(posted 3/23/06)
Reading assignment:  Chapter 12

Opening example:  grain -weighing problem

• A farmer had a 40 pound stone that he used to weigh 40 pounds of feed using a balance scale.
• A neighbor borrowed the stone, and when she returned it she said, “I'm terribly sorry, I dropped the stone and it broke into four pieces.”
• The farmer said “Don't worry, you actually did me a favor,”  and he explains that the pieces of the broken stone could now be used to weigh any item, assuming those items were at one pound increments from one pound to 40 pounds.
• What were the weights of the four individual stones? And how does the farmer’s system work?

How do people solve problems?

• Form a problem representation, including a _______
• Take steps to move toward goal
• Use a strategy (or _________) for selecting each step

• ________-_________ representation
• Problem has a starting state
• Goal state
• Set of __________
• For example, consider the rats and cats problem
• Number scrabble and the importance of problem representation
• Problem Solving Strategies
• _________ - computation guaranteed to find the solution, but may be labor intensive
• _________ - simple strategies that often work, but not always

• __________________-
• Take a step, see if you are closer to the goal. If no, undo step, if yet, take another step & repeat
• Example: tuning a TV or radio
• Problem: could reach ________________
• Example: Stage in rats & cats problem where one has to move 2 animals back across the river.
• ____________ analysis
• Compare starting state to goal state
• Notice any discrepancy
• Think of a move that will decrease the discrepancy
• Make that move
• Repeat
• Working ____________
• Start with goal and work backwards to the starting state.
• Figure out what ____________ you need to achieve in order to reach the goal.
• Then figure out how to reach those subgoals.
• Example: Tower of Hanoi
• Subgoal is to get the big ring to post C
• Previous subgoal is to get middle and small rings on post B
• Previous subgoal is to get small ring on post C

• 2 players who take alternate turns
• Row of 10 pennies
• At each turn, player picks up 1, 2, or 3 pennies
• Goal is to make the other player pick up the last penny.
• What is your solution?
• Working forward: any move would get us closer to the last penny, but hard to know how to force the other player to pick up the last one.

• You have
• A candle
• A box of nails
• A pack of matches
• Your goal: Attach the candle to the wall so that, when lit, it does not drip wax on the table below
• Your solution? From Dunker (1945)

• ________________ = Einstellung = mental set
• Assume ______________ to the problem that aren't there.
• Represent the problem using familiar categories
• ________________ = restructuring the problem

• Simmels (1953) coin problem
• You have 8 coins and a balance
• Your job is to find the one counterfeit (lighter) coin by using the balance only twice
• 9 dot problem
• Connect all dots with four straight lines without lifting pencil
• Mental set is to stay ____________; solution requires _________________
• Maiers two string problem
• Use object in room (such as hammer) as a weight to swing one string like a pendulum
• Requires using the object ___________________

Transfer

• Can we apply what we learned in one problem to another problem?
• Does the problem solver learn only the solution to this particular problem, or is some general principle learned that could be transfered to new problems?
• Most-studied way to transfer a principle from one problem to another is through an ___________ .

• Problem
• Army general wants to attack a fortress in center of forest.
• Many roads lead to fortress like spokes of a wagon wheel.
• All the roads have land mines that are triggered by heavy traffice.
• Aren't triggered by light traffic, so allow supplies to be delivered.
• If entire army travels down a road, land mines will go off.
Solution?

• Problem
• Patient has an abdominal tumor (in center of body)
• Radiation will kill tumor
• But rays strong enough to kill tumor will also destroy healthy tissue that rays pass through on way to tumor
• Solution
• Point rays  . . .
• Rays  . . .
• All other tissue gets  . . .
• This is a _______________ solution (see schematic deptition presented in class)

The analogy

• tumor : _________
• X-rays: ________
• multiple rays : multiple _________
• converge on tumor : converge on __________

• Represent _______ problem
• Retrieve a useful source __________
• Abstract a _________ from source problem
• Map _____ onto ________

Some of the following encourage transfer and some do not. Which ones do?

• ____ Solving one base problem
• ____ Solving multiple base problems
• ____ Instructions to use the analogy
• ____ Explicit statement of the problem solving principle + 1 base problem
• ____ Explicit statement of the problem solving principle + 2 base problems

____ Surface & structural similarity between base and transfer problem

• Subjects learned about a principle, illustrated by an example. E.g.,
• Principle: ____________
• Example: 16 golfers compete in a tournament.  There are money prizes for the top 9 golfers. In how many different ways might the prizes (1st through 9th) be given?
• Solution: _________________