Problem Solving

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Problem Solving

• Kimberley Clow
• kclow2_at_uwo.ca
• http//instruct.uwo.ca/psychology/130

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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

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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

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Learning vs. Thinking
Learning Curve
Puzzle Box
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First Trial
Exploring
Sniffing
Grooming
Tendency to perform
Reaching
Scratching
Reaching with paw
Lever-Pressing
Reward
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Later Trial
Lever-pressing
Exploring
Grooming
Tendency to perform
Reaching
Scratching
Reaching with paw
Sniffing
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Classic Problem Solving

• Archimedes
• Is the kings golden crown pure?
• Köhlers Chimps
• How do I get at the banana that is outside my
  cage and beyond my reach?

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Insight
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Stages of Creative Thinking

• Preparation
• Incubation
• Illumination
• Verification

• Formulating problem making initial attempts to
  solve it
• Leaving the problem while considering other
  things
• Achieving insight to the problem
• Testing and/or carrying out the solution

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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

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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?

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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 full-scale direct attack
• How can the general attack the fortress?

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Using Analogies
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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
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Evidence for Incubation
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Problems in Arrangement
Unscramble xbo

• Heuristic
• If 2 consonants and a vowel, vowel probably in
  the middle
• Few words start with x, so put b first
• box

• Algorithm
• Generate all possible letter combinations and
  find which one(s) is(are) real words
• xbo
• xob
• oxb
• obx
• bxo
• box

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Try Some

• Set One
• verba
• luppi
• bagler
• thrize

• Set Two
• prega
• rogena
• pleap
• viole

• Set Three
• broin
• arancy
• chifn
• relbawr

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Evidence for Illumination

• Draw through all 9 dots with 4 straight lines,
  without lifting your pencil.

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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

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Verification
Water Jug Problem
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• Support the candle on the wall so that it doesnt
  drip on the table below

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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

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Looking at the Process

• Sequence of problem solving
• Identify the problem
• Represent the problem
• Plan the solution
• Execute the plan
• Evaluate the plan
• Evaluate the solution

• Characteristics of Problem Solving
• Goal Directedness
• Sequence of Steps
• Cognitive Operations
• Subgoal Decomposition

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Definitions

• Problem Space
• Includes the initial, intermediate and goal
  states of the problem
• Also includes the problem solvers 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

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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

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The Tower of Hanoi
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The Problem Space
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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!

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Hill Climbing
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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

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Get the Orcs and Hobbits to the other side of the
river BUT Cant have more Orcs than Hobbits
anywhere Boat will only take 2 passengers at a
time Someone must steer the boat
Destination
Initial state
Boat
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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

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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!

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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

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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.

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• 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?

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Symbol Problem

• Solve the following problem
• Mary is ten years younger than twice Susans age.
• Five years from now, Mary will be eight years
  older than Susans age at that time.
• How old are Mary and Susan?

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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?