Post on 04-Jan-2016
Introduction to Problem Solving
Psychology 355: Cognitive Psychology
Instructor: John Miyamoto
05/26/2015: Lecture 09-2
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There is no Lecture 09-1 because Monday was Memorial Day (no lecture on that day.)
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Outline
• Definition of “problem”
• Information processing versus Gestalt approach to problem solving.
Algorithmic problems & insight problems
• Tower of Hanoi – an example of an algorithmic problem
• Insight problemso Problem representationo Problem restructuringo Problem isomorphs
Definition of Problem Solving
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Definition of Problem Solving
• A problem exists when the present state differs from a goal state.
The problem is to change the present state into the goal state. o Initial stateo Goal stateo Permissible "moves" – ways to change the problem state from the initial
state towards the goal state.
• Interesting problems are situations where it is not obvious
how to change the initial state into the goal state.
• Cognitive psychology of problem solving –
how do people solve problems.
Examples of Problem Solving Situations
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Problem Solving - Examples
• Math problems, physics problems, science problems generally.o Initial state: The given information in the problem.o Goal state: The “answer” or solution to the problem.
• Practical problems, e.g., arranging furniture, building a mechanical
device.
• Winning strategies in games, business, public health, law & war.
Key Ideas in Theory of Problem Solving
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Key Ideas in the Psychology of Problem Solving
• Problem representation –
The mental representation of the problem that the problem solver
manipulates while trying to solve the problem. o Initial stateo Goal stateo Moves or transformations. Constraints and rules.-------------------------
• Insight problems & algorithmic problems
• Restructuring a problem representation
-------------------------
• Set
• Functional fixedness
Algorithmic vs Insight Problems
6
Algorithmic Problems versus Insight Problems
• Algorithmic problems: The initial problem state can be transformed to the goal state by a systematic procedure.
♦ Example: The Tower of Hanoi ♦ Example: Solving a long division problem
• Insight problems require mental restructuring of the problem representation to get a solution.
♦ Circle problem ♦ Mutilated checkerboard problem
• Algorithmic and insight problems require somewhat different psychological processes to solve them.
Psych 355, Miyamoto, Spr '15 Tower of Hanoi – Example of an Algorithmic Problem
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The Tower of Hanoi (A Problem with an Algorithmic Solution)
• Tower of Hanoi is an algorithmic problem – there is a logically
adequate strategy that will always solve this problem.
General Idea of an Insight Problem
8
General Idea of an Insight Problem
• The solution of insight problems usually depends on finding a new way to represent the problem.
Ideas from Gestalt Psychology
• The mind searches for structure in perception
• The mind searches for structure in problem solving
Psych 355, Miyamoto, Spr '15
Mental Representation of a Problem
The Problem Representation=
Finding a New Way to Represent a Problem
Restructuring the Problem
Representation
=
Solving the Circle Problem by Restructuring the Problem Representation
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The Circle Problem: An Example of an Insight Problem
Given:
• radius r = 1
• length of a = 0.9
• line b is perpendicular to line a
Question: What is the length of x?
Hint:
Change the problem representation.
ax
r
b
Initial Representation
Restructuring the Representation of the Circle Problem
10
ax
r
b
Psych 355, Miyamoto, Spr '15
Restructuring the Representation of the Circle Problem
If r = 1, a = 0.9, and a and b are
perpendicular, what is the
length of x?
• Solution: Add dashed line that
connects the opposite corners.o Alternative representation:
The answer is obvious: x = r = 1.
• Alternative problem representation
makes the solution obvious.
• Solutions to insight problems often
depend on a “trick”.o Here the trick is to change the problem
representation.
Alternate Representationfor the Circle Problem
Another Insight Problem – the Mutilated Checkerboard Problem
11
Another Insight Problem – Mutilated Checkerboard Problem
Problem: Cover the mutilated checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible.
The domino pieces must always be perpendicular or parallel to the sides of the board - they cannot be placed in a diagonal position.
Psych 355, Miyamoto, Spr '15
= domino piece
Failed Attempt to Solve the Mutilated Checkerboard Problem
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Failed Attempt at Solving the Mutilated Checkerboard Problem
Problem: Cover the mutilated checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible.
Psych 355, Miyamoto, Spr '15
= domino piece= domino piece= domino pieceFailure!
• This is not a solution!
• FACT: It is impossible to cover the mutilated checkerboard with dominoes.
• Why is it impossible?
Solution to the Mutilated Checkerboard Problem
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Solution to the Mutilated Checkerboard Problem
• Problem: Cover the checkerboard with domino pieces so that every
domino covers two squares OR if this is impossible, explain why it
is impossible.
A Solution is Impossible!
• A domino piece always covers one dark square and one light square. Therefore any solution covers an equal number of dark and light squares.
• The mutilated checkerboard has 30 dark squares and 32 light squares so it is impossible to cover an equal number of dark and light squares.
= domino piece
Easy Version of the Mutilated Checkerboard Problem – The Matchmaker Problem
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Easy Version of the Mutilate Checkerboard ProblemThe Russian Marriage Problem (a.k.a. the Matchmaker Problem)
Hayes, 1978: [wording slightly altered below]
In a small Russian village, there
were 32 bachelors and 32 unmarried
women. A matchmaker arranges
32 highly satisfactory marriages. The
village was happy and proud. One
night, two bachelors got drunk and
killed each other. Can the matchmaker
come up with heterosexual marriages
(one man, one woman) among the
62 survivors?
Woman Man Woman Man Woman Man Woman Man
Woman Man Woman Man Woman Man Woman Man
Woman Man Woman Man Woman Man Woman Man
Woman Man Woman Man Woman Man Woman Man
Man Woman Man Woman Man Woman Man Woman
Man Woman Man Woman Man Woman Man Woman
Man Woman Man Woman Man Woman Man Woman
Man Woman Man Woman Man Woman Man Woman
Mutilated Checkerboard Problem & Russian Marriage Problem Are Isomorphs
There are 30 men and 32 women. Obviously there is no way to match them into a complete set of heterosexual couples.
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Mutilated Checkerboard Problem & Russian Marriage Problem
• The multilated checkerboard problem and the Russian marriage problem are problem isomorphs.
• Problem Isomorphs: Problems that differ superficially but have identical logical structure.
Woman Man Woman Man Woman Man Woman Man
Woman Man Woman Man Woman Man Woman Man
Woman Man Woman Man Woman Man Woman Man
Woman Man Woman Man Woman Man Woman Man
Man Woman Man Woman Man Woman Man Woman
Man Woman Man Woman Man Woman Man Woman
Man Woman Man Woman Man Woman Man Woman
Man Woman Man Woman Man Woman Man Woman
Concept of Problem Isomorphs
= domino piece
Mutilated Checkerboard Problem Russian Marriage Problem
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Concept of Problem Isomorphs
• Problem isomorphs – structurally identical versions of a problem.
• Basic fact about problem isomorphs: Some versions of a problem
are harder to solve than other versions of the problem.
• What is the psychological difference between the mutilated
checkerboard problem and the matchmaker problem?
• Kaplan and Simon: It is easier to solve the Russian marriage
problem than the mutilated checkerboard problem, presumably
because the Russian marriage version makes the importance of
pairing men with women obvious. (See next slide)
Four Isomorphic Versions of the Mutilated Checkerboard Problem
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Kaplan & Simon: Four Isomorphic Versionsof the Mutilated Checkerboard Problem
• Blank board is
hardest problem.
• “Bread”/“Butter”
word labels are
easiest problem.
• Colored &
“Pink”/“Black”
word labels are
intermediate
difficulty.
• The salience of the
pairing affects
difficulty.
Blank(hardest)
Colored(intermediate)
“Pink” & “Black”
Word Labels(intermediate)
“Bread” & “Butter” (easiest)
Conclusions re Problem Representation
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Conclusion re Problem Representation
• Some problem representations make problem solving easier than
other problem representations.
• Solving an insight problem often depends on finding a problem
representation that make it obvious how to find the solution.
Examples that support these claims:
• Mutilated checkerboard problem; Russian marriage problem;
other isomorphic versions.
• Circle problem. .
Cheap Necklace Problem – An Example of a False Constraint
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Cheap Necklace Problem (Chain Problem)
Cheap Necklace Problem: Convert these 4 strands of chains into a
single loop by opening and closing only 3 links. (Insight problem)
• This is an example of a problem that is difficult because people
apply a false constraint to the problem representation.
Stop Here?
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Problem Definition for the Chain Problem
Initial state: 4 strands of
chains, initially separated.
Goal state:
One unbroken loop.
Moves:
Open and close links.
Constraint: Only 3 links can
be opened and closed.
Initial State Goal State
What series of permissible moves will transform the initial
state into the goal state?
Solution to the Chain Problem
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
(Next – see how this would
work)
Show How to Visualize the Solution
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
Show how to visualize the solution
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
Show how to visualize the solution
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
Show how to visualize the solution
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
Show how to visualize the solution
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
Show how to visualize the solution
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Solution to the Chain Problem
• Open all three links of one
strand.
• Use these open links to link
together the other three
strands.
Summary re Solution to the Cheap Necklace Problem
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Summary re Solution to the Chain Problem
• Open all three links of one strand.
• Use these open links to link together the
other three strands.
Why is this solution hard to discover?
• False constraint: People assume that
they can only open the links at the ends
of existing chains.o Often we have difficulty solving a problem
because we add a requirement to the solution that is not a true requirement (false constraint).
Nine Dot Problem
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Nine-Dot Problem
Make a diagram that has 9 dots as shown below.
Draw 4 straight lines that connect all of the dots
without lifting the pencil or pen from the paper.
• The Nine-Dot Problem is difficult because people tend to assume a
false constraint. (Same difficulty as with the Cheap Necklace
Problem.)Failed Attempt at a Solution to the Nine-Dot Problem
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Nine-Dot Problem (cont.)
• Dead-end thinking.This is NOT a solution (5 lines are used).
Psych 355, Miyamoto, Spr '15 Solution to the Nine-Dot Problem
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Solution to the Nine-Dot Problem
• "Thinking inside the box" – People impose constraints on the
problem that aren't there. To solve this problem, you have to
“think outside the box.”
• False constraint: In a failed solution, people assume that they must
stay within the boundaries of the square.
So Far: Two Obstacles to Problem Solving
It can be useful to "think outside the box" – discard false constraints on the problem solution.
Psych 355, Miyamoto, Spr '15 32
Tuesday, May 26, 2015: The Lecture Ended Here
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So Far: Two Obstacles to Problem Solving
• Obstacle #1: A poor initial problem representation makes it difficult
to solve a problem.o Example: The Circle Problemo Example: The Mutilated Checkerboard Problemo Remedy: Change the problem representation
(sometimes a radical change is helpful.)
• Obstacle #2: People sometimes place a false constraint
on the permissible ways to solve the problem.o Example: Cheap Necklace Problem.o Example: Nine-Dot Problemo Remedy: Examine the constraints – are you imposing a false constraint?
END: Time Permitting, a Fishing Story
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Time Permitting: An Example of a Real False Constraint – A Fishing Story
• Time permitting, give practical example of a false constraint.
END