Arrangement problems

• People are given some objects and asked to arrange them in a way that satisfies some criterion.

  Anagrams: KEROJ = JOKER

• Skills needed to solve arrangement problems
  • Fluency in generating possibilities
  • Retrieving prior solution attempts
  • Knowledge of principles that constrain search(certain letter combinations are unlikely to co-occur; Vowels are often in the second position)

• Functional Fixedness in arrangement problems: Tendency not to see alternative uses for common tools.
  • The candle problem:
    • Subjects are put in a room in which there is a table with three boxes containing candles and some tacks.
    • The goal is to place the three candles at eye level on a door in the room using only the objects on the table.
    • The solution is to tack the boxes to the door and use them as platforms for the candles.
    • People are more likely to solve the problem if in the initial state the candles are out of the box than if they are in the box.
  • The string problem:
    • There are two strings hanging from a ceiling but are too far apart to allow a person to hold one and walk to the other.
    • On the floor are a book of matches, a screwdriver, and a few pieces of cotton.
    • The goal is to tie the two pieces of string together
    • The solution is to tie the screwdriver to one of the strings and swing it like a pendulum. Then while the string is swinging, walk over to the other string.
  • Functional fixedness limits creative problem solving because it causes people to ignore possible solutions to problems.

Structure inducing problems

• Some objects are given that have a certain relationship to one another. Subject is asked find out what this relationship is.

  • What's the next number:

    1 2 8 3 4 6 5 6 ?

  • Simple Analogies
    • Merchant is to Sell as Customer is to ?
    • Washington is to 1 as Lincoln is to 10 or 5?

Transformation Problems

• In transformation problems there is an initial state, a group of legal moves which change the state, and a goal state.
• Water jar problem
  • Let's say you have a 8 gallon pail and a 5 gallon pail. How could you obtain exactly 2 gallons of water?

    Initial state: Both pails are empty.

  • Legal moves:
    • You can added water to either pail from an external source
    • You can transfer water between the two pails.
  • Goal state: One of the pails should have exactly 2 gallons

Newell & Simon's Theory of Problem Solving: The General Problem Solver.

• What is problem solving: They described all problem solving as being transformational. There is some initial state, some set of legal moves and a goal state.
• The problem space: These are the set of all possible paths that a problem solver could take in trying to solve a problem.
  • Let's say you want to win a game of tic-tac-toe.
    • The initial state is that there are no Xs and no Os.
    • The goal state is that you should get three Xs in a row before the other player gets three Os in a row.
    • The legal moves are that you are allowed to place an X on a square when it is your turn and you can put an X in any square which is empty.

  • The problem space is the set of all the possible ways the game could get played.
  • COMPUTATIONAL COMPLEXITY: The problem with the problem space is that even for a simple game the problem space is huge.

    X		O
    	X
    		O

  • In tic-tac-toe there are 9 ititial moves possible
  • For each of the 9 initial moves there are 8 subsequent moves resulting in 72 possible boards just two moves into the game
  • For each of those 72 possible boards, there are 7 possible moves for a total of 504 possible boards just three moves into the game.
  • And so on!
    OR CONSIDER THIS GAME

    	DONALD
    +	GERALD
    =	ROBERT

  • Where each letter represents a number between 0 - 9 and no letter represents the same number. And D = 5

  • A dumb way to solve this problem would be to try out every possible letter number combination until you found one that worked.

  • This would require trying out
    8 x 7 x 6 x 5 x 4 x3 x2 x1 = 362,880 possibilities.

  • THAT IS A LARGE PROBLEM SPACE

• Constraining the search: Successful problem solvers are successful because they search only part of the problem space, ie, only the parts of the problem space which will be fruitful. (i.e. R is an odd number which is more than 5 so only search the paths where R is 7 or 9)
  • Ways of Constraining the Search
    • Algorithms: A method of solving a problem which is guarenteed to work for an entire class of problems. Algorithms provide for certainty but are often time consuming.
    • Heuristics: Methods of problem solving which tend to work on average but are not guarenteed to work.
      • Means/Ends analysis: Reduce the distance between the goal state and the current state.

        There are 5 guards and 5 prisoners on one side of a river.

        There is a single boat which is bigenough to hold 3 people.

        The goal is to have all five guards and all 5 prisoners on the other side of the river.

        But if the prisoners out number the guards at any point they will be killed.

        How can the people get across the river without anyone being killed.

        • Play the game on-line.

        • The optimal solution to this problem requires 11 moves but most people take between 20-30 moves.
        • Most people proceed by trying to reduce the difference between the goal state, all ten people on the other side of the river and whatever the current state is.

      • Subgoals: Subgoals are similar to means-ends analysis only when using subgoals there is some specific state that someone is trying to achieve that should make solving the rest of the problem easier.

        Tower of Hanoi problem

        • Puzzle consists of a series of 3 pegs and rings of varying sizes.
        • The initial state all the rings are on peg A and the end state is that all the rings are on peg C.
        • But a larger ring can never be placed on top of a smaller ring.
        • A subgoal people use when solving this problem is the goal of having the largest ring on the bottom of peg C

        **Think-out-loud protocols indicate that people often spontaneously mention subgoals when working on problems. **Subjects who are given subgoals solve problems more quickly than those without the subgoals **However its not always obvious what the subgoals are in a problem
      • Working Backwards: Sometimes it is easiest to solve a problem if you work backwards from the goal state towards the goal state.

Analogy in problem solving

Analogies are comparisons based on shared relational structures.
• Consider the following problems

  A fortress was located in the center of the country. Many roads

  radiated out of the fortress. A general wanted to capture the fortress

  with his army. The army was to large to attack the general using a

  single road but if the general does not send the whole army at once his

  troops will be defeated. What should the general do?

  A tumor was located in the interior of the patient's body. A doctor

  wanted to destroy the tumor with rays. To destroy the tumor the

  doctor would have to use a high intensity ray. But a high intensity ray

  would also destroy too much healthy tissue. What should the doctor do?

• People who know the solution to the first problem could apply it to the second problem but most fail to without specific hints.
• Analogical Problem Solving: The Structure Mapping Approach (Gentner, 1983)
  • When solving a problem by Analogy you use a previous problem to solve a problem which is analogically related. How does this occur
    • Retrieve from memory the previous problem
    • Align the representation of the old problem with the representation of the new problem
    • Evaluate the analogical goodness of the two problems
    • Derive candidate inferences
  • The alignment process
    • Problem representations
      • Attributes (HOT, sun)
      • Relations(REVOLVE, earth, sun)
      • Higher order relations(CAUSE, (HOT, sun), (SUNBURNED, man))
    • Alignment involves trying to set up a 1-to-1 correspondence between the two representations in which the relations are preserved (especially the higher order relations)

      Atom	Solar System
      Nucleus	Sun
      Electrons	Planets
      Electrical force	Gravitational force

  • Candidate inferences: Candidate inferences are made by assuming that when two problems are good analogs that unstated relations might also hold.
  • Retrieving analogs: There's the rub
    • Analogy is a useful way to solve problems. It turns out thought that whereas analogy is governed by relations memory retrieval seems to be heavily influenced by attributes.
    • Gick and Holyoak (1983): Subjects given the stories about the fortress and then about the tumor. Even though the fortress story immediately preceded the tumor story only 30% of the subjects successfully solved the problem. When they were told to think of the previous story, 90% solved the problem.
    • MAC/FAC: Many Are Called/Few Are Chosen.
      • MAC
        • In the MAC phase relations and attributes are treated as if they were identical. Both attributes and relations contribute equally to the probability that something will be recalled.
        • In the MAC phase a relatively large number of memories are retrieved.
      • FAC
        • In the FAC stage the similarity of the two problems are compared. The one that's most similar wins out. (Here similarity considers more than just a list of features and can also consider relationships between features).
      • How does MAC/FAC explain the results?
        • Remindings based on attributes is more likely because typically there will be more of these than there will systematic relations. Thus the MAC stage will tend not to retrieve analogies
        • But when analogies are retrieved they will be preferred by the FAC stage.

Insight in Problem Solving

• Kohler(1925) Mentality of Apes:
  • Apes were put in a room with a banana hanging from the ceiling.
  • Various objects were placed in the room. Correct solution required combining various objects.
  • Apes started off fairly randomly but solution seemed to become suddenly apparent to apes (based on Gestaltist's interpretation of the way the apes acted.)
• Insight(Gestalt): In arrangement problems correct solution often seems to appear in a sudden flash of insight.
• Metcalfe (1986): Subjects given anagrams to solve. Every 10 seconds as they were solving the problem they were asked to indicate on a scale how close they thought they were to a solution. Results showed that subjects gave low ratings until immediately before finding the answer.

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