Problem Solving,
Problem Solving
Human ability to solve novel problems greatly surpasses that of any other species,
and this ability depends on the advanced evolution of the prefrontal cortex in
humans. We have already noted the role of the prefrontal cortex in a number of
higher-level cognitive functions: language, imagery, and memory. It is generally
thought that the prefrontal cortex performs more than these specific functions, however,
and plays a major role in the overall organization of behavior. The regions of the
prefrontal cortex that we have discussed so far tend to be ventral (toward the bottom)
and posterior (toward the back), and many of these regions are left lateralized.
In contrast, dorsal (toward the top), anterior (toward the front), and right-hemisphere
prefrontal structures tend to be more involved in the organization of behavior. These
are the prefrontal regions that have expanded the most in the human brain.
Goel and Grafman (2000) describe a patient, PF, who suffered damage to his
right anterior prefrontal cortex as the result of a stroke. Like many patients with damage
to the prefrontal cortex, PF appears normal and even intelligent, and he scored in
the superior range on an intelligence test. In fact, he performed well on most tests,
although he did have difficulty with the Tower of Hanoi problem described later in this
chapter. Nonetheless, for all these surface appearances of normality, there were profound
intellectual deficits. He had been a successful architect before his stroke but
was forced to retire due to loss of the ability to design. He was able to get some work
as a draftsman. Goel and Grafman gave PF a problem that involved redesigning their
laboratory space. Although he was able to speak coherently about the problem, he
was unable to make any real progress on the solution. A comparably trained architect
without brain damage achieved a good solution in a couple of hours. It seems that the
stroke affected only PF’s most highly developed intellectual abilities.
This chapter and Chapter 9 will look at what we know about human problem
solving. In this chapter, we will answer the following questions: • What does it mean to characterize human problem solving as a search of a
problem space? • How do humans learn methods, called operators, for searching the problem
space?
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210 | Problem Solving
• How do humans select among different operators for searching a problem
space? • How can past experience affect the availability of different operators and the
success of problem-solving efforts?
•The Nature of Problem Solving
A Comparative Perspective on Problem Solving
Figure 8.1 shows the relative sizes of the prefrontal cortex in various mammals
and illustrates the dramatic increase in humans. This increase supports the
advanced problem solving that only humans are capable of. Nonetheless, one
can find instances of interesting problem solving in other species, particularly
in the higher apes such as chimpanzees. The study of problem solving in other
species offers perspective on our own abilities. Köhler (1927) performed some
of the classic studies on chimpanzee problem solving. Köhler was a famous
German gestalt psychologist who came to America in the 1930s. During World
War I, he found himself trapped on Tenerife in the Canary Islands. On the
island, he found a colony of captive chimpanzees, which he studied, taking
particular interest in the problem-solving behavior of the animals. His best
participant was a chimpanzee named Sultan. One problem posed to Sultan was
FIGURE 8.1 The relative proportions of the frontal lobe given over to the prefrontal cortex in
six mammals. Note that these brains are not drawn to scale and that the human brain is really
much larger in absolute size. (After Fuster, 1989. Adapted by permission of the publisher. © 1989 by Raven Press.)
Squirrel monkey Cat Rhesus monkey
Dog Chimpanzee Human
Brain Structures
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The Nature of Problem Solving | 211
to get some bananas that were outside his cage. Sultan had no difficulty when
he was given a stick that could reach the bananas; he simply used the stick to
pull the bananas into the cage. The problem became harder when Sultan was
provided with two poles, neither of which could reach the food. After unsuccessfully
trying to use the poles to get to the food, the frustrated ape sulked in
his cage. Suddenly, he went over to the poles and put one inside the other, creating
a pole long enough to reach the bananas (Figure 8.2). Clearly, Sultan had
creatively solved the problem.
What are the essential features that qualify this episode as an instance of
problem solving? There seem to be three:
1. Goal directedness. The behavior is clearly organized toward a goal—in
this case, getting the food.
2. Subgoal decomposition. If Sultan could have obtained the food simply
by reaching for it, the behavior would have been problem solving, but
only in the most trivial sense. The essence of the problem solution is that
the ape had to decompose the original goal into subtasks, or subgoals,
such as getting the poles and putting them together.
3. Operator application. Decomposing the overall goal into subgoals is
useful because the ape knows operators that can help him achieve these
subgoals. The term operator refers to an action that will transform the
problem state into another problem state. The solution of the overall
problem is a sequence of these known operators.
Problem solving is goal-directed behavior that often involves setting subgoals
to enable the application of operators.
FIGURE 8.2 Köhler’s ape,
Sultan, solved the two-stick
problem by joining two short
sticks to form a pole long
enough to reach the food
outside his cage. (From Köhler, 1956.
Reprinted by permission of the publisher.
© 1956 by Routledge & Kegan Paul.)
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The Problem-Solving Process: Problem Space and Search
Often, problem solving is described in terms of searching a problem space,
which consists of various states of the problem. A state is a representation of the
problem in some degree of solution. The initial situation of the problem is
referred to as the start state; the situations on the way to the goal, as intermediate
states; and the goal, as the goal state. Beginning from the start state, there are
many ways the problem solver can choose to change the state. Sultan could reach
for a stick, stand on his head, sulk, or try other approaches. Suppose he reaches
for a stick. Now he has entered a new state. He can transform it into another
state—for example, by letting go of the stick (thereby returning to the earlier
state), reaching for the food with the stick, throwing the stick at the food,
or reaching for the other stick. Suppose he reaches for the other stick. Again, he
has created a new state. From this state, Sultan can choose to try, say, walking on
the sticks, putting them together, or eating them. Suppose he chooses to put the
sticks together. He can then choose to reach for the food, throw the sticks away,
or separate them. If he reaches for the food, he will achieve the goal state.
The various states that the problem solver can achieve define a problem
space, also called a state space. Problem-solving operators can be thought of as
ways to change one state in the problem space into another. The challenge is to
find some possible sequence of operators in the problem space that leads from
the start state to the goal state.We can think of the problem space as a maze of
states and of the operators as paths for moving among them. In this model, the
solution to a problem is achieved through search; that is, the problem solver
must find an appropriate path through a maze of states. This conception of
problem solving as a search through a state space was developed by Allen
Newell and Herbert Simon, who were dominant figures in cognitive science
throughout their careers, and it has become the major problem-solving approach,
in both cognitive psychology and AI.
A problem space characterization consists of a set of states and operators
for moving among the states. A good example of problem-space characterization
is the eight-tile puzzle, which consists of eight numbered, movable tiles set
in a 3 _ 3 frame. One cell of the frame is always empty, making it possible to
move an adjacent tile into the empty cell and thereby to “move” the empty cell
as well. The goal is to achieve a particular configuration of tiles, starting from
a different configuration. For instance, a problem might be to transform
212 | Problem Solving
The possible states of this problem are represented as configurations of tiles in
the eight-tile puzzle. So, the first configuration shown is the start state, and the second
is the goal state. The operators that change the states are movements of tiles
into empty spaces. Figure 8.3 reproduces an attempt of mine to solve this problem.
into
2 1 6
4 8
7 5 3
1 2 3
8 4
7 6 5
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My solution involved 26 moves, each move being an operator that changed the
state of the problem. This sequence of operators is considerably longer than necessary.
Try to find a shorter sequence of moves. (The shortest sequence possible is
given in the appendix at the end of the chapter, in Figure A8.1.)
Often, discussions of problem solving involve the use of search graphs or
search trees. Figure 8.4 gives a partial search tree for the following, simpler
eight-tile problem:
The Nature of Problem Solving | 213
(a) (b) (c) (d) (e) (f) (g)
(o) (p) (q) (r) (s) (t) (u)
(n) (m) (l) (k) ( j) (i) (h)
