Transcript Problem Solving - Meredith College

“If we can really understand the
problem, the answer will come out of it,
because the answer is not separate from
the problem.”
Jiddu Krishnamurti
Indian philosopher
Problem Solving
Definition
• Problem solving refers to active efforts to
discover what must be done to achieve a
goal that is not readily attainable
• In problem solving situations, one must go
beyond the information given to overcome
obstacles and reach a goal
Types of Problems
1) Problems of inducing structure – the person
must discover the relationships among the parts
of the problem (e.g., series-completion,
analogies)
2) Problems of arrangement – the person must
arrange the parts in a way that satisfies some
criterion (e.g., string problem)
3) Problems of transformation – the person must
carry out a sequence of transformations in order
to reach a specific goal (e.g., hobbits & orcs
problem)
Barriers to Effective Problem
Solving
You’re driving a bus that is leaving from
Pennsylvania and ending in New York.
To start off with, there were 32
passengers on the bus. At the next bus
stop, 11 people get off and 9 people get
on. At the next bus stop, 2 people get off
and 2 people get on. At the next bus stop
12 people get on and 16 people get off.
At the next bus stop, 5 people get on and
3 people get off.
What color are the bus
driver’s eyes?
In the Thompson family there are five
brothers, and each brother has one sister.
If you count Mrs. Thompson, how many
females are there in the Thompson
family?
Answer = 2
The number of brothers is irrelevant in
determining the number of females in the
Thompson family
15% of the people in Topeka have
unlisted telephone numbers. You select
200 names at random from the Topeka
phone book. How many of these people
can be expected to have unlisted
telephone numbers?
Answer = None
The numerical information is irrelevant, since
all the names came out of the phone book
Irrelevant Information
Effective problem solving requires
that you attempt to figure out what
information is relevant and what is
irrelevant before proceeding.
Functional Fixedness
The tendency to perceive an item
only in terms of its most common
use.
For each problem in the series, you must
work out how you could measure out the
quantities of liquid indicated on the right
by using jars with the capacities shown
on the left.
Jar A
1. 21
2. 14
3. 18
4. 9
5. 20
6. 28
Jar B
127
163
43
42
59
76
Jar C
3
25
10
6
4
3
Obtain
100
99
5
21
31
25
Mental Set
When people persist in using
strategies that have worked in the
past. Similar to functional fixedness
in that it involves rigid thinking.
Without lifting your pencil from the
paper, try to draw four straight lines that
will cross through all nine dots.
Unnecessary Constraints
Effective problem solving requires
specifying all the constraints
governing a problem without
assuming any constraints that don’t
exist.
Approaches to Problem Solving
Strategies we often use
• Trial and Error
• Algorithm (e.g., matchstick problem)
• Heuristics
Types of Heuristics
Means/Ends Analysis
Involves identifying differences that
exist between the current state and the
goal state and making changes that
will reduce these differences.
Forming Subgoals
Intermediate steps toward a solution.
When you reach a subgoal, you’ve
solved part of the problem.
The water lilies on the surface of a small
pond double in area every 24 hours.
From the time the first water lily appears
until the pond is completely covered
takes 60 days. On what day is half of the
pond covered with lilies?
Answer = Day 59
Working Backwards
If you’re working on a problem that has a
well-specified end point, you may find the
solution more readily if you begin at the
end and work backwards. This is a good
strategy when you have many available
options at the beginning, but relatively few
options near the end.
The Tumor Problem
Given a human being with an inoperable
stomach tumor, and rays that destroy
organic tissue at sufficient intensity, by what
procedure can one free him of the tumor by
these rays (and ONLY these rays) and at the
same time avoid destroying the healthy
tissue that surrounds it?
The Story of the General
A small country was ruled from a strong fortress by a dictator.
The fortress was situated in the middle of the country,
surrounded by farms and villages. Many roads led to the
fortress through the countryside. A rebel general vowed to
capture the fortress. The general knew that an attack by
his entire army would capture the fortress. He gathered his
army at the head of one of the roads, ready to launch a fullscale direct attack. However, the general then learned that
the dictator had planted mines on each of the roads. The
mines were set so that small bodies of men could pass over
them safely, since the dictator needed to move his troops
and workers to and from the fortress. However, any large
force would detonate the mines. Not only would this blow
up the road, but it would also destroy many neighboring
villages. It therefore seemed impossible to capture the
fortress.
However, the general devised a simple plan. He divided his
army into small groups and dispatched each group to the
head of a different road. When all was ready he gave the
signal and each group marched down a different road.
Each group continued down its road to the fortress so that
the entire army arrived together at the fortress at the same
time. In this way, the general captured the fortress and
overthrew the dictator.
Using Analogies
If you can spot an analogy between
problems, you may be able to use the
solution to a previous problem to
solve a current one.
A dog has a 6 foot chain fastened around
his neck and his water bowl is 10 feet
away. How can he reach the bowl?
Changing the Representation of
the Problem
Whether you solve a problem often
hinges on how you envision it. When
you fail to make progress with your
initial representation of a problem,
changing your representation may be
a good strategy.
This problem involves using the numbers
1,2,3,4,5,6,7,8,9. The goal is to pick any
three of these numbers which, when added
together, must equal 15. Each person
alternates choosing and cannot take longer
than 1-2 seconds to make a choice. Once a
number is used it cannot be chosen again.
The police entered the gym containing
five wrestlers just as the dying man
looked at the ceiling and mumbled the
words, “He did it.” They immediately
arrested one of the wrestlers. How did
they know which one?
The other 4 wrestlers were women.
Remove six letters from
ASIPXPLETLTERES.
What word is left?
Answer: Remove S-I-X L-E-T-T-E-R-S
and you get APPLE.
Expertise: Knowledge and
Problem Solving
What do experts know that makes
problem solving more effective?
• Use of schemas
– Chase and Simon (1973)
• Organization of schemas
• Representation of the problem
Creativity
Four Principles of Creativity
• Flexibility – thinking of different kinds of
ideas.
• Fluency – thinking of many, many ideas.
• Elaboration – adding details to an already
existing idea.
• Originality – thinking of unusual or novel
ideas.
What facilitates creative
thinking?
•
•
•
•
•
Effective use of analogies
Incubation
Insight
Broad knowledge base
Motivation and persistence
How can we judge creativity?
• Convergent vs. Divergent thinking tasks
• Abstract or unconventional thinking
• Cultural heritage may limit the creative
process
The Ping-pong Ball Problem
Assume that a steel pipe is embedded in the concrete
floor of a bare room as shown in the picture
below. The inside diameter is 0.6 inches larger
than the diameter of a ping-pong ball (1.50 inches)
that is resting gently at the bottom of the pipe.
You are one of a group of six people in the
room, along with the following objects:
100 feet of clothesline, a carpenter’s
hammer, a chisel, a box of Wheaties, a file,
a wire coat hanger, a monkey wrench, and a
light bulb. How can you get the ball out of
the pipe without damaging the ball, the
pipe, or the floor?
Are highly creative people more
intelligent than others?
Not necessarily. While people who score
high in creativity also tend to score
high in intelligence, many intelligent
people are not very creative.
Example
The following is reportedly an actual question given
on a University of Washington chemistry mid
term. The answer was so “profound” that the
professor shared it with colleagues, which is why
we now have the pleasure of enjoying it as well.
Bonus Question: Is Hell exothermic (gives off heat)
or endothermic (absorbs heat)?
Most of the students wrote proofs of their beliefs
using Boyle’s Law (gas cools off when it expands
and heats up when it is compressed) or some
variant. One student, however, wrote the
following:
First, we need to know how the mass of Hell is changing in
time. So we need to know the rate that souls are moving
into Hell and the rate they are leaving. I think that we can
safely assume that once a soul gets to Hell, it will not
leave. Therefore, no souls are leaving.
As for how many souls are entering Hell, let’s look at the
different religions that exist in the world today. Some of
these religions state that if you are not a member of their
religion, you will go to Hell. Since there are more than
one of these religions and since people do not belong to
more than one religion, we can project that all souls go to
Hell. With birth and death rates as they are, we can expect
the number of souls in Hell to increase exponentially.
Now, we look at the rate of change of the volume in Hell
because Boyle’s Law states that in order for the
temperature and pressure in Hell to stay the same, the
volume of Hell has to expand as souls are added. This
gives us two possibilities:
1) If Hell is expanding at a slower rate than the rate at
which souls enter Hell, then the temperature and
pressure in Hell will increase until all Hell breaks loose.
2) Of course, if Hell is expanding at a rate faster than the
increase of souls in Hell, then the temperature and
pressure will drop until Hell freezes over.
So which is it? If we accept the postulate given to me by Ms.
Teresa Banyan during my freshman year, “…that it will
be a cold day in Hell before I sleep with you,” and take
into account the fact that I still have not succeeded in
having sexual relations with her, then, #2 cannot be true,
and thus I am sure that Hell is exothermic and will not
freeze.
This student received the only ‘A’ given.