“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 Jar B Jar C Obtain

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 full-scale 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.