## Figure 148

A pictorial representation of the initial state of the monsters-globes problem. Adapted from Simon and Hayes (1976).

other studies, Hayes and Simon used isomorphs to the Tower of Hanoi that involved monsters and globes. In the basic, monster-globe problem there are three monsters of different sizes (small, medium, and large), each holding different-sized globes (that are small, medium, and large). The small monster is holding the large globe; the medium-sized monster the small globe, and the large monster the medium-sized globe. The goal is to achieve a state in which each monster is holding a globe proportionate to its size (e.g., the small monster holding the smallest globe). However, monster etiquette demands that (i) only one globe is transferred at a time, (ii) if a monster is holding two globes, only the larger of the two is transferred, and (iii) a globe may not be transferred to a monster who is holding a larger globe. This monster-globe problem is, thus, a move problem with an isomorphic problem space to the Tower of Hanoi problem.

Simon and Hayes also had a change version of the monster-globe problem with the same monsters and globes but subjects had to shrink and expand the globes held by the monsters rather than moving them; the rules were that (i) only one globe may be changed (i.e., shrunk or expanded) at a time, (ii) if two globes have the same size, only the globe held by the largest monster may be changed, and (iii) a globe may not be changed to the same size as the globe of a larger monster.

Hayes and Simon's rule-application hypothesis predicted that the move version of the monster-globe problem should be easier than the change version, because the rules in the latter were more difficult to apply (i.e., they involved complex tests to determine legal operations). What they found was that the move problem was twice as easy as the change problem (see Hayes & Simon, 1977; see Figure 14.8). However, apart from Hayes and Simon's rule-application hypothesis, Kotovsky, Hayes, and Simon (1985) proposed that a rule-learning hypothesis could also account for the data. That is, that some rules can be learned more easily than others and this contributes to the ease with which the problem is solved. In fact, Kotovsky et al. found evidence, from a task in which subjects simply learned the move and change rules, that they took longer to learn the change rules than the move rules. They also showed that the general ease of rule learning and rule application was likely to be influenced by (i) the extent to which the rules are consistent with real-world knowledge, (ii) the memory load inherent in the problem; that is, how much of the problem solving could be performed in an external memory (e.g., on paper) rather than in working memory, (iii) whether the rules could be easily organised in a spatial fashion or more easily imagined.

Further work on the sources of difficulty in problems has emerged from tests of isomorphism of another problem: the Chinese ring puzzle (see Kotovsky & Simon, 1990). In the original version, this puzzle involves a complex arrangement of five interconnected metal rings on a bar, the task being to remove the rings from the bar (see Afriat, 1982). This puzzle has two important characteristics: (i) what constitutes a move is not immediately obvious, because the rings can be twisted and turned in a number of ways, (ii) the problem space of moves, once found, is linear (i.e., a straight line of moves with no branching). The latter ensures that problem difficulty—which is considerable for this problem—cannot emerge from searching the problem space, but must be due to discovering how to make moves. Kotovsky and Simon developed a digitised version of this puzzle, involving the moving of five balls out of boxes. Their study showed that the maj or source of difficulty lay in discovering what a legal move was, rather than navigating through the problem space.

### Solving the missionaries and cannibals puzzle

Problem-space theory has also been applied, with some success, to the missionaries-cannibals puzzle. In this problem, you are given the task of transferring three missionaries and three cannibals across a river in a boat. As the boat is fairly small, only two or fewer people can be taken across in it at a time and someone must always accompany the boat back to the other side. Furthermore, at no point in the problem can there be more cannibals than missionaries left on one bank of the river or else the cannibals will have a religious feast. Figure 14.9 shows the legal search space for reaching the goal state. Researchers have argued that people use a variety of heuristics to solve different variants of this problem.

Thomas (1974) used a variant of this problem involving J.R.R.Tolkien's (1966) hobbits and orcs, in which orcs have a proclivity for gobbling up hobbits. He showed that at some points in the problem— especially states 5 and 8 in Figure 14.9 —subjects took considerably longer and produced more errors than at other points. Thomas maintained that the difficulties experienced at these states had different cognitive sources. In the case of the state 5, the difficulty lies in the many alternative moves that are possible at this point (five in number). Only two of these moves are illegal and of the remaining three legal moves, only one is really helpful. In the case of state 8, subjects are misled because they need to move away from the goal state in order to get closer to it. As can be seen in Figure 14.9, in going from state 8 to state 9, one enters a state that seems further away than closer to the goal. At this point, subjects typically think that they have reached a blind alley and start to backtrack. Figure 14.10 shows the distribution of incorrect responses by Thomas' subjects at each state in the problem.

Thomas also suggested that subjects made three or four major planning decisions in solving the problem, and having made each of these decisions, carried out whole blocks of moves with increasing speed. Then, at the beginning of each planned sequence of moves, there would be a long pause before the next decision was made. Thomas' statistical analysis of the distributions of subjects' times-to-move supported this hypothesis.

Other researchers have looked at more complex versions of the problem and noted strategic changes in subjects' behaviour. Simon and Reed (1976) investigated a version of the missionariescannibals problem, involving five missionaries and five cannibals. This problem is more complex in that it has many more legal states even though it can be solved in just 11 moves. However, on average, subjects take 30 moves to solve the problem. Simon and Reed suggested that there were three main strategies used in solving the problem. Initially, subjects adopted a balancing strategy whereby they simply tried to ensure that equal numbers of missionaries and cannibals remained on either side of the river. This strategy avoids illegal moves, resulting in more cannibals than missionaries on either bank of the river. At a certain point, subjects become more oriented towards the goal state and adopt a means-ends strategy. This strategy is manifested by a tendency

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