Introduction to Problem Solving Psychology 355: Cognitive Psychology Instructor: John Miyamoto 05/26 /2015: Lecture 09-2 This Powerpoint presentation may

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.)

2Psych 355, Miyamoto, Spr '15

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


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


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


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


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


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


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.


= domino piece

Failed Attempt to Solve the Mutilated Checkerboard Problem

12

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.


= 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


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


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




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.


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.









Concept of Problem Isomorphs

= domino piece

Mutilated Checkerboard Problem Russian Marriage Problem


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


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


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


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?


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



• 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




strand.



strands.

Show how to visualize the solution




strand.



strands.





strand.



strands.





strand.



strands.





strand.



strands.





strand.



strands.

Summary re Solution to the Cheap Necklace Problem


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


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

30

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


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


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


Time Permitting: An Example of a Real False Constraint – A Fishing Story

• Time permitting, give practical example of a false constraint.

END

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Introduction to Problem Solving Psychology 355: Cognitive Psychology Instructor: John Miyamoto 05/26 /2015: Lecture 09-2 This Powerpoint presentation may