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					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
               ► Ariseswhen a goal is blocked
                 and the solution is not
                 obvious
      Learning vs. Thinking


                   Learning Curve

Puzzle Box
             First Trial
                     Exploring
                     Sniffing
                     Grooming
Tendency
                     Reaching
to perform
                     Scratching
                     Reaching with paw
                     Lever-Pressing


                                 Reward
             Later Trial

                     Lever-pressing
                     Exploring
                     Grooming
Tendency
                     Reaching
to perform
                     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
                  Achieving insight to the problem

► Verification    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
 full-scale direct attack
  How can the general attack the fortress?
                                       Using Analogies
                                 0.8
Proportion of People Accessing


                                 0.7
                                                               Same Theme
                                 0.6

                                 0.5                           Different Theme
            Analogy




                                 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   Long Prep, Long
                       Break               Break
      Problems in Arrangement
                Unscramble: xbo

► Algorithm                 ► Heuristic
   Generate all possible      If 2 consonants and a
    letter combinations         vowel, vowel probably
    and find which one(s)       in the middle
    is(are) real words         Few words start with x,
     ► xbo                      so put b first
     ► xob                       ► box
     ► oxb
     ► obx
     ► bxo
     ► box
               Try Some
► Set   One
     verba
     luppi
               ► Set   Two
     bagler
                    prega
     thrize
                    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):

                        Water Jug Problem
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
        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
               of
►Characteristics        ►Sequence of
 Problem Solving         problem solving
   Goal Directedness     Identify the problem
   Sequence of Steps     Represent the
   Cognitive              problem
    Operations            Plan the solution
   Subgoal               Execute the plan
    Decomposition         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
         ► Thisis 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            Destination
O

        O
O

        H
    H
        H          Boat
          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

           brain regions associated with the
► Identified
  mapping process in analogical reasoning
   Medial Frontal Cortex
   Left Prefrontal Cortex
   Left Inferior Parietal Cortex
► Solvinginsight 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.
► Monsteretiquette 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.
► Bywhat 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.
► Whatdisease does Ms. Anderson have, and in
 what room is she?

				
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