Richard Griffiths - Lecture Notes

Problem Solving Phenomena 

This lecture describes a range of phenomena observed to occur when humans are engaged in problem solving, and which affects the outcome.

Issues of representation

The way that a problem is represented can influence the ease by which it is solved.

Correct representation

Mutilated Checkerboard problem  (Wickelgren 1974)

A problem that demonstrates the significance of representation in finding a solution.

You are presented with a checkerboard in which two diagonally opposite corner squares have been cut out.  62 squares remain.  You are also given 32 dominoes, each of which covers exactly two squares on the board.

The problem is to show if there is an arrangement of the dominoes that will exactly cover all of the squares on the board.  If it can not be done, prove it.

Solution
It can not be done.
Each domino must cover one black and one white square.
The mutilated board contains more black than white squares.  QED!

When the problem is presented as "... dominoes, each of which covers one black and one white square" it is comparatively easy to solve.

Functional fixedness

Difficulty in solving a problem when the conventional or normal properties or use of an object cause an alternative use, which would solve the problem, not to be perceived.

Solution can be primed by previous experience in which the relevant property is pointed out.

Experimental demonstrations:  Maier’s two string problem (Maier 1931), and Dunker’s candle problem (Dunker 1945).

Set Effects

People may become biased by experience to prefer certain approaches to a problem, which may block the solution in a particular case — the einstellung effect (mechanization of thought).

Luchins’ water-jug experiment (Lurchin 1942, 1959)

The subject is given a set of jugs of various stated capacities, and is asked to measure out a desired quantity of water.
 
 Problem
Capacity of Jug A
Capacity of Jug B
Capacity of Jug C
Desired quantity
1
21
127
3
100
2
14
163
25
99
3
18
43
10
5
4
9
42
6
21
5
20
59
4
31
6
23
49
3
20
7
15
39
3
18
8
28
76
3
25
9
18
48
4
22
10
14
36
8
6
All problems except 8 can be solved by B - 2C - A.
For problems 1 through 5 this solution is simplest.
For problem 7 and 9 the simpler solution is A + C.
Problem 8 cannot be solved by B - 2C - A, but can be solved by A - C.
Problems 6 and 10 can be solved more simply as A - C.

Subjects who worked through all problems in order:

83% used B- 2C - A on problems 6 and 7.
64% failed to solve problem 8.
79% used B - 2C - A on problems 9 and 10.
Subjects who saw only last 5 problems.
Fewer than 1% used B - 2C - A.
Only 5% failed to solve problem 8.
Problem can be overcome by warning subjects.
After problem 5, Lurchins told some subjects “Don't be blind”, which caused more than 50% to find the simpler solution on the remaining problems.

Incubation effect

Where interruption of the task improves eventual success rate.

Examples from scientific literature, Poincaré (mathematical discovery whilst taking a walk on the beach), Tesla (invention of the alternating current motor after years of thinking about the problem, while quoting Goeth, inspired by watching a sunset), Kekulé (discover of benzene rings, who dreamed of carbon atoms dancing in a circle and joining hands).
 

The cheap-necklace problem experiment (Silveira 1971)

“You are given four separate pieces of chain that are each three links in length.  It costs 2¢ to open a link and 3¢ to close a link.  All links are closed at the beginning of the problem.  Your goal is to join all 12 links of chain into a single circle at a cost of no more than 15¢.”

Control group:

Worked on the problem for half an hour.
55% solved the problem.
Experimental group 1:
Worked for half an hour, interrupted by a half-hour break in which other activities were performed.
64% solved the problem.
Experimental group 2:
As 1, but with a 4 hour break.
85% solved the problem.
Subjects were asked to talk as they worked on the problem.  They came back to the problem where they left off, and did not have preformed solutions.

Could be explained by set effects.  Subjects will bring particular knowledge structures to bear on solving the problem.  If however they are not appropriate, the subject may be stuck with them through the process of spreading activation.  Taking a break may allow the activation to subside, and other structures get a chance.

References

Anderson, J. R.  1985  "Cognitive Psychology and Its Implications" (Second Edition).  W.H. Freeman and Co.

Duncker, K.  1945  "On problem solving" (translated by L. S. Lees)  Psychological Monographs, 58, No. 270.

Lurchins, A. S.  1942  "Mechanization in problem solving"  Psychological Monographs, 54, No. 248.

Lurchins, A. S., & Lurchins, E. H.  1959  "Rigidity of behaviour: A variational approach to the effects of einstellung"  University of Oregon Books.

Maier, N. R. F.  1931  "Reasoning in humans II.  The solution of a problem and its appearance in consciousness"  Journal of Comparative Psychology, 12, 181-194.

Silveira, J.  1971  "Incubation: The effect of interruption timing and length on problem solution and quality of problem processing"  unpublished doctoral dissertation, University of Oregon, reported in Anderson 1985.

Wickelgren, W. A. 1974  "How to solve Problems"  W. H. Freeman & Co.
 
 
 

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