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.