Problem Solving Concepts

Defining a Problem

  • What is a problem?
    • A problem can be defined as a situation where there is a goal to achieve but is impeded by obstacles needing resolution.
    • Examples of problems:
    • Mathematical: Find the mean of the following numbers: 3, 6, 7, 11.
    • Logical: "All humans are mortal, Socrates is a human. Is Socrates mortal?"
    • Scientific: Investigate the role of mitochondria.
    • Engineering: Determine the maximum weight a bridge can bear.
    • Programming: Implement a stack data structure using two queues.
    • Business: Strategies to improve customer loyalty.
    • Social: Solutions for combating climate change.
    • Philosophical: Ethical considerations of using AI for academic work.
    • Creative: Design a logo for a new enterprise.
    • Personal/Emotional: Ways to enhance time management skills.

Types of Problem-Solving

  • SAT/GRE Preparation Problems - Verbal Analogies
    • Examples of word analogies:
    • Merchant : Sell :: Customer : __ ?
    • Lawyer : Client :: Doctor : __ ?
    • Mason : Stone :: Carpenter : __ ?
    • Apprentice : Master :: Student : __ ?
  • Logic Problems:
    • Missionaries and Cannibals Problem:
    • Five missionaries and five cannibals need to cross a river using a boat carrying only three at a time.
    • The solution needs to ensure missionaries are never outnumbered by cannibals.

River Crossing Problems

  • Water Jug Problem
    • Given a 5-gallon jug and a 3-gallon jug, how to measure out exactly 4 gallons of water?
    • Solution steps:
    1. Fill the 5-gallon jug completely.
    2. Pour water from the 5-gallon jug into the 3-gallon jug until it is full, leaving 2 gallons in the 5-gallon jug.
    3. Empty the 3-gallon jug.
    4. Pour the 2 gallons from the 5-gallon jug into the 3-gallon jug.
    5. Fill the 5-gallon jug again.
    6. Pour into the 3-gallon jug until it is full, leaving exactly 4 gallons in the 5-gallon jug!

Representations of Problems

  • Newell & Simon (1972): Four Features of a Problem:
    1. Goal: A solution definition.
    2. Relevant Objects: Items/information needed for a solution.
    3. Operations/Actions: Steps to reach a solution.
    4. Constraints: Limitations that can't be violated during the solving process.

Well-defined vs. Ill-defined Problems

  • Well-defined Problems:
    • Example: Arithmetic problems, Tower of Hanoi puzzle, jigsaw puzzles.
    • Clear goal, defined objects, operations, and constraints.
  • Ill-defined Problems:
    • Example: Solving global issues like world hunger or defining meaning of life.
    • Lack specific objects, operations, or constraints.

Problem Representation Example

  • Problem: Sandwich costs 5 times as much as soda ($2 less than the sandwich).
    • Let S = Sandwich, D = Drink (Soda).
    • Relationships:
    • S = 5D
    • Set up equations to find costs.
    • Solutions yield Sandwich = $2.50, Drink = $0.50

Algorithms vs. Heuristics

  • Algorithm: A defined procedure to solve a problem (example: finding a painting at the Louvre).
  • Heuristics: General strategies for solving problems. Examples include:
    • Inferences, means-end analysis, working backwards, and using analogy.

Impediments to Problem Solving

  • Functional Fixedness (Duncker, 1945):
    • Candle Problem: Finding a method to attach the candle to the wall without using it as a simple candle.
  • Set Effects:
    • Influences of previous experiences impacting problem-solving efforts.

Insight and Incubation

  • Insight Problems: Problems that seem suddenly solvable with an "Aha!" moment, such as water lily problems or riddles.
  • Incubation Effect: Taking a break from a problem can lead to fresh insights upon return; helpful in complex problems that might benefit from subconscious processing.

Test-Taking Advice

  • Get sufficient rest, eat a healthy breakfast, read instructions carefully for clarity, pause for difficult problems to allow for incubation, as new insights may arise after a break.