## Figure 147

The initial state of the five-disk version of the Tower of Hanoi problem used by Anzai and Simon (1979).

### Learning different strategies

Egan and Greeno's work illustrates the effects that experience can have on subjects' ability to solve the problem. Egan and Greeno's subjects adopted the strategy of partitioning a complex problem into several simpler sub-problems and then solving each of these in turn. Anzai and Simon (1979) examined other strategies adopted by a single subject in four successive attempts to solve a five-disk version of the Tower of Hanoi (see Figure 14.7).

On each of the four attempts the subject used a different strategy, becoming progressively more efficient at solving the problem. Initially, the subject seemed to explore the problem space without much planning of moves. Search at this stage seemed to be guided by avoidance of certain states rather than moves towards definite goal/subgoal states. Anzai and Simon argued that the subject was using general domain-independent strategies. These strategies included a loop-avoidance strategy to avoid returning to previously visited states and a heuristic strategy that preferred shorter sequences of moves to achieve a goal, to longer sequences. These general strategies allowed the subject to learn better sequences of moves, and these sequences were carried forward to be used in later attempts to solve the problem. Anzai and Simon developed an adaptive production system model that learned in the same manner. This model could create new production rules that were used to solve the problem on a later attempt. From Anzai and Simon's research, the general course of learning in these situations hinges on the initial use of general, domain-independent heuristics which then allow one to learn domain-dependent or domain-specific heuristics (see also Anderson's, 1993, 1996 ACT-R theory).

Isomorphic problems: Understanding and problem representation

Clearly, the way you understand a problem should influence your ability to solve it. This intuition is specified in problem-space theory and has been borne out by a variety of findings on solving isomorphic problems. Two things are isomorphic if they have the same form or relational structure. So, in problemspace terms, two problems can be isomorphic if there is a one-to-one correspondence between the states and operators of the problems such that whenever two states are connected by an operator in one problem space, corresponding states are connected by the corresponding operator in the other problem space. Research has shown that slight differences in the way isomorphic problems are presented have significant effects on subjects' problem-solving success, presumably because the presentational differences affect their understanding of the problem. Furthermore, problem-space theory is specific enough to allow one to pinpoint what it is about subjects' understanding of the problem that causes these effects.

Several studies of this type have been performed on variants of the Tower of Hanoi (see Hayes & Simon, 1974, 1977; Simon & Hayes, 1976). In one study, Simon and Hayes (1976) used problem isomorphs about a tea ceremony involving three different people (corresponding to the three pegs) carrying out three ritual tasks for one another (like the three disks) in differing orders of importance (like disks of different sizes). In

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