When insight just won't come: the failure of visual cues in the nine-dot problem.

The nine-dot problem is a classic in the field of human problem solving. Cognitive accounts of the problem's difficulty have been criticized on the grounds that the experimental methods on which they rely for support involve a qualitative change to the task requirements of the problem. The three experiments reported here utilize visual and visual-procedural hints to examine the notion that its difficulty is rooted in a mismatch between problem shape and solution shape. Experiment 1 demonstrated that a perceptual cue to the shape of the solution in the form of shading gave rise to only minimal improvements in performance; an additional hint about the relevance of the shading gave rise to modest, but not statistically significant, improvements. Experiment 2 replicated these findings against an additional control condition in which a solely verbal hint to violate the perceptual boundary of the problem shape was given. Furthermore, when both the verbal and visual hints were provided, performance improved only slightly. Experiment 3 provided participants with experience in producing the shape of the correct solution in problem variants closely related to the nine-dot problem. Performance on the transfer task, the basic nine-dot problem, remained at floor, however. These data suggest that visual constraint relaxation is unlikely to be the sole process by which the insight required to find a solution is achieved. The results are interpreted in terms of a previously proposed computational model of performance.

Information processing and insight: a process model of performance on the nine-dot and related problems.

The 9-dot problem is widely regarded as a difficult insight problem. The authors present a detailed information-processing model to explain its difficulty, based on maximization and progress-monitoring heuristics with lookahead. In Experiments 1 and 2, the model predicted performance for the 9-dot and related problems. Experiment 3 supported an extension of the model that accounts for insightful moves. Experiments 4 and 5 provided a critical test of model predictions versus those of previous accounts. On the basis of these findings, the authors claim that insight problem solving can be modeled within a means-ends analysis framework. Maximization and progress-monitoring heuristics are the source of problem difficulty, but also create the conditions necessary for insightful moves to be sought. Furthermore, they promote the discovery and retention of promising states that meet the progress-monitoring criterion and attenuate the problem space.

A model of human performance on the traveling salesperson problem.

A computational model is proposed of how humans solve the traveling salesperson problem (TSP). Tests of the model are reported, using human performance measures from a variety of 10-, 20-, 40-, and 60-node problems, a single 48-node problem, and a single 100-node problem. The model provided a range of solutions that approximated the range of human solutions and conformed closely to quantitative and qualitative characteristics of human performance. The minimum path lengths of subjects and model deviated by average absolute values of 0.0%, 0.9%, 2.4%, 1.4%, 3.5%, and 0.02% for the 10-, 20-, 40-, 48-, 60-, and 100-node problems, respectively. Because the model produces a range of solutions, rather than a single solution, it may find better solutions than some conventional heuristic algorithms for solving TSPs, and comparative results are reported that support this suggestion.

Evaluating the importance of the convex hull in solving the Euclidean version of the traveling salesperson problem: reply to Lee and Vickers (2000).

Lee and Vickers (2000) suggest that the results of MacGregor and Ormerod (1996), showing that the response uncertainty to traveling salesperson problems (TSPs) increases with increasing numbers of nonboundary points, may have resulted as an artifact of constraints imposed in the construction of stimuli. The fact that similar patterns of results have been obtained for our "constrained" stimuli, for a stimulus constructed under different constraints, for 13 randomly generated stimuli, and for random and patterned 48-point problems provides empirical evidence that the results are not artifactual. Lee and Vickers further suggest that, even if not artifactual, the results are in principle limited to arrays of fewer than 50 points and that, beyond this, the total number of points and number of nonboundary points are "diagnostically equivalent." This claim seems to us incorrect, since arrays of any size can be constructed that will permit experimental tests of whether problem difficulty is influenced by the number of nonboundary points, or the total number of points, or both. We present a reanalysis of our original data using hierarchical regression analysis which indicates that both factors may influence problem complexity.

Global perceptual processing in problem solving: the case of the traveling salesperson.

The traveling salesperson problem (TSP) consists of finding the shortest tour around a set of locations and is an important task in computer science and operations research. In four experiments, the relationship between processes implicated in the recognition of good figures and the identification of TSP solutions was investigated. In Experiment 1, a linear relationship was found between participants' judgments of good figure and the optimality of solutions to TSPs. In Experiment 2, identification performance was shown to be a function of solution optimality and problem orientation. Experiment 3 replicated these findings with a forced-pace method, suggesting that global processing, rather than a local processing strategy involving point-by-point analysis of TSP solutions, is the primary process involved in the derivation of best figures for the presented TSPs. In Experiment 4, the role of global precedence was confirmed using a priming method, in which it was found that short (100 msec) primes facilitated solution identification, relative to no prime or longer primes. Effects of problem type were found in all the experiments, suggesting that local features of some problems may disrupt global processing. The results are discussed in terms of Sanocki's (1993) global-to-local contingency model. We argue that global perceptual processing may contribute more generally to problem solving and that human performance can complement computational TSP methods.

Spatial and contextual factors in human performance on the travelling salesperson problem.

The travelling salesperson problem (TSP) provides a realistic and practical example of a visuo-spatial problem-solving task. In previous research, we have found that the quality of solutions produced by human participants for small TSPs compares well with solutions from a range of computer algorithms. We have proposed that the ability of participants to find solutions reflects the natural properties of human perception, solutions being found through global perceptual processing of the problem array to extract a best figure from the TSP points. In this paper, we extend the study of human performance on the task in order to understand further how human abilities are utilised in solving real-world TSPs. The results of experiment 1 show that high levels of solution quality are maintained in solving larger TSPs than had been investigated previously with human participants, and that the presence of an implied real-world context in the problems has no effect upon performance. Experiment 2 demonstrated that the presence of regularity in the point layout of a TSP can facilitate performance. This was confirmed in experiment 3, where effects of the internality of point clusters were also found. All three experiments were consistent with a global, perceptually based approach to the problem by participants. We suggest that the role of perceptual processing in spatial problem-solving is an important area for further research in both theoretical and applied domains.

Enhancing the usability of a task analysis method: a notation and environment for requirements specification.

This paper presents a notation and computer-based tool for the Sub-Goal Template (SGT) method, a task-analytic approach to specifying information requirements. In particular, it focuses upon the ergonomic redesign of a notation used in the SGT method for specifying contingent sequences in operators' tasks. Two experiments are reported in which two notations for redescribing sequences that involve making a choice between task alternatives are compared: a disjunctive form (Either ... or ...)-used in the original SGT scheme, and a conditional form with a redundant negative (If ... then ... if not ...), which has been promoted as a notation for capturing procedural sequences in computer programming. In experiment 1, performance with the conditional notation was better than with the disjunctive notation for redescribing simple and moderately complex task plans, although there was no difference for the most complex task plans. In experiment 2, a computer environment for specifying task plans using the SGT method was compared with a paper-based equivalent. In general, the computer environment was found to be easier to use and led to greater accuracy in plan redescription, although it was slower perhaps as a result of the constraints that it placed on participants. This paper argues for the necessity of ergonomic design in developing notations and tools to support task analysis methods.