Honors Seminar
Problem Solving
Modified: 2023-10-03 9:00 PM CDST
One aspect of critical thinking is problem solving. Solving abstract problems is more difficult than solving concrete ones. The section concludes with some classic problems.
Problem Solving
Play MasterMind in class as example of abstract problem solving
MasterMind Game
Play Mastermind online
- Wason Selection Problem: Read the problems below. Then decide which minimum number of cards you must turn over and why?
- Wason Selection Problem (Abstract)
- The secretary you replaced may have mistakenly misfiled some documents. The rule was: if a card has a D on one side, it should have a 3 on the other.
- Wason Selection Problem (Concrete)
- You are the new bouncer at a bar. Your job is to be sure the following rule is not violated: If someone is drinking beer, they must be 21 or over. The cards represent drinkers. On one side of the card it says their age and on the other what they are drinking.
- Scientists and Wason Selection task:
- Physicists (21-25% correct), Biologists (8-13% correct),
Psychologists (13-17% correct)
- Problem Solving Definition
- "gap that separates the present state and the goal state"
(Hayes, 1978)
- I need to get to Little Rock.
- I'm in Magnolia = present state
- I need to be in Little Rock = goal state
- Understanding the Problem
- Understanding the problem
- Context
- Why the need to go to Little Rock?
- Emergency?
- Routine?
- Entertainment
- Shopping
- Operators (rules, moves, legality, reality)
- Well defined vs. Fuzzy
- Chess and other games
- Real world rules
- Walking?
- Driving
- Flying?
- Teleporting?
- Goal State
- Games
- Part of game
- Chess = capturing King
- Football = scoring most points
- Golf = taking least strokes
- Real world
- Location in Little Rock
- School
- Work
- Life
- Solving the problem
- Problem Space
- Path from initial state to goal state
- Number of paths
- Types of Solutions
- Algorithms
- Guarantee a solution
- How many desks in Peace Hall?
- How many desks at SAU?
- How many desks in Arkansas?
- How many desks in the World?
- All of the above problems solved by same
algorithm: count them
- Some algorithms do not work in real time
- Heuristics
- Do not guarantee a solution (hunches, guesses, or
experential attempts)
- Traveling Salesman Problem
- Subgoaling
- Divide problem into smaller parts
- get vehicle
- fill with fuel
- establish route
- drive to Prescott
- drive to second Rest Stop
- drive to Little Rock
- Means-Ends Analysis
- Reduce the distance between initial and goal
states
- From the island of Magnolia sail NE basically to the island of Little Rock in the Sea of Arkansas
- Walk to California from Magnolia? Go west
- Working
- Start at goal and work back to initial state
- Need to be in Little Rock by 5:00
p.m.
- It takes 2.5 hours to drive
- I need to get gas first (add 30
minutes)
- I need to take kids to school (add 30
minutes)
- Leave by 1:30 p.m.
- Analogies
- Similar problems may help solve current
problem
- I have never driven to Little Rock, but I
have driven to Dallas. Much will be the same,
but some things (direction, roads) will be
different.
- Think about how all US Interstates follow the same design, use the same signage, and rules of the road
- Restructuring
- Gestalt idea
- Mental Set
- How are these numbers arranged?
- 8, 5, 4, 9, 1, 7, 6, 3, 2, 0
- Solve the following problem:
- Solution?
- Functional Fixedness
- Scheerer (1963) conducted an experiment where
he manipulated the salience of a piece of string.
The more salient it was, the more likely students
were to solve the problem.
- Nearly all students knew they needed string
to solve the problem. The solution was to tie
the two sticks together to make them effectively
longer. Picture
- Insight
- Learning characterized by sudden realization
about solution
- Incubation
- Delaying the problem solving process
- Works by:
- loss of detail and subsequent focusing on
important details
- better integration of recent and
pre-existing memories
- weakening of mental sets
- relaxation
- take a day to plan trip to Little
Rock
More Problems
- Cheap necklace problem:
- “You are given four separate pieces of chain that are each three links in length. It costs 2¢ to open a link and 3¢ to close a link. All links are closed at the beginning of the problem. Your goal is to join all 12 links of chain into a single circle at a cost of no more than 15¢.”
- Meier's two string problem: How can you tie two strings together if you cannot reach both at the same time. You also have: a chair, a pliers, some copier paper, and a jar of tacks.
- Missionaries and cannibals problem:
- Three missionaries and three cannibals all wish to cross a river. But there are some constraints. Their boat can only hold two people at a time. If ever the cannibals outnumber the missionaries then the missionary will end up as lunch. Get everyone across the river without anyone being eaten.
- Video
- Mutilated chessboard:
- Can the chessboard shown be covered by 32 dominoes? Why or why not?
- King and his daughter:
- Once upon a time there was a beautiful princess named Portia. Portia's father, the King, wanted to be sure his daughter married an intelligent man. To test his daughter's suitors the King hid Portia's picture in one of three boxes. The suitor had to be able to select the box with Portia's picture on one try and within twenty seconds. On the gold box was the message "Portia's picture is in this box". The silver box had the message "Portia's picture is not in this box." "Portia's picture is not in the gold box" was written on the bronze box. The King would tell each suitor "Only one of the three messages is correct." Which box contained Portia's picture?
- Solution and an additional problem
- Anagrams (easy):
- kmli
- graus
- teews
- recma
- foefce
- ikrdn
- Why did they get easier?
- Anagrams (hard):
- Find an anagram for: DRY OXTAIL IN REAR
- In 2009 when I presented this problem one student (Daniel Tye) shouted out the answer in a few seconds
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