(state 7), and we have almost solved the problem! This procedure of setting
subgoals and looking slightly ahead often results in an effi cient solution to
a problem.
Why is the Tower of Hanoi problem important? One reason is that
it illustrates means-end analysis, with its setting of subgoals, and this
approach can be applied to real-life situations. For example, I recently had
to plan a trip from Pittsburgh to Copenhagen. Remember that in Newell
and Simon’s terminology, an operator is the action to get from one state to
another. The operator for getting from Pittsburgh to Copenhagen is to take
a plane, and one of the rules governing this operator is that if there isn’t
a direct fl ight (there isn’t!), it is important to have enough time between
fl ights to ensure that passengers and baggage can get from the fi rst fl ight to
the second one. Another rule is that the cost of the fl ights have to be within
my budget.
My fi rst step was to try to reduce the distance between myself and
Copenhagen by taking the direct fl ight from Pittsburgh to Paris, and then
transfer to a fl ight to Copenhagen. But there was only a gap of 90 minutes between
fl ights, which violated the fi rst rule, and waiting for a later fl ight increased the fare,
which violated the second rule. The failure of the Pittsburgh-to-Paris idea led me to
set a new subgoal: Find a city that has a direct fl ight to Copenhagen. The answer:
Atlanta. So the new routing was Pittsburgh to Atlanta, then Atlanta to Copenhagen
(● Figure 12.13). As it turned out, fl ights that met both of the rules were available, so
the problem was solved. Notice that for this solution, the fi rst subgoal involved travel-
ing away from Copenhagen. Just as for subgoal 4 in the Tower of Hanoi example we
had to move a disc away from the right peg to eventually get it there, I had to fi rst fl y
away from Copenhagen to position myself to achieve my goal.
One of the main contributions of Newell and Simon’s approach to problem solving
is that it provided a way to specify the possible pathways from the initial to goal states.
But research has shown that there is more to problem solving than specifying the prob-
lem space. As we will see in the next section, this research has shown that two problems
with the same problem space can vary greatly in diffi culty.THE IMPORTANCE OF HOW A PROBLEM IS STATED
How a problem is stated can affect its diffi culty. We can appreciate this by considering
two similar problems: the acrobat problem and the reverse acrobat problem.Subgoal 3: Move large disc onto third peg.
Subgoal 2: Free up third peg.
Subgoal 1: Free up large disc.
(a)(b)(c)(d)●FIGURE 12.12 Initial steps in solving the
Tower of Hanoi problem, showing how the
problem can be broken down into subgoals.
(Source: Based on K. Kotovsky, J. R. Hayes, & H. A. Simon,
H. A., “Why Are Some Problems Hard? Evidence From
Tower of Hanoi,” Cognitive Psychology, 17, 248–294, 1985.)
AtlantaPittsburghCopenhagenParis●FIGURE 12.13 Two possible routes from Pittsburgh to Copenhagen. The
route through Paris (solid black line) immediately reduces the distance to
Copenhagen, but doesn’t satisfy the rules of the problem. The route through
Atlanta (dashed red line) involves some backtracking but works because it
satisfi es the rules.Modern Research on Problem Solving • 335Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.