Problem Solving Techniques

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Most Important Ideas
Prerequisite: Are you solving the right problem?
State initial Problem
The Steps of Problem Solving
Articulating Assumptions around your Problem
Assumptions
are often hidden and unrecognized
set limits to the problem
Reflect desired values 
Simplify the problem
Pick an entry-point
Have multiple hypothesis
Tips
Solve the problem that really exists, not just the symptoms of a problem, not the problem you already have a solution for, not the problem you wish existed, and not the problem someone else thinks exists.
When the goal state is clear but the present state is ambiguous, try working backwards.
 Be careful not to look for a solution until you understand the problem, and be careful not to select a solution until you have a whole range of choices.
Take time to examine and explore the problem thoroughly before setting out in search of a solution. Often, to understand the problem is to solve it.

As with creative thinking, flexibility is a crucially important feature in problem solving. Many of these techniques you will begin to use regularly for each major problem you address. Others you will use selectively.

Assumption Articulation
A first and frequently overlooked step in problem solving is to identify the assumptions you are making about the situation. Many of the assumptions will be hidden and unrecognized until a deliberate effort is made to identify them. Often it is the unrecognized assumption that prevents a good solution. However, before we get too critical of assumptions, we should note their value and necessity. So we begin there.

Assumptions are Necessary
Assumptions and constraints are necessary for three reasons:

1. They set limits to the problem and thus provide a framework within which to work. These limits might include constraints of possibility, economics, or some other desired narrowing.

2. Assumptions reflect desired values, values that should be maintained throughout the solution. For example, in punishing criminals, we assume that we are still concerned about their humanity, so that, say, torture with electric prods will not be considered as a possibility for punishment.

3. Assumptions simplify the problem and make it more manageable by providing fewer things to consider and solve. A problem with no assumptions is usually too general to handle.

Assumptions are Often Self-imposed
In spite of the necessity of having assumptions, many assumptions produce self-imposed limits. That is, the impossibilities or fixed constraints in a problem are often not imposed by nature or the laws of physics, but by ourselves through our understanding of the situation or through the desire to focus the problem.

In assumption articulation, then, our goal is to identify the assumptions being made and to determine the following:

1. Is the assumption necessary? If not, can or should it be dispensed with?

2. If the assumption is not necessary, is it appropriate? That is, many rather arbitrary assumptions and constraints are nevertheless desirable.

For example, when we say, “We have only two weeks to solve this problem,” those two weeks may be entirely appropriate as an outside time limit for generating and implementing the solution, simply because the problem’s importance in relation to the rest of life warrants no more than those two weeks.

Examine the Assumptions Behind your Problem
1. Make a list of assumptions. As you think about your problem, force to the surface every given, taken for granted, assumed fact about the situation you can think of. Many, if not most, assumptions do not really fit into categories like those in the checklist below. Instead, most assumptions are statements about reality that we believe to be true. Many of them are “obvious” and we normally would not think to question them. Yet that is exactly why we so often get blocked when we try to solve a difficult problem.

For example, the design of women’s swim suits was long constrained by limited technology. How can we make a new design that will stand up to the rigors of swimming in salty or highly chlorinated water? Only a few fabrics are strong enough and printing or decorations don’t hold up well. The completely obvious and absolutely unquestionable assumption being made here is that most women do a lot of swimming in their swim suits. Of course, dummy, why else would they buy them? Some brave soul, who was probably called a fool, decided to question this assumption and do some research. It was discovered that 90% of women’s swim suits never get wet (except perhaps in the laundry). This was quite a revelation for suit designers, because it opened up a whole new world of materials and designs that would stand up to sunning but wouldn’t take swimming. Who would have thought that anyone would buy a swim suit marked “dry clean only”?

When you have thought of all the miscellaneous assumptions you can, you might find it helpful to use a checklist of assumption areas like this:

A. Time. How quickly or slowly am I assuming it will take? Can the solution be sped up or can more time be found somewhere?

B. Money. Are the limits of money I’m assuming necessary? Can I find more money? Or, more creatively, can I do it for less money or no money? Can I get someone else to pay? Money is a common block to the solution of many problems. We say, If only I had the money, I could do it. Often, however, we can find ways of accomplishing the same thing with less money or with none or with other people’s money. Don’t let the money psychology block you. Example: We need computers and hard disks but we don’t have the money. Possibilities: donated funds, find lower price, get manufacturers or dealers to donate the parts.

C. Cooperation. Am I assuming that certain people will be in favor of the solution, support it, help implement it, when in fact they might not? Or am I assuming that certain people will be against it when they might not be?

D. Physics. Are the laws of physics interfering? The problem is “impossible” of solution? What at first seems physically impossible may on reflection not be so after all. Remember the pear in the bottle, “moving” the Statue of Liberty, or even launching rockets out of the atmosphere.

E. Law. Is the solution blocked by law? Can the law be changed, circumvented (for moral purposes only), or even broken (for the right cause)? Maybe it can be reinterpreted to permit the solution. Example: Bible clubs in high schools. According to one high school’s interpretation, the Freedom of Association law permits students to get together to pray but not to advertise their prayer group. Can this regulation be skirted by word of mouth advertising or by holding a prayer meeting right after another non-prayer meeting?

F. Energy. We can devote only so much energy to any given solution. Is the amount assumed to be appropriate or maximum really so? It’s better to expend a little more energy to solve a problem well the first time than to have to redo the entire thing after a half-energetic solution.

G. Cost/Benefit. How much is it worth to solve the problem? Costs can include an investment of time, energy, money, emotion, or other resource–mental effort, eyesight, whatever.

H. Information. Is the information available correct? This assumption often proves wrong. Double check the so-called facts surrounding the problem. Note that in most cases, more information can always be obtained. Are we assuming that all available information or all pertinent information is at hand? New information often changes the entire appearance of the problem?

I. Culture Binding. Is the solution being limited because of attitudes in the culture or practices of recent history? How did or do other peoples solve the problem? These ideas that are socialized into us often go unexamined. Why do we balk at eating squid or dogs? Up until about seventy-five years ago, it was common for men to marry women fifteen or twenty years younger than themselves. Now we consider that unusual and some people even consider it wrong, just as we consider older women marrying younger men unusual.

2. Focus your assumption identification on the crux or sticking point of the problem. You may be making an unnecessarily limiting assumption about something right at the point of blockage.

For example, let’s say your problem is to clean the mineralization off the water faucets in the bathrooms of your house. You have gone to a hardware store or home center and tried every cleaner in the housewares department but nothing has been satisfactory. You think, “I’ve used every household cleaner I can find.” Examine your assumptions: I’m assuming that household cleaners are found in the housewares department. Is that true or necessary? What about other kinds of cleaner that might be found in the automotive, plumbing, hardware, or garden department? Also, what about products not even described as cleaners but that might clean off the mineralization? The solution you finally come up with is to use an automotive chrome bumper cleaner or perhaps some household vinegar to clean off the mineralization and then to apply some car wax to the chrome to protect it from future build up. Your assumptions about store locations, product names and types and uses have all been challenged and found not necessary.

3. Look over your written statements of the problem and your lists of constraints and write out a list of the assumptions behind each item. In these three steps, you’ll have a three-part list:

A. General assumptions. These are the assumptions you make without thinking or realizing that you have made them. Some of them are necessary, but some may not be. Write out even the most obvious ones.

B. Assumptions at the crux. These assumptions are usually made consciously, but are not often examined critically to determine whether they are necessary or not. Again, write them out so that each one may be examined and tested individually.

C. Assumptions determining the constraints. These are the assumptions about cost, time, effort, size, results and so forth that you make in order to establish the boundaries of the solution. Most of them are desirable. Sometimes one or more of them will be made too hastily, though, so that they deserve reexamination as well as the other kinds.

An Example
Let’s say you are the manager of a factory that makes portable electric generators. Your product is largely bolted together at final assembly by workers using air wrenches. The wrenches, like those you hear screaming in auto repair shops, make a lot of noise, hurting the workers’ hearing and job satisfaction. Your problem is, “How can we reduce the noise made by these air wrenches?”

Note that as with most problem statements, the problem as stated implies certain solutions. If you simply accepted the problem as stated, you would probably think of some possible alternatives like these:

put silencers or mufflers on the wrenches
build a sound proof room for the wrench assembly
install lead curtains around the assembly area to soak up the noise
install a sound “canceler”
But instead of this, you decide to do some assumption articulation. Here are some of the assumptions being made:

1. Air wrenches are noisy.
2. We must use air wrenches to put the parts together.
3. People must use the air wrenches.
4. We must use wrenches.
5. The fastening must take place in this area or in this factory.
6. Bolts must be used to hold the pieces together.
7. The employees don’t like the noise.

As you think about these assumptions, some new ideas come to you:

1. Air wrenches are noisy. Are all air wrenches equally noisy? Can we buy a quieter brand? Is there a “silent air wrench” being sold?
2. We must use air wrenches to put the parts together. Why not use manual wrenches, or electric wrenches, or hydraulic wrenches?
3. People must use the air wrenches. Why not use robots? Can we use the wrenches less? Rotate employees so that each one uses the wrenches just a little each day.
4. We must use wrenches. Why not use other tools? Nut drivers?
5. The fastening must take place in this area of the factory. Why not move it outside? Subcontract it? Put it in a special soundproof room?
6. Bolts must be used to hold the pieces together. Why not rivets? Spot welding? Adhesive? Screws? Clamps? Mold some of the pieces together so they need not be bolted or fastened at all?
7. The employees don’t like the noise. Get employees who like noise? Who don’t hear it (like deaf people)? Give them ear muffs? Play loud music to mask the noise?

Note that ideas like robots, deaf employees, adhesive bonding and so on would not be suggested by the original form of the problem statement, which is based on several perhaps unnecessary assumptions. A little assumption articulation breaks our thinking out of these restraints and allows us to see some new possibilities.

Techniques for Approaching a Problem
Here are several ways to attack a problem, each way designed to clarify the problem, suggest alternatives, or break a fixation. You will want to experiment with the applicability of these for various situations.

Entry Points
An entry point is, as Edward de Bono has said, “the part of a problem or situation that is first attended to.” In our linear, traditional problem solving mindset, this usually means a particular point–usually the most obvious–on the front end of the problem. However, there is no reason that some other point cannot be chosen as an entry point, nor is there any reason that the problem cannot be approached from the middle or even the end. Let’s look at each of these.

1. Front end entry points. Most problems are attacked on the front end first, which is to say, by stating the problem. However, there is really more than one front end because a give problem can be attacked from any one of several angles. Too often we assume that the first front-end angle that comes to mind is the method of approach, the only way to attack the problem. But that is not so.

Example problem: How to keep rain off of you while you walk on the street.
Possible entry points:
1. Inadequacies of current umbrellas. (Suggests “improve the umbrella” as a problem direction.)
2. Irritation of having to carry an umbrella. (Suggests “develop easily portable umbrella.)
3. Let the government do it. (Suggests public works items like awnings, free taxis, underground corridors.)
4. Let the individual do it. (Why not just get wet? Why does getting wet matter? What are the problems? Do they really need to be solved?)
5. Walking. (Why walk? Why not ride? Conveyances?)
6. Street. (Why go out on the street in the first place? Why not stay at home? Keep out of the rain? Solve the problem that made you go onto the street in the first place. E. g. to get a video, why not TV or cable movie or read a book or make popcorn and talk about rainy days?)

Notice here that what seems to be just one problem actually has several possible entry points, and depending on the point chosen, entirely different solutions will result. Edward de Bono comments about the importance of choosing an entry point:

Usually the obvious entry point is chosen. . . . There is no way of telling which entry point is going to be best so one is usually content with the most obvious one. It is assumed that the choice of entry point does not matter since one will always arrive at the same conclusions. This is not so since the whole train of thought may be determined by the choice of entry point.
Example problem: ATC’s cause many injuries and deaths each year.
Possible entry points:
1. They tip over easily. (redesign them?)
2. They are not toys. (license users? require age minimums?)
3. Riders don’t know how to use them safely. (educate riders?)
4. Many head and spinal injuries result. (roll bars? seat belts?)
Problem: How to have secret conversations in the bugged embassy in Moscow. Possible entry points:
1. conversations can be heard (notes, sign language, special room)
2. diplomats must share information (disinformation?)
3. the whole building is bugged (leave building? erect internal room?)

2. Beginning at the end. When a particular solution state is clearly defined, a problem can often be more easily solved by starting with the solution and working backwards toward the problem, filling in the necessary steps along the way.

The classic example is the problem: Divide a triangle into three parts so that the parts can be put together to form a square. That’s very hard. But if you start from the solution end, with a square, it’s easy to divide it into three parts all of which form a triangle.

Example: How do you count the number of people in a stadium that’s over ninety percent full? Count the number of empty seats and subtract from the number of seats in the stadium. Easier than counting people.

Example: How do you improve your relationship with your parents when you’re not quite sure what’s wrong with it–what the problem is? Start at the end, with the solution. Envision how you want the relationship to be and work backwards toward a discovery of the problem.

Whenever the solution or goal state is clearer than the problem, then changing the entry point to the end may be the best approach. Start with the goal or solution and look for ways to work back to the problem.

3. Somewhere between the beginning and the end. After all, there’s no law that says you have to start at one end or the other. So why not start in the middle?

Ancient Greek epics typically start in medias res, in the middle of things, and later go on to fill out preceding and succeeding action. You can do this in problem solving. It’s, again, sort of the “ready, fire, aim” approach.

For example, say you want to put up a new building. Why not assume that the funding and planning have already been done and begin with the construction phase, which contractors to hire, etc. Then work in both directions–backward toward planning where to put the building and how to get the money, and forward toward arranging for tenants.

Note that you can really begin at any point on this alleged continuum, with location, tenants, architect, and work in both directions:
building type—architect—location—contractors—tenants

Movies are put together this way all the time. The “obvious” order is
idea—script—producer—actors—studio—filming

but many movies get actors first, then a producer, then a script, etc.

Beginning in the middle has some risks, but it’s especially good for getting things done quickly and for beginning to do something even when you’re not quite sure of either the problem or the solution. It’s the kind of thing that will sometimes get you labeled as rash and hasty and sometimes as brilliant and visionary.

Rival Hypotheses
A hypothesis is a proposed explanation for a collection of data. A rival hypothesis is an alternative explanation for the same sets of data, another way of explaining the same results or events. Often the hypothesis is a statement about causation: the data indicate that X caused Y or that B occurs when A is present. It is critically important to remember, however, that in the realm of hypothesis and explanation, the data do not speak for themselves; they must be interpreted. The act of interpretation involves many difficulties, including those of experimenter bias, the confusion between correlation and cause, and non-random sampling.

Dangers of Having only One Hypothesis
The danger of limiting ourselves to one hypothesis to explain a collection of phenomena is twofold.

1. Some evidence will be ignored. If we are focused on a single hypothesis, we will overlook as not relevant any information that does not bear on the truth or falsity of the hypothesis. However, such information might bear on the truth or falsity of some other hypothesis.

For example, if our hypothesis is that suspect X burglarized the Turner’s house, we will focus on evidence that helps to establish or disprove our theory. As a result, we will probably overlook the fact that the story told by the Turner’s son does not add up. That’s just an ignorable anomaly. If, on the other hand, one of our hypotheses is that the Turner’s son might have faked a burglary and stolen the missing items himself, then the difficulties in his story will not be overlooked.

2. We may become emotionally committed to our hypothesis. The idea of falling in love with a pet theory is not limited to problem solving, of course. Wherever it happens, the lover begins to search for and select out only the evidence that supports the hypothesis, ignoring or subconsciously filtering out information that argues against the pet.

For our example, here’s a story: An experimenter carefully conditioned a flea to jump out of a box when a bell was rung. Then he pulled off the first pair of the flea’s legs. The flea still jumped out of the box. So he pulled off the second pair of legs. The flea could still jump out. Finally, he pulled off the last pair of legs. This time, when the bell was rung, the flea didn’t jump our of the box. The experimenter concluded that his theory was correct: “When all the legs of a flea have been removed, it will no longer be able to hear.”

To avoid these two problems, then, we should attempt to generate as many rival hypotheses as possible for each set of data, and then test each of them against the known facts.

Rules for Generating and Testing Hypotheses
1. The hypothesis should account for all possibly relevant data. An explanation that covers only part of the data or that is in conflict with a major fact, is not a good explanation. Remember, though, that especially early on, all explanations will have problems and will fact some seemingly conflicting data. Facts are refined and clarified as better information becomes available. So don’t throw out all but “perfect” explanations; you won’t have any.

2. Simpler explanations are usually to be preferred over more complex explanations. This is the principle of Occam’s razor, discussed inHuman-Factor Phenomena in Problem Solving.

3. More probable explanations are usually to be preferred over less probable ones. Many things are possible; fewer things are probable. It is possible that ancient astronauts built the pyramids, but it is more probable that the Egyptians did.

4. The consequences following from the truth of the hypothesis must match the facts. If, for example, you hypothesize that a bomb destroyed an airplane and caused it to crash, you will expect to find bomb residue as a consequence of this hypothesis.

When you first read how facts match a theory, you might be tempted to think, “Why, yes, that must be it.” However, when you make the effort to research (or even take a few moments to generate on your own) a few rival hypotheses–alternative explanations–the original hypothesis becomes suddenly less persuasive. As with many other things in life, When you have a choice of only one, it seems to be the right choice; but when you have a choice of many, your taste improves. There is even a Biblical passage relevant to this issue: “The first to present his case seems right, till another comes forward and examines him” (Proverbs 18:17).

When you begin to examine a proposed explanation for some data, ask yourself, “What other variables are involved that might also account for the result?

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Try It Yourself
Rival Hypotheses. What rival hypotheses can you think of for each of these explanations?

1. Speed Kills? In 1973, when the national speed limit was 65 miles per hour, there were 55,000 automobile-related deaths. In 1974, when the speed limit was reduced to 55 mph, deaths declined 20 percent. In 1975, they declined 2 percent more. However, in 1976, as motorists began to ignore the speed limit and drive at 65 once again, deaths increased. The conclusion is clear: lower speed limits save lives.

2. Wedded Bliss? Many studies over long periods have established that married people are generally healthier than single (never married, widowed, divorced) people. Lung cancer, stroke, and coronary heart disease are all lower in married people. Married men live longer than men who do not marry. One researcher attributes these facts to the harmful consequences of loneliness. Are there any other possible explanations for these differences?

3. Coffee Coffin? A recent study has found that men who drink more than six cups of coffee per day have a much higher heart attack rate than those who drink fewer than six cups a day. Clearly, drinking coffee causes heart attacks. Or is there a rival hypothesis?

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Role Playing
Role playing consists of several techniques, having in common the use of the mind to imagine a different reality, to change what you have to what you want.

1. Mental Practice. Before attempting a solution or doing something–taking a test, driving to a new area, writing a paper, asking for a raise–practice the situation mentally.

For example, Abraham Lincoln imagined what he would do and say as president before he was ever elected. Dr. Charles Mayo of Mayo Clinic fame always mentally practiced his surgical operations before doing them–he would find a quiet spot and then go through the whole procedure in his mind: cutting, asking for instruments, examining, suturing. Many athletes rehearse their upcoming performances mentally to gain confidence and familiarity with the moment of performance.

Visualize the problem and your solution to it and you’ll be able to solve it or do it better. One woman imagined driving on the left side of the road, turning, passing, merging, etc. before taking a trip to England. When she finally got to England, she found that she could drive easily–it was already a familiar experience.

2. Becoming another person. The second form of role playing is to imagine that you are someone else–involved in either the solution or the problem.

A. Problem Person. Imagine that you are the litterbug, the reckless, drinking driver, or the short tempered, hard to live with friend. What makes you this way? What might improve you? What are the nuances of your personality?

B. The Solver. Imagine that you are an expert who can solve the problem with your special knowledge. What do you know and what do you do? Solutions take direction from past experience. They derive from what is already done or known. We go with the familiar and use what we have learned–or what we imagine we have learned or experienced.

For example, suppose you must build a canal. Imagine first that you are not a canal builder but a pipeline maker. How would he build the canal? (Perhaps by using reinforced half pipeline sections?) Now imagine that you are a tunnel maker. Now how would you solve the problem? (Perhaps by using an inverted tunnel?) Now imagine that you are a swimming pool builder. How would you solve it? (Perhaps by using steel rebar and spray-on gunite?)

3. Mental metamorphosis. In this kind of role playing, you change yourself into the problem thing–become a bearing, a helicopter, an electric current, a germ. Michael Faraday imagined that he was an atom under pressure and thereby developed his electromagnetic theory.

For example, suppose you want to find a solution for rusty and leaking gasoline tanks. Imagine all the attributes of the situation: the metal tank, its color, temperature, touch, the leak in it, the sound of the dribble of gasoline as is plops to the sandy soil under the tank. What does it feel like to be a tank in the sun, to feel your side leaking, to smell the wet sand/gasoline combination under you? What do you taste like? When the service man puts the wrench on your valve, how does it feel? Do your insides itch as they rust? What would help that? A coating? Does the gasoline running down your side bother you? What would soak that up or seal it off?

Modeling
A model is a representation or pattern of an idea or problem. That is, a model is a way to describe or present a problem in a way that aids in understanding or solving the problem. Models serve several purposes:

The Purpose of Modeling
1. To make an idea concrete. This is done by representing it pictorially or symbolically. We are very visually oriented creatures, and it is easy to bring about understanding or conceptualization through an image–much the way analogy works, only now you use a picture, drawing, map, boxes, circles. A drawing can show a relationship, connection, arrangement, hierarchy, and so forth much more quickly than words alone can.

Another use of representative modeling is to enhance creativity by converting an idea into something that can be experienced by the senses. “Okay, this salt shaker is our blocked plan, and these French fries are the people opposing the plan by holding up the rules–this napkin–in front of it. Well, what can we do? Lift the salt shaker, move it around, over, through, empty it.”

Many a problem solver has drawn on a napkin, arranged the food on his plate, scratched a stick in the sand, sketched a form of some sort, or even played with some children’s blocks.

2. To reveal possible relationships between ideas. Relationships of hierarchy, support, dependence, cause, effect, etc. can be revealed by constructing a visual model.

For example, what is the relationship between faith and reason? This can be shown by one block on top of another (a hierarchy), one circle inside another (one concept as part of the other), two blocks side by side, one each on a balance, and so on. Each model suggests a different relationship, each easy to remember.

A fact that needs special emphasis is that the model one uses for understanding will have a profound effect on perception and conceptualization. In fact, to a large extent, a model will determine your perception of an idea or problem and control your thinking about possibilities, relationships between parts, and so on. That’s why multiple models are often highly desirable: they allow a person to think of the same concept in several different ways without the unconscious controlling influence that a single model might have.

Another example: The saying, “Ready, fire, aim” seems funny and illogical to most people because they automatically assume a rifle or pistol or arrow model, and with such a model, the saying doesn’t make sense. These people are trapped by their own thought processes and automatic modeling. However, if we construct a different model–that of a machine gun, fire hose, laser beam, flame thrower, heat gun, fire extinguisher, blowtorch, hammer drill or whatever, then the saying makes great sense after all.

We have to be careful, then, how much we let our models control our thinking.

3. To simplify the complex to make it manageable or understandable. Almost all models are simplifications because reality is so complex. The whole economy, weather system, human personality, geological structure of the earth, air flow over airplane wings–all are too complex to be treated as is, so models are constructed that present simplifications that can be treated. Simplification is both benefit and danger, and when dealing with a model, one must always be sure not to forget that the model and reality might not match perfectly–and sometimes not well at all.

4. The main purpose of modeling, which often includes all of the above three purposes, is to present a problem in a way that allows us to understand it and solve it. That is, by seeing the problem in a different form or from a different angle, we can gain the insight necessary to find a solution. We take a problem and simplify it, make it visual, and provide a familiar pattern.

Types of Models
1. Categories. Models can be put into one of two categories, conceptual and structural. Of the types listed below, many of them can fall into either category depending on the use made of them.

A. Conceptual. Models used for concretizing or reifying an idea, used to aid conception or understanding. These can be ultimately symbolic or arbitrary, whatever is necessary or useful. Also models to aid memory or teaching and relationship models.

B. Structural. Physical models of physical structures–oil refineries, DNA helixes, buildings, architectural model, a new kind of record player or bicycle. A model is almost always constructed before a prototype is made for a product and models are usually made for all large construction projects.

2. Types. These are not fixed and exclusive boxes–they often overlap, as in visual symbolic.

A. Visual. Draw a picture of it. If the problem is or contains something physical, draw a picture of the real thing–the door, road, machine, bathroom, etc. If the problem is not physical, draw a symbolic picture of it, either with lines and boxes or by representing aspects of the problem as different items–like cars and roads representing information transfer in a company.

Visual models are among the most effective because we are highly visually oriented beings. Remember Confucius’ saying that is now a cliche but a true statement nonetheless: A picture is worth a thousand words.

B. Physical. The physical model takes the advantages of a visual model one step further by producing a three dimensional visual model. Again, you can use a real model or a symbolic one.

C. Mathematical. Many problems are best solved mathematically, by using calculations for speed, area, projected income, national unemployment. Thinking beyond three dimensions visually or four dimensions physically is very difficult. But with math, ten or fifteen dimensions are no problem. Ideas of speed, acceleration, and accelerating acceleration are often more understandable mathematically.

Example problem: Whom to hire. A mathematical model, such as a decision matrix, enables the thinker to quantify subjectivity and to be sure that all considerations (or criteria) are taken into account to the degree desired. The expected value calculation is another mathematical method of making a choice based on probable effects and preferred outcomes.

D. Metaphorical or Symbolic or Analogical. Remember what we said about metaphor and analogy, that the unfamiliar becomes understandable by comparing it to the familiar. That’s how this kind of modeling works. Both understanding and structure can be established for a problem by using a metaphor or symbol. Here are some examples useful kinds:

General Paradigms
1. System model. A system is a collection of interrelated elements working together to accomplish a common goal. The parts are input, processing, [storage], output, feedback, and control. Example systems are house heating system with thermostat, circulatory system.

Example problem: Interpersonal relationship improvement.
input: words, actions
processing: reactions
output: happiness, mutual support or discontent
feedback: communication (words actions)
control: change of processing (reactions and actions and output)

2. Design model. Design is planning with a concern for pattern and overall harmony. Component parts are identified and worked together into a whole. The key to design consideration is to plan so that the result to be an effective presentation. (For more details on design, see Chapter 7.)

Example problem: Vacation. Design a vacation
Sketch out parts–what should be included in a vacation? How will one part affect other parts? How does travel method affect sightseeing? Boat, rail, plane, care, walk, bike ride, etc.

3. Construction model. This model emphasizes sequential building. Part by part.

Example problem: Term paper. How can I build this paper? Foundation? Walls? Roof? or Beginning, ending, drawings, outline, other parts? Order of information?

4. Recipe model. This model emphasizes ingredients and proportions, with perhaps some consideration given to minor items that add “spice” or “flavor” to a project. The Japanese seem to use the recipe model in making many of their consumer products, from stereos to cars. Many cars include a toolkit, first aid kit, sometimes a trouble light–things that American manufacturers sometimes think of negatively as gimmicks or gadgets. The recipe model could be a list or formula for success. Great in advertising, products with features, certain kinds of fiction, etc.

Specific Metaphors:
1. Garden model. How is problem or solution like a garden? Vegetative, growing, expansive, fruitful, weedy, nurturant, bug infested, etc.

2. Machine model. How is problem like a machine? Parts working together, parts worn or broken, energy input or driving force, work output?

3. Symphony model. How like a symphony? Conductor? Harmony? Soloists? Percussion? What is the music they are playing? What orchestrates the interaction of the parts?

4. Human body model. How like a body? What makes it move? What is life energy? What are hands, feet, mouth, eyes, ears?

5. Vehicle model. Ship, plane, boat, car, train, blimp, bike, skateboard. What powers it? Who are passengers? Where going? What are its wheels?

Other metaphors useful for modeling are sculpting, movie making, an island, the ocean, a computer.

Using Criticism and Suggestion
Making use of the observations of critics to improve a plan or idea is a fairly obvious technique, but one that is not often used simply because most people don’t like criticism. Our ideas are our precious children and to be told that they are ugly or defective is painful and offensive.

However, it is possible to work around the ego sensitivity we have by renaming our criticism seeking into “suggestion seeking” and by viewing the procedure as a formal technique for exploiting the minds, experiences, and ideas of other people. What better way to get other viewpoints than to ask real, other people?

Basic Guidelines
Remember that in problem exploration it was suggested to talk over a problem with others to get insight into it. Well, now we come to the preliminary solution idea and do the same thing. Here are some suggestions:

1. Choose in advance a fixed number of people you will talk to, to reduce fear and make the process more formulaic (which will make it less ego damaging). Four to six is usually a good number.

2. Frame your request for criticism in a positive way, so that the criticizer will have to suggest improvements rather than just point out defects.

For example:
A. I have an idea to sell concentrated or dehydrated apple juice. Can you think of some ways to improve it?
B. I’m asking several of my most thoughtful friends how I can improve this idea for making concentrated fruit drinks. Can you think of anything?
C. I’m working on the problem of reducing shipping costs for drinks by concentrating or dehydrating them. I wonder if you could help me find a solution? Here’s what I’ve come up with so far. (This puts the other person in a solution mindset rather than a criticism mindset.)

3. Ask all kinds of people, not just people knowledgeable in the area. Ask children, even. Remember the value of mind stimulation, where an idea may not be directly useful but may suggest something else.

4. When you get more confidence, you can ask for an analysis of defects or inadequacies.

For Example:
A. What am I missing? What am I not thinking of? What am I not taking into account?
B. What don’t you like about this? What’s wrong with it? How would you have done it differently?

5. Use the dual method of asking for suggestions. There are two ways to operate the idea and suggestion technique.

A. Ask each person to improve the original plan or suggestion. Go to several people and propose the same plan and ask for input about it. This way you will get several different responses to the original.

B. After each suggestion, alter the idea to incorporate the suggestions and criticisms, and then present the new idea to the next person for suggestion and criticism. That way, the idea builds and improves with each criticism. The drawback is that certain other fundamental suggestions may be eliminated because the subsequent suggesters don’t see the original idea.

It is important for you as a creative thinker to see yourself as independent and separate from your ideas. Don’t get your ego so involved in an idea that you will be unwilling to alter it if you discover or are told about needed changes. And don’ be unwilling to abandon it if you discover a better idea. Keep a whole sackful of possibilities that can be rotated or combined to form the best solution, and put your pride in solving the problem, the result, not in the particular solution path you are currently thinking of.

Searching Techniques
Heuristic Methods
A heuristic is a guide, a rule of thumb, a learn as you go strategy, typified by trial and error. It involves choice, hunch, knowledge, and a lot of creativity. It’s the way most education works. However, no heuristic can guarantee a solution. A heuristic simply increases the probability of finding a solution. An example heuristic method follows.

1. Trial and error. The trial and error search involves the non use of directional information. That is, the search proceeds without any sense of choice or likelihood of one path over another. Trial and error can be made much more efficient if it is systematic rather than blind, that is, when a record of attempts and failures is kept so that the same path or solution is not tried more than once. So take good notes.

Algorithmic Methods
There is another kind of technique called an algorithm that can guarantee a solution. An algorithm is a list of set procedures, a recipe, a formula, or set of exact directions–computer programs and math formulas for finding volumes and areas are algorithms. There are a couple of common search algorithms:

1. The maze algorithm. This algorithm guarantees that you will be able to solve or walk through a maze. All you have to do is follow the same wall all the way through. In practical terms this means put your hand on the wall and keep it there as you walk through. Either hand and either wall.

2. The split-half method. This powerful technique is used for finding a problem or phenomenon along any linear system. It is used by electricians, plumbers, mechanics, electronics technicians and others to find trouble in equipment. (e.g. faulty doorbell, leak in pipe). The method involves going immediately to the halfway point in the linear system and checking to see if the problem or a symptom of the problem appears there. If it does, the problem is in the first half of the system. If it doesn’t, the problem appears in the second half. Next, the investigator goes to the half of the system where the problem is now know to occur and checks at its halfway point to see if the problem or symptom appears there. The answer eliminates another quarter of the system. Note that in just two steps, two checks, three quarters of the system has been eliminated from possibility. The halving continues until the problem is located. This is much faster than random checking or than by starting at one end of the linear system and proceeding toward the other end.

Example uses:

Someone is stealing oil from our transdesert pipeline; where?
Our packages are arriving from Germany all beat up; where are they being damaged?
Somewhere along the manufacturing line our product is getting dented on the corner; where is this occurring?
Somewhere between here and Sacramento the river is being polluted; where is this happening?
Somewhere in our spy network information is leaking to the Soviets; who is doing it?
Somewhere in the process of transmitting data from the factory floor to the main office, information is being lost or garbled. Where is this happening?
Note that many systems are or can be perceived as linear, whether the thing moving through them is water, paint, food, information, television sets, smog, whatever.

Other Techniques
Here are some general techniques for help in solving problems.

1. Public Solution. Post the problem on a bulletin board or circulate it in a newsletter, memo, or whatever written medium is in use in your organization or group. Make a note that suggestions and solutions are solicited and that ideas should be sent to you.

This technique causes public discussion of the problem at an intellectual rather than personal level. If your problem is employee absenteeism, poor quality parts, financial difficulty, or something similar, the public discussion will tend to focus on solutions rather than on blame attribution. If the problem does not derive from people difficulty, as in how to pack light bulbs more safely or how to hold books upright on partially filled library shelves, posting the problem can hook solutions that may have been applied to a similar problem elsewhere. And of course, the basic strategy behind posting a problem is that it gets several minds working on the problem, both independently and in discussion with others. People in the organization will talk about the problem in their idle moments.

During group problem solving discussions, posting a problem on the board is useful because it (1) stimulates interest and discussion in the problem, (2) makes people willing to take responsibility for the problems of others, and (3) develops problem solving attitudes in all members of the group.

Summary
State the Perceived Problem
Articulate Assumptions
Pick a point from start to finish to approach the problem 
Generate Alternative Hypothesis
Roleplay
Model the Problem
Get an outsiders opinion
Use Problem Solving Heuristics

Problem Solving Hints and Wisdom
1. Take time to examine and explore the problem thoroughly before setting out in search of a solution. Often, to understand the problem is to solve it.
2. Breaking the problem into smaller parts will often make solving it much easier. Solve each part separately.
3. The resources for problem solving are immense and ubiquitous.
4. You can always do something.
5. A problem is not a punishment; it is an opportunity to increase the happiness of the world, an opportunity to show how powerful you really are.
6. The formulation of a problem determines the range of choices: the questions you ask determine the answers you receive.
7. Be careful not to look for a solution until you understand the problem, and be careful not to select a solution until you have a whole range of choices.
8. The initial statement of a problem often reflects a preconceived solution.
9. A wide range of choices (ideas, possible solutions) allows you to choose the best from among many. A choice of one is not a choice.
10. People work to implement their own ideas and solutions much more energetically than they work to implement others’ ideas and solutions.
11. Remember the critical importance of acceptance in solving problems. A solution that is technologically brilliant but sociologically stupid is not a good solution.
12. When the goal state is clear but the present state is ambiguous, try working backwards.
13. Procrastinators finish last.
14. Denying a problem perpetuates it.
15. Solve the problem that really exists, not just the symptoms of a problem, not the problem you already have a solution for, not the problem you wish existed, and not the problem someone else thinks exists.
16. A maker follows a plan; a creator produces a plan.
17. Creativity is the construction of somethings new out of somethings old, through effort and imagination.

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