The dynamics of search, impasse, and representational change provide a coherent explanation of difficulty in the nine-dot problem

Abstract

The nine-dot problem is often used to demonstrate and explain mental impasse, creativity, and out of the box thinking. The present study investigated the interplay of a restricted initial search space, the likelihood of invoking a representational change, and the subsequent constraining of an unrestricted search space. In three experimental conditions, participants worked on different versions of the nine-dot problem that hinted at removing particular sources of difficulty from the standard problem. The hints were incremental such that the first suggested a possible route for a solution attempt; the second additionally indicated the dot at which lines meet on the solution path; and the final condition also provided non-dot locations that appear in the solution path. The results showed that in the experimental conditions, representational change is encountered more quickly and problems are solved more often than for the control group. We propose a cognitive model that focuses on general problem solving heuristics and representational change to explain problem difficulty.Keywords: Nine-Dot Problem, Insight, Heuristics, Search, Representational Change Nine-dot problem 3There is growing evidence that an insight to the solution of a problem can be characterized by a representational change (Knoblich, Ohlsson, Haider, & Rhenius, 1999;Ohlsson, 1984aOhlsson, , 1992Öllinger, Jones, & Knoblich, 2008;Thevenot & Oakhill, 2008). This evidence makes it difficult to explain insight problem solving within the classical information-processing account (Newell & Simon, 1972), where problem solving is understood as search within a well-defined problem space (problem representation). The problem space account has no mechanism to implement a representational change for instances when the current search gets stuck, is insufficient, or does not reduce the distance to the desired goal.There are a few accounts that attempt to remedy this omission. One suggestion is to claim that insight problems are nothing special and therefore representational change plays only a marginal role. For such explanations, problem difficulty relates either to the size of the problem space being overly large and preventing exhaustive search (Kaplan & Simon, 1990), or that problem solvers apply inappropriate heuristics when searching the problem space Ormerod, MacGregor, & Chronicle, 2002). Both accounts miss a cognitive process that addresses the change of the search space. In the first, an additional process is necessary that re-focuses on particular areas of the search space by changing the problem representation; in the second an additional cognitive process is required that changes the search space after repeated failures of the problem solving process.Ohlsson (Knoblich et al., 1999;Ohlsson, 1984aOhlsson, , 1984bOhlsson, , 1992) provided a detailed framework that stressed the importance of a representational change for insight problem solving, and identified impasse as a crucial pre-condition. Moreover, Ohlsson (1992) identified at least three ...

Insight and Search in Katona’s Five-Square Problem

“…Taken together, the data suggest that under no time restrictions, the ending location hint is likely to be stronger than that of the starting location hint. This assumption is in line with studies that showed that providing the key cue for a representational change can increase solution rates when the search space is adequately restricted (Kershaw & Ohlsson, 2004;Öllinger et al 2012;Öllinger, Jones & Knoblich, 2013). Moreover, the comparison between the 4to5 and the 5to4 conditions convincingly demonstrated that the main source of problem difficulty is to break the perceptual grouping, which was fostered by the "E" hints and not by the "S" hints.…”

“…The results provide further evidence for our dynamic constraint and representational change model (Öllinger, Jones & Knoblich, 2013), that postulates that a representational change is sometimes not sufficient to solve insight problems due to the fact that modifying a problem representation often results in an even larger problem space that has to be constrained by efficient search strategies. Recently, Lee and Johnson-Laird (2013) provided a more general model for strategy changes in problem solving and tested their assumptions by using matchstick problems that do not necessarily require insight.…”

“…The study aimed at testing the predictions of the dynamic search and constraint model that was recently introduced by Öllinger, Jones & Knoblich (2013), by using the Five-Square insight problem introduced by Katona (1940). In two experimental conditions, we provided hints that highlighted the location of starting and ending moves that are crucial for solving the problem.…”

“…Recently, we suggested a model (Öllinger, Jones, & Knoblich, 2013, see also Jones, 2003) that elaborates that of Ohlsson (1992), characterizing insight problem solving as a dynamic search process where heuristics constrain the search space before and after an insight (Kaplan & Simon, 1990;MacGregor et al, 2001;Ormerod et al, 2002) and representational change provides the insight to enable a change in the problem representation (Knoblich et al, 1999;Ohlsson, 1992). The model explains insight via a strict sequence of events.…”

The Roles of Internal Representation and Processing in Problem Solving Involving Insight: A Computational Complexity Perspective

“…This model underlies many subsequent and current models of human problem solving (Kaplan & Simon, 1990;Knoblich, Ohlsson, Haider, & Rhenius, 1999;Öllinger, Jones, & Knoblich, 2014;Ormerod, MacGregor, & Chronicle, 2002). In this model, a representation of a problem's givens and goals is chosen based on previous experience and search is then performed (possibly aided by heuristics) within the space of problem states associated with this representation until a state is encountered that satisfies the problem's goals, i.e., a goal state.…”