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Chapter 12: Problem Solving
WHAT IS A PROBLEM?
• Problem: When there is an obstacle between a present state and a goal, and it is not immediately
obvious how to get around the obstacle (it is difficult, without an obvious solution)
• Well-Defined Problem: One which usually has a correct answer, which will be produced by certain
procedures applied correctly. This includes math or physics problems.
• Ill-Defined Problem: These do not necessarily have one “correct” answer, and the path to their
solution is often unclear. This includes relationship problems, picking a career, or other life problems
THE GESTALT APPROACH: REPRESENTATION & RESTRUCTURING
• Problem solving is about 1) how people represent a problem in their mind, and 2) how solving a
problem involves a reorganization or restructuring of this representation
Representing a Problem in the Mind
• Problems can be presented in many ways, e.g. crossword puzzles as a diagram with clues about how
to fill in the open squares. However, this problem is represented in the mind differently – for example,
only a small part of the puzzle at the time by starting with horizontal words, or find horizontals and
verticals that fit together.
• Success in problem solving is influenced by how a problem is represented
in one’s mind. E.g. Solving for x, representing this as “a small triangle in the
upper left quadrant”, vs. “a small rectangle with x being the diagonal”
Restructuring and Insight
• Restructuring: The process of first perceiving the object, and then changing its
representation. This is associated with insight, the sudden realization of a
problem’s solution, mostly due to suddenly discovering a crucial element leading to
• This is supported by some “Aha!” experiences where one seems to suddenly arrive at a solution.
Others believe there is a lack of concrete evidence to support the specialness of the insight experience.
• Metcalfe & Wiebe (1987): Predict there is a basic difference in how subjects feel they are progressing
toward a solution when working on an insight vs. noninsight problem. Specifically, subjects working on an
insight problem should not be very good at predicting how near they are to a solution since the
answer appears suddenly; subjects working on the noninsight problem would be more likely to know
when they are getting closer.
• As subjects solve the two problems, they are asked to rate “warmth” judgements every 15 seconds,
with “hot” (7 out of 7) indicating being very close to a
• Triangle Problem: Show how you can move three of the
circles to get the triangle to point to the bottom of the page.
Chain Problem: A woman has four pieces of chain, each
made of three links. She wants to join the pieces in a single
closed loop of chain. To open a chain costs 2 cents, and to close one
costs 3 cents; she only has 15 cents. How does she do it? These are
both insight problems
• Noninsight problems include algebra problems: solve for X, 0.2X + 10
= 25. The systematic way one tackles this problem, based on top-down
knowledge and experience with algebra, makes this non-insight. Page 326-351, 25 pages Page 2 of8
• For insight problems, warmth ratings remain low until at 2 or 3 seconds just before the problem is
solved. In contrast, for algebra problems the ratings gradually increased until the problem was solved.
Thus the solution for insight problems does occur suddenly.
• Insight in Birds: Pigeon which cannot reach a small treat hung from the top of the box initially walks
around and tries to reach a few times. Eventually it realizes it can stand on a cube found at the other side
of the box, and nudges the cube so it can be used to reach the treat.
Obstacles to Problem Solving
• Fixation: People’s tendency to focus on a specific characteristic of the problem. This can prevent
arriving at a solution, e.g. by focusing on familiar uses of an object in functional fixedness.
• Candle Problem (Duncker): Illustrates
how functional fixedness can hinder
problem solving. “You are in a room with a
corkboard on the wall. You are given some
candles, matches in a matchbox, and some
tacks. Your task is to mount a candle on the
corkboard so it will burn without dropping
wax on the floor.”
• The solution occurs when the person
realizes the matchbox can be used as a support rather than as a container.
• If one group were presented with small cardboard boxes
containing the materials, and another were presented with
the same materials and empty boxes, the first group was
twice as worse at solving the problem. Seeing the boxes
as containers inhibited the first group from using them as
• Wine Bottle Opening: A bottle of wine can be opened
without a corkscrew by putting it in a shoe, and then
tapping the system against a wall so the cork will rise out
of the bottle. The shoe is used as a tool beyond its typical
• Two-String Problem (Maier): Task to tie together two strings hanging from the ceiling, but difficult since
the strings are separated so it is impossible to reach one while holding the other. There is also a chair
and a pair of pliers.
• The solution requires tying the pliers to one of the strings to create a pendulum which can be swung to
within one’s reach. 60% of subjects did not solve the problem because they focused on the usual
function of pliers and did not think of them as weights.
• If the experimenter accidentally set the string into motion, subjects who didn’t solve the problem after
10 minutes can now solve it within 60 seconds – seeing the swing triggered the insight that the pliers
could be used for a pendulum. This thus involved a
restructuring of the function of pliers.
• Functional fixedness is a type of mental set, a preconceived
notion about how to approach a problem, largely determined by
past experience and knowledge [top-down].
• Water-Jug Problem (Luchins): A mental set can arise out of
the situation created as a person solves a problem. The subject
is given three jugs of different capacities and must use them to
measure out a specific quantity of water.
• All problems in the set can be solved by (B – A – 2C), but
problems 7 and 8 can be solved more simply [(A+C) and (A-C) Page 326-351, 25 pages Page 3 of8
respectively]. If subjects start with problem 1 and do each problem in sequence, they are not able to see
the shorter solution than if a no mental set subject solved only problems 7 and 8.
• A mental set, with carryover of knowledge or skills from one problem situation to another, has both
positive and negative effects. Positive: Solution of an earlier problem can aid in solving a new one
(analogical transfer). Negative: Makes it more difficult to solve a problem that requires a novel approach.
THE INFORMATION-PROCESSING APPROACH
Newell & Simon’s Approach
• Newell & Simon saw problems in terms of an
initial state (conditions at the beginning of the
problem) and a goal state (solution of the
• Tower of Hanoi Problem: Three discs
stacked on the left peg, with goal state as
stacked on the right peg.
• Operators are actions that take the problem
from one state to the other. For the Tower of
Hanoi, rules specify which operators are
allowed: moved one at a time from one peg to
another; a disc can be moved only when there are no discs on top of it; a larger disc can never be placed
on top of a smaller disc.
• There are many ways to move the discs as
one tries to reach the goal state; these
sequences of choice of steps create
intermediate states. The initial state, goal
state, and all possible intermediate states
create a problem space.
• There are many paths to get from the initial
state to the goal state, but one is shorter than
the others (7 moves).
• Means-End Analysis: The person has to
search the problem space to find a solution,
and they can do so by using the route to
reduce the difference between the initial and
goal states. This is achieved by creating a
series of subgoals, each of which may involve several moves (e.g. free up large disc needs 2 moves),
and which often need one to look slightly ahead.
The Importance of how a Problem is Stated
• The Acrobat Problem: Three circus
acrobats develop an amazing routine in
which they jumped to and from each other’s
shoulders, on tall flagpoles, to form human
towers. The large acrobat weighed 400
pounds, the medium 200 and the small
acrobat 40 pounds. (Same rules as Tower of
Hanoi). They moved from the initial state to
the goal state; how did they manage to do
this while obeying the safety rules? Page 326-351, 25 pages Page 4 of8
• Reverse Acrobat Problem: Same as above, except the last rule is changed to state that a smaller
acrobat cannot stand on a larger one.
• Subjects take almost twice as long to solve the reverse acrobat
problem despite both problems being able to be solved in 5 moves.
This may be because the rule is inconsistent with our knowledge
of the world, increasing the load on the problem-solver’s memory –
• Mutilated Checkerboard Problem (Kaplan & Simon): A
checkerboard consists of 64 squares, which can be completely
covered by placing 32 dominos each of which covers two squares. If
we eliminate two corners of the checkerboard, can we cover the remaining squares with 31 dominos?
• The way a problem is framed affects its difficulty, four conditions: 1) blank – a board with all blank
squares; 2) colour – a board with alternating black and pink squares; 3) black and pink – a board with
the words black and pink; 4) bread and butter – a board with the words bread and butter alternating on
• The key is to realize that when a board covers two squares, it always covers two that are different –
one cannot cover two black or two pink squares. Therefore, for 31 dominos to cover the mutilated board,
there must be 31 pink and 31 black squares; this cannot
happen since two pink squares from the corners were
• All four versions have the same board layout and same
solution, but what’s different is the information used to
provide participants with the insight that a domino must
cover two different squares.
• Subjects given versions 2, 3, and 4 (condition 4 emphasized
the difference the most) found the problem easier to solve,
taking less time and requiring fewer hints. Condition 4 subjects
• Think-Aloud Protocol: Subjects are asked to say aloud
what they are thinking while doing a problem. They are
instructed not to describe what they are doing, but to verbalize
new thoughts as they occur. This helps determine what
information the person is attending to while solving a problem.
• The protocol reveals a shift in how the person perceived elements of the problem, arriving at the
solution once they realized bread and butter were important – similar to Gestalt idea of restructuring.
• The Russian Marriage Problem: “In a small Russian village, there are 32 bachelors and 32 unmarried
women. The village matchmaker succeeded in arranging 32 highly satisfactory marriages; the village was
happy. Then one drunken night, two bachelors, in a test of strength, stuffed each other with perogies and
died. Can the matchmaker, through some quick arrangements, come up with 31 heterosexual marriages
among the 62 survivors?”
• In this case the answer is obvious; reading this story enables one to form a connection with the
mutilated checkerboard problem. Analogy is the process of noticing connections between similar
problems and applying the solution for one problem to others.
• Analogical Problem Solving: Technique of using the solution to a similar problem to guide solution of
a new problem, such as with the Russian Marriage Problem and the Mutilated Checkerboard
Analogical Transfer Page 326-351, 25 pages Page 5 of 8
• Analogical Transfer: The transfer of experience from solving one problem to another. To study this
phenomenon, participants who are trying to solve a target problem are presented with a source
problem/story that shares some similarities and illustrate a way to solve the target problem.
• Evidence of analogical transfer oc