Key Actions in Insight Problems: Further Evidence for the Importance of Non-Dot Turns in the Nine-Dot Problem

Key actions are single actions or behaviors that can be singled out as leading to the solution of a problem. In the nine-dot problem (Maier, 1930), Kershaw and Ohlsson (2004) proposed that non-dot turns are the key action necessary for solution. In two experiments, non-dot turns are further analyzed as the key action necessary for solving the nine-dot problem and its variants. Non-dot turns are found to be predictive of solution, while the classic conception of drawing lines outside the dots does not distinguish between solvers and non-solvers. Key Actions in Problem Solving Problem solving in everyday life, as well as the laboratory, can be quite difficult. Often a problem or activity can seem unduly difficult when one does not know the key action necessary for completing the problem. A key action can be defined as a single action or behavior that can be singled out as the key to the solution. Examples of key actions abound in everyday life. A proper roux cannot be made without engaging in continual stirring. Algebra problems become routine once one understands how to balance the equation and isolate the variables. Finally, as I have learned one too many times, data will invariably disappear if I have not completed the key action of backing it up! In laboratory problem solving, many insight problems can be solved through the production of a key action. For example, using the pliers as the pendulum weight is the key action necessary for solving Maier’s (1931) two-string problem, and moving objects in three-dimensional space is necessary for the six matches problem (Scheerer, 1963) and the eight-coin problem (Ormerod, MacGregor, & Chronicle, 2002). As a third example, the key action necessary for solving the prisoner and rope problem (Metcalfe & Wiebe, 1987) is to unravel the rope into two strands, then tie the ends of the two strands together to escape the tower. Sometimes the key action can be realized without much struggle, depending on an individual’s prior knowledge. A friend (and fellow insight researcher) worked on a farm growing up, where the splitting of rope to make it longer was a common occurrence. He instantly knew how to solve the prisoner and rope problem; the key action was easy to discover. In other insight problems, however, the key action is not easy to discover. The common use of pliers hinders their use as a pendulum weight in Maier’s (1931) two-string problem (cf. Birch & Rabinowitz, 1951). Additionally, other insight problems may have multiple factors of difficulty preventing the discovery of the key action, such as in the nine-dot problem (Maier, 1930). Finding the Key Action in the Nine-Dot Problem The nine-dot problem (Maier, 1930; see Figure 1) is quite possibly the most difficult insight problem that has been studied, with a typical solution rate for unaided participants of 0% (MacGregor, Ormerod, & Chronicle, 2001). Problem solvers are required to connect all the dots in a 3 x 3 matrix by using four straight lines, without lifting their pens from the page or retracing any lines. Figure 1: The Nine-Dot Problem and its Solution The classic conception of the key action necessary for solving the nine-dot problem is that participants should draw lines that extend beyond the dots (Maier, 1930; Maier & Casselman, 1970; Scheerer, 1963). In a related conception, Lung and Dominowski (1985) claimed that the key action was drawing lines that did not begin or end on dots. Kershaw and Ohlsson (2001) hypothesized that the key action necessary for solving the nine-dot problem was making non-dot turns, or turns that occur in the empty space between dots. The conception of non-dot turns as the key action came about through an inspection of two of MacGregor et al.’s (2001) nine-dot problem variants. The variant with no non-dot turn had a solution rate of 88% after four attempts, while the variant that required one non-dot turn only had a solution rate of 27% after four attempts. Kershaw and Ohlsson (2004) continued in this line of reasoning by explaining that the likelihood of producing a key action is dependent on the cognitive factors that underlie that action. In the nine-dot problem, multiple factors of difficulty are operating that each lower the probability of making a non-dot turn. Kershaw and Ohlsson (2004) distinguished three classes of difficulty: perceptual, knowledge, and process. Perceptual factors include Gestalt properties of the ninedot problem such as goodness of figure and figure-ground relationships. Making a non-dot turn requires that one both breaks the good figure of the square and views the white space beyond the dots as part of the problem. Knowledge factors refer to an individual’s prior knowledge. Making a non-dot turn is hindered by people’s prior experience with dot puzzles, such as connect-the-dot games played by children (cf. Weisberg & Alba, 1981). Process factors include the size of the search space, the specificity of the goal state, and the amount of mental lookahead necessary to find the solution. Making a non-dot turn is difficult because it is not obvious where to draw the first line or what the end state of the problem will be. In addition, people vary in the amount of mental lookahead they possess (cf. MacGregor et al., 2001), which affects the process of making non-dot turns. Kershaw and Ohlsson (2004) showed that perceptual, knowledge, and process factors interact to suppress the probability of producing the key action of non-dot turns in the nine-dot problem. In the following experiments, non-dot turns are again examined as the key action necessary to solve the nine-dot problem. Experiment I follows up on Kershaw and Ohlsson (2004) but adds an additional possible facilitating factor: giving participants the first line of the solution, which should narrow the search space. Experiment II uses a think aloud methodology to explore what behaviors precede the production of non-dot turns.