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The results of Weisberg and Alba (1981) and Lung and Dominowski (1985) indicate that, for most people, solving the Nine-Dot problem may require a large amount of relevant information. Even with a large amount of relevant information, however, not nearly everyone solves the problem. And when we examine the performance of those who solve the problem, we do not see anything like a sudden Aha! experience, where the solution simply falls into place. Even a relatively knowledgeable person has much difficulty solving the Nine-Dot problem. It seems that in this problem fixation is not simply blocking the solution from occurring.
A recent set of studies on the Nine-Dot problem provides additional information on why it usually is so difficult to solve and points further toward the importance of analytical processes in solving it. MacGregor, Ormerod, and Chronicle (2001) analyzed the problem from a cognitive perspective, looking at heuristic methods that might be applied to it and the planning capacities that are needed to carry out such methods. They assumed that people begin by trying to formulate a solution plan, based on a heuristic whereby each line that one draws should cover as many as possible of the remaining dots. That means that the first line will cover three dots, and the next three lines will cover two. After drawing four lines, one finds that there are still dots left to be covered, and so one fails. In order to solve the problem, one must be able to plan far enough ahead to realize that simply using the maximum-coverage strategy is doomed to failure.
The Question of Insight in Problem Solving
A. Weisberg and Alba practice problems
B. Lung and Dominowski practice problems
C. Lung and Dominowski strategy instructions
In order to solve each of these problems, you must extend some of the solution lines beyond the dots; that is, you should not always regard a dot as the place where you stop a line just drawn and where you start a new line. On the contrary, sometimes it is necessary to extend a line beyond the last dot on the line, to a point where you can start a new line connecting other dots. At least one line must end beyond the last dot on the line, and the next line will start beyond the dots on that line.
D. Lung and Dominowski results: Performance on the Nine-Dot problem
Group Solvers (%) Mean trials
Strategy + practice 59 8.5
Practice 22 13.4
Strategy 34 18.9
Control 9 19.7
E. MacGregor and colleagues problems
In each problem, connect all the dots with four straight lines (leftmost problem is the Nine-Dot problem).
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Figure 6.8 Studies of Nine-Dot problem Source: A, Weisberg and Alb (1981); B, Lung and Dominowski (1985); E, MacGregor, et al. (2001).
Creativity: Understanding Innovation
MacGregor and colleagues (2001) assume that most people cannot go beyond the maximum-coverage strategy, because doing so would require an ability to imagine the consequences of drawing four lines (which they refer to as a “look-ahead” of four moves). Most people do not have a large enough working-memory capacity to look ahead that far. That is, most people are not able to imagine in advance the consequences of carrying out their chosen strategy, and so they must actually carry it out before they can see that it will not work. MacGregor and colleagues provided support for their view by designing new versions of the Nine-Dot problem, which were structured so that the participant did not have to imagine as many lines in advance (see Figure 6.8E), and under those circumstances solution was much more frequent, a finding that supports their analysis.
The analysis of the Nine-Dot problem by MacGregor and colleagues
(2001), if it was pointed in the correct direction, also raises problems for the conclusion drawn by Lavric and colleagues (2000) that planning is not important in solution of insight problems. A recent study by Murray and Byrne (2005) also provided evidence that planning is important in insight: They found that people who performed best on a set of insight problems also performed well on a test of working-memory capacity. Although that study was correlational in design and therefore did not test for the causal influence of working memory on problem solution, the fact that a correlation was found between the two types of performances supports the hypothesis that planning is important in achieving insight.
One potentially interesting piece of information that MacGregor and colleagues (2001) did not obtain was an actual measure of the visual working memory capacities of their participants. People with larger visual working-memory capacities ought to be more likely to solve the Nine-Dot problem in its usual form. A related prediction is that if one could teach people strategies for visualizing, and in so doing increase their visual working-memory capacities, one would expect an increase in solutions of the Nine-Dot problem. Neither of these sorts of studies has been carried out as yet, so for the present the analysis of the Nine-Dot problem proposed by MacGregor and colleagues has received only indirect support, although it may be a promising way to analyze the problem. It is also of note that the analysis of MacGregor and colleagues does not assume that fixation plays a role in making the Nine-Dot problem difficult. Rather, in their view, what makes that problem difficult is that it puts heavy demands on the cognitive capacities of the would-be solver, and most people cannot meet those demands.