Problem Solving
Kimberley Clow
kclow2@uwo.ca
http://instruct.uwo.ca/psychology/130
Outline
► Learning
vs. Thinking
► Stages in Creative Thinking




Preparation
Incubation
Illumination
Verification
► Hindrances
to Problem Solving
► Process of Problem Solving
► Methods of Problem Solving
► The Brain
What is Problem Solving?
► Problem
 A situation in which one is
trying to reach a goal
► Problem
Solving
 Finding a means for arriving
at a goal
► Arises
when a goal is blocked
and the solution is not
obvious
Learning vs. Thinking
Learning Curve
Puzzle Box
First Trial
Exploring
Sniffing
Grooming
Tendency
to perform
Reaching
Scratching
Reaching with paw
Lever-Pressing
Reward
Later Trial
Lever-pressing
Exploring
Grooming
Tendency
to perform
Reaching
Scratching
Reaching with paw
Sniffing
Classic Problem Solving
► Archimedes
 Is the king’s golden
crown pure?
► Köhler’s
Chimps
 How do I get at the
banana that is
outside my cage and
beyond my reach?
Insight
Stages of Creative Thinking
► Preparation
 Formulating problem & making
initial attempts to solve it
► Incubation
 Leaving the problem while
considering other things
► Illumination
► Verification
 Achieving insight to the problem
 Testing and/or carrying out the
solution
Preparation
►Different
types of problems need
different skills and approaches
 Problems in Inducing Structure
►Analogy
Problems
 Problems in Transformation
►Cheap
Necklace
 Problems in Arrangement
►Anagrams
Problems in Inducing Structure
► Suppose
you are a doctor faced with a
patient who has an inoperable stomach
tumour. You have at your disposal rays that
can destroy human tissue when directed
with sufficient intensity.
 How can you use these rays to destroy the
tumour without destroying the surrounding
healthy tissue?
Analogy
►A
general wishes to capture an enemy
fortress. Radiating outward from the fortress
are many roads, each mined in such a way
that the passing of any large force will
cause an explosion. This precludes a fullscale direct attack
 How can the general attack the fortress?
Proportion of People Accessing
Analogy
Using Analogies
0.8
0.7
Same Theme
0.6
Different Theme
0.5
0.4
0.3
0.2
0.1
0
Close Situation
Remote Situation
Situational Similarity
Problems in Transformation
►
Cheap Necklace Problem
 Make a necklace out of 4 chains. It costs 2 cents to open a link and
3 cents to close a link. Make the necklace without spending more
than 15 cents.
Given
Goal
Evidence for Incubation
90
85
80
75
70
65
60
55
50
45
40
Short Prep
Long Prep, Brief
Break
Long Prep, Long
Break
Problems in Arrangement
Unscramble: xbo
► Algorithm
 Generate all possible
letter combinations
and find which one(s)
is(are) real words
► xbo
► xob
► oxb
► obx
► bxo
► box
► Heuristic
 If 2 consonants and a
vowel, vowel probably
in the middle
 Few words start with x,
so put b first
► box
Try Some
► Set




One
verba
luppi
bagler
thrize
► Set




Two
prega
rogena
pleap
viole
► Set




Three
broin
arancy
chifn
relbawr
Evidence for Illumination

Draw through all 9 dots with 4 straight lines,
without lifting your pencil.
Individual Differences
► Remote
Associations Test (RAT)
 What single word is an associate of these three?
►
Results
 Correlation between RAT performance and the
generation of associates
 RAT improves following training in the generation of
word associations
 RAT is a better predictor of implicit learning than IQ
►IQ
is a better predictor of intentional learning
 High and low RAT scorers deploy their attention in
different ways
Verification
The water jar problem (Luchins, 1942):
How would you use 3 jars with the indicated capacities to measure out the desired
amount of water?
Desired Quarts of
Problem #
Jar A
Jar B
Jar C
water
Water Jug Problem
1
21
127
3
100
2
14
163
25
99
3
18
43
10
5
4
9
42
6
21
5
20
59
4
31
6
23
49
3
20
7
15
39
3
18
8
28
76
3
25
► Support
the candle on
the wall so that it
doesn’t drip on the
table below
Hindrances to Problem Solving
► Persistence
of Set
 Old strategies continue to be used -- even if they
are less efficient -- if we fail to perceive that the
situation has changed
► Functional
Fixedness
 A tendency to use objects in their customary way
► Negative
Set
 A tendency to solve problems in one particular
way, even when a different approach might be
more productive
►Why
are these numbers arranged in this order?
8, 5, 4, 9, 1, 7, 6, 3, 2, 0
Looking at the Process
►Characteristics
of
Problem Solving
 Goal Directedness
 Sequence of Steps
 Cognitive
Operations
 Subgoal
Decomposition
►Sequence
of
problem solving
 Identify the problem
 Represent the
problem
 Plan the solution
 Execute the plan
 Evaluate the plan
 Evaluate the
solution
Definitions
► Problem
Space
 Includes the initial, intermediate and goal
states of the problem
►Also
includes the problem solver’s knowledge at
each of these steps
► Operators
 The set of legal moves that can be
performed during problem solving
► Goal
 Ultimate solution to the problem
Methods of Problem Solving
► Brute
Force Search
 explore entire problem space
► Hill
Climbing
 always move toward a better state
► Work
Backward
 start with goal and work back to start
► Means-Ends
Analysis
 combination of hill climbing and working
backward
The Tower of Hanoi
The Problem Space
Brute Force Search
► Go
through all possible states until solution
is found
 Guaranteed to work (eventually)
 Impractical with large problem spaces
► Example
 A 4-letter word meaning “not smart”
 456,976 possible states!
Hill Climbing
Extreme Hill Climbing
► Dog
cannot stop hill
climbing toward bone
 needs to back up
 by going away from the
goal for a bit, it will
eventually solve the
problem
► This
is a problem for
people too…
Get the Orcs and Hobbits to the other side of the river
BUT
Can’t have more Orcs than Hobbits anywhere
Boat will only take 2 passengers at a time
Someone must steer the boat
Initial state
O
O
O
H
H
H
Boat
Destination
Means-End Analysis
► Five
Steps
 Set up a goal or subgoal
 Look at the difference between the current
state and the goal/subgoal state
 Look for an operator that will reduce or
eliminate the difference
 Apply the operator
 Apply steps 2 to 4 repeatedly until all goals
have been achieved
Improving Your Problem Solving
► Increase
Domain Knowledge
► Automate Components
► Have a systematic Plan
► Draw Inferences
► Develop Subgoals
► Work Backwards
► Search for Contradictions and Relations Among
Problems
► Use Different Problem Representations
► Practice!
Back to the Brain
► Identified
brain regions associated with the
mapping process in analogical reasoning
 Medial Frontal Cortex
 Left Prefrontal Cortex
 Left Inferior Parietal Cortex
► Solving
insight problems involves the right
hemisphere
For the Logic Lover
► Three
5-handed monsters were holding three
crystal globes.
► Because of the quantum-mechanical peculiarities
of their neighbourhood, both monsters and
globes come in exactly three sizes:
 small, medium, and large.
► The
medium-sized monster was holding the
small globe; the small monster was holding the
large globe; and the large monster was holding
the medium-sized globe.
► Because
this situation offended their keenly
developed sense of symmetry, they proceeded
to transfer globes from one monster to another
so that each monster would have a globe
proportionate to his own size.
► Monster
etiquette complicated the solution of the
problem because it requires:
 that only one globe may be transferred at a time
 that if a monster is holding two globes, only the larger
of the two may be transferred
 that a globe may not be transferred to a monster who is
holding a larger globe.
► By
what sequence of transfers could the monsters
have solved this problem?
Symbol Problem
► Solve
the following problem:
 Mary is ten years younger than twice Susan’s
age.
 Five years from now, Mary will be eight years
older than Susan’s age at that time.
 How old are Mary and Susan?
List Person
► We








know that
1. The person with asthma is in Room 101.
2. Ms. Jones has heart disease.
3. Ms Green is in Room 105.
4. Ms. Smith has tuberculosis.
5. The woman with mononucleosis is in Room 104.
6. Ms. Thomas is in Room 101.
7. Ms. Smith is in Room 102.
8. One of the patients, other than Ms. Anderson, has
gall bladder disease.
► What
disease does Ms. Anderson have, and in
what room is she?