Problem Solving

Everyone must have felt at least once in his or her life how wonderful it would be if we could solve a problem at hand preferably without much difficulty or even with some difficulties. Unfortunately the problem solving is an art at this point and there are no universal approaches one can take to solving problems. Basically one must explore possible avenues to a solution one by one until one comes across a right path to a solution. Thus generally speaking, there is guessing and hence an element of luck involved in problem solving. However, in general, as one gains experience in solving problems, one develops one's own techniques and strategies, though they are often intangible. Thus the guessing is not an arbitrary guessing but an educated one.

In this chapter we are going to learn a framework for problem solving and get a glimpse of strategies that are often used by experts. They are based on the work of Polya. For further study,

The following four phases can be identified in the process of solving problems:

(1) Understanding the problem (see below)

(2) Making a plan of solution (see below)

(3) Carrying out the plan

(4) Looking back i.e. verifying

Each of the first two phases is going to be explained below a little more in detail. Phases (3) and (4) should be self-explanatory.

Needless to say that if you do not understand the problem you can never solve it. It is also often true that if you really understand the problem, you can see a solution.

Below are some of the things that can help us understand a problem.

(1) Extract the principal parts of the problem.

The principal parts are:

For "find" type problems, such as "find the principal and return for a given investment", UNKNOWNS, DATA and CONDITIONS, and

for "proof" type problems HYPOTHESIS and CONCLUSION.

For examples that illustrate these, see

Be careful about hidden assumptions, data and conditions.

(2) Consult definitions for unfamiliar (often even familiar) terminologies.

(3) Construct one or two simple example to illustrate what the problem says.

**2. Devising a Solution Plan**

**Where to start ? **

Start with the consideration of the principal parts: unknowns, data, and conditions
for "find" problems, and hypothesis, and conclusion for "prove" problems.

To see examples that illustrate these

**What can I do ?**

Once you identify the principal parts and understand them, the next thing you can do is
to consider the problem from various angles and
seek contacts with previously
acquired knowledge. The first thing you should do is to try to find facts that are related
to the problem at hand. Relevant
facts usually involve words that are the same as or similar to those in the given problem.
It is also a good idea to try to recall previously solved similar problems.
See

To see examples that illustrate these

**What should I look for ?**

Look for a helpful idea that shows you the way to the end. Even an incomplete idea should be considered. Go along with it to a new situation, and repeat this process.

There is no single method that works for all problems. However, there is
a wealth of
heuristics we can try. Following are some of the often used heuristics.

Add your own heuristics as you gain experience.

(1) Have I **seen it before ?**

That is, do I know similar or related problems ? Similar/related problems are ones with
the same or a similar unknown or unknown may be different but the settings are the same or similar.
See

(2) Do a little **analysis** on relationships among data, conditions and unknowns,
or between hypothesis and conclusion.

(3) What facts do I know **related** to the problem on hand ?

These are facts on the subjects appearing in the problem. They often involve the same or similar
words.
An example can be found

It is very important that we know inference rules.

(4) **Definitions:** Make sure that you know the meaning of technical terms. This is obviously crucial
to problem solving at any level. But especially at this level,
if you know their meaning and understand the concepts, you can see a solution
to most of the problems without much difficulty. See for example

(5) Compose a **wish list** of intermediate goals and try to reach them.

(6) Have you used all the **conditions/hypotheses** ? : When you are looking for paths
to a solution or trying to verify your solution, it is often a good idea to check
whether or not you have used all the data/hypotheses. If you haven't, something is
often amiss. See

(7) **Divide into cases:** Sometimes if you divide your problem into a number of
separate cases based on a property of objects appearing in the problem, it simplifies
the problem and clear your mind. For example if the problem concerns integers, then
you may want to divide it into two cases: one for even numbers and the other
for odd numbers as, for example, you can see in

(8) **Proof by contradiction:** If you make an assumption, and that assumption
produces a statement that does not make sense, then you must conclude that
your assumption is wrong. For example, suppose that your car does not start.
A number of things could be wrong. Let us assume for simplicity's sake that either
the battery is dead or the starter is defective. So you first assume that the battery is dead
and try to jump start it. If it doesn't start,
you have a situation that does not
make sense under your assumption of dead battery.
That is, a good battery should start the car but it doesn't. So you conclude that
your assumption is wrong. That is the battery is not the cause.
Proof by contradiction follows that logic.

In this method we first assume that the assertion to be proven is not true
and try to draw a contradiction i.e. something that is always false. If we produce
a contradiction, then our assumption is wrong and therefore the assertion we are trying
to prove is true.

When you are stuck trying to justify some assertion, proof by contradiction is always
a good thing to try.

(9) **Transform/Restate** the problem, then try (1) - (3) above.

(10) **Working backward:** In this approach, we start from what is required, such as
conclusion or final (desired) form of an equation etc., and assume what is sought has been found. Then we ask from what
antecedent the desired result could be derived. If the antecedent is found, then
we ask from what antecedent that antecedent could be obtained. ... We repeat this
process until either the data/hypotheses are reached or some easy to solve problem
is reached.

(11) **Simplify** the problem if possible. Take advantage of symmetries which often exist.

Keep in mind that your first try may not work. But don't get discouraged.
If one approach doesn't work, try another. You have to keep trying different approaches,
different ideas. As you gain experience, your problem solving skills improve and you tend
to find the right approach sooner.

Let us now look at some examples to illustrate the topics discussed above.

**Example 1**

This is an example in which you can find a solution once you analyze and understand
the unknowns and data.

**Problem:** A survey of TV viewers shows the following results:

To the question "Do you watch comedies ?", 352 answered "Yes".,

To the question "Do you watch sports ?", 277 answered "Yes", and

To the question "Do you watch both comedies and sports ?", 129 answered "Yes".

Given these data, find, among people who watch at least one of comedies
and sports,
percentages of people who watch at least one of comedies
and sports watch only comedies, only sports, and both comedies and sports.

Let us try to solve this problem following the

**Understanding the Problem:** This is a "find" type problem. So we try to identify
unknowns, data and conditions.

The **unknowns** are the percentage of people who watch only comedies,
the percentage of people who watch only sports, and
the percentage of people who watch both comedies and sports.

The **data** are the three numbers: 352, 277 and 129, representing the number
of people
who watch comedies, sports, and both comedies and sports, respectively.
Note that 352 includes people who watch both
comedies and sports as well as people who watch only comedies.
Similarly for 277.

The **conditions** are not explicitly given in the problem statement. But
one can see that the percentages must add up to 100, and they must be nonnegative.

**Devising a Solution Plan:** Here we first examine the principal parts in
detail.

First let us consider the unknowns in more detail. To calculate the percentage
of the people who watch only comedies, for example, we need the number of people
who watch at least one of comedies and sports, and the number of people who watch only
comedies. Thus actually two unknowns are involved in each of the required percentages,
and the real unknowns are the number of people in each of the categories,
and the number of people
who watch at least one of comedies and sports.

Next let us look at the data. First the number 352 is the number of people who watch
comedies. But that is not necessarily that of the people who watch only comedies.
It includes that and the number of people who watch both comedies and sports.
Similarly for the second number 277.

Let us use symbols to represent each of the unknowns: Let C represent the number
of people who watch only comedies, S that of the people who watch only sports, and T
that of the people who watch at least one of those programs.

Then we have the following relationships among the unknowns:

*C* + *129* = 352

*S* + *129* = 277

*C* + *S* + *129* = T

From these equations we can easily obtain ** C = 223,
S = 148**, and

Thus the required percentages are

All we had to do to solve this problem is to analyze relationships between the data and the unknowns, that is, nothing much beyond "understanding the problem".

This is a problem which you can solve using similar known results.

Again let us try to solve this problem following the

The

Before proceeding to the next phase, let us make sure that we understand the terminologies.

First a rectangular parallelepiped is a box with rectangular faces like a cube except that the faces are not necessarily a square but a rectangle as shown below.

Next a diagonal of a rectangular parallelepiped is the line that connects its two vertices (corner points) that are not on the same plane. It is shown in the figure below.

One of the facts that immediately comes to our mind in this problem is Pythagoras' theorem. It has to do with right triangles and is shown below.

Let us try to see whether or not this theorem helps. To use this theorem, we need a right triangle involving a diagonal of a parallelepiped. As we can see below, there is a right triangle with a diagonal x as its hypotenuse.

However, the triangle here involves two unknowns:

Applying Pythagoras' theorem again, we can obtain the value of

Thus

is obtained from the second triangle, and

is derived from the first triangle.

From these two equations, we can find that

This is a proof type problem and

The

The hypothesis is straightforward. In the conclusion, "rational root" means a root, that is, the value of

Following the "proof by contradiction", let us assume that the conclusion is false, that is the equation

Let us try to derive a contradiction from this.

First let us make this equation simpler, that is, let us get rid of fractions.

Since

Since

Let us first consider the case when

Then

Next let us consider the case when

By an argument similar to the previous case, we can see that

If

Thus by assuming that the conclusion is false, we have arrived at a contradiction, that is

This is another proof type problem and

The

First by multiplying out the left hand side of the inequality,

Hence if

Next, to see what we can try, note that we have not used the hypothesis yet, and see if it can help here.

It is well known that the sum of two sides of a triangle is greater than the third side.

Hence

From these we can obtain

By adding these three inequalities, we get

Hence

Hence

Hence

This is a find type problem and

The problem is to obtain 6 quarts of water in the 9 quart pail using 4 quart and 9 quart pails as measures. You can fill either pail from the water source or from the other pail, and you can empty the pails any time.

Our solution to the original problem is obtained by traversing this process backward to the desired state.

Let us denote the 9 quart pail by

In this problem, the desired state is to have 6 qts in

Thus in the first step of "working backward", we ask how we could get to the desired state with one operation.

As one can easily see if we could dump 3 qts from 9 qts in

In the second step, the question we ask is how to get 1 qt in

It does not look easy to get 1 qt in

In the third step, the question we ask is how to get 1 qt in

This is relatively easy to accomplish because all you have to do is to get rid of 8 qts from a full

Since this state can be easily reached (all you have to do to get to this state is to fill

Thus first we fill up

**Reference**

**Polya:** G. Polya, How to Solve It, *A New Aspect of Mathematical Method*,
Second Ed., Princeton University Press, Princeton, NJ, 1985 -- in ODU Library

**Larson:** L. C. Larson, Problem-Solving Through Problems, Springer-Verlag,
New York, NY, 1983 -- in ODU Library.