis are useful if there is no previous specialised
knowledge which enables us to solve a problem (Smyth et al.,1994). After all, if there were a fool-proof method (= algorithm) for getting that ‘A’- grade in an exam, somebody would have sold it for a lot of money by now!

However, in many real-world problem-solving situations, the problem space is not easily defined. Think of various workplace scenarios, for example detectives in a murder enquiry, medical doctors trying to find effective treatments when dealing with varied and ambiguous features in their clinical practice (Norman, 2005), and many more. The actual formulation and mental representation of the problem, the selection of heuristics, recognising when a goal or sub-goal is reached and many other aspects of problem-solving are not well defined. The mental representation of such problem spaces is affected by the solver’s experience and expertise.
Evaluation of Newell & Simon information-processing theories of problem-solving:
Positive aspects of this approach:
•    It emphasised the need to be explicit about the mental operations (and sequences of operations) by which people solve problems, in marked contrast to the Gestaltists’ “sudden insight”.
•    It provides a normative theory of problem-solving: the notion of problem space allows us to specify the idealised structure of a problem and the ideal solution path. This provides a clear basis from which to empirically study how and why people’s actual problem-solving behaviour deviates from the ideal.
•    Theories are explicit about the kinds of heuristic strategies that people adopt to tackle problems and therefore make strong predictions about what they will do, when they will succeed and when they will fail.
•    Many of these predictions are borne out in early problem research.
Aspects of this approach that have been criticised include:
•    Early problem-solving research focused on a very narrow class of problems (i.e., puzzle-like lab problems).
•    For many problems (in the real world), there is no easily defined problem space. How and when heuristics are applied in complex real-world problem-solving is a matter for further on-going research.

Problem-Solving Success (and Failure)
Finally, it’s worth pointing out that a wide range of research has looked into successful problem-solving, with obvious potential impact for everyday problem-solving. This brief section aims to give you a flavour of the many aspects such research investigated.

Previous experience and knowledge can be assumed to make solving of a current problem faster and
easier. Research into skilled performance provides evidence for this (for example research into expert chess players, see Charness et al., 2001). However, past experience sometimes disrupts and slows down current problem-solving, as we have discussed in the context of functional fixedness. Clearly, we need a better understanding of when previous knowledge and experience help and when they are detrimental. Eysenck & Keane (2010) and Robinson-Riegler & Robinson-Riegler (2012) provide good discussions of this research.

Problem representation is a critical component of successful problem-solving, as Robinson-Riegler &
Robinson-Riegler (2012) point out. This applies to all aspects of the problem space, i.e., the mental
representation of the initial and goal states and of the legal operators. Failure in problem representation might result from many factors, such as insufficient attention being directed to certain problem elements, or lack of understanding of problem elements, or, again, functional fixedness, i.e., an overly rigid representation of problem elements due to previous experience. Individual differences in problem representation might be at play, such as anxiety due to stereotype threat: the expectation that a negative stereotype (‘girls are bad at maths’) is going to be used to judge one’s performance can adversely affect problem-solving (Robinson-Riegler & Robinson-Riegler, 2012).

The format of presenting problems can have effects on the solutions being produced (see framing effects in decision-making, Eysenck & Keane, 2010).