## The Joy and Art of Problem Solving

24 09 2009

Since……

Problems are inevitable and unavoidable.

They are the means by which we grow.  They are not necessarily “bad.”

There is no such thing as a problem without a gift in it.

Problem solving is one of the critical and central activities in one’s life.

Problems come in all shapes, sizes, varieties, and levels of difficulty.

Problems grow more complex each year.

Problem solving can be easier, more effective, and more fun if you have a flexible system for solving problems.

There is no substitute for experience.  If you want to become a better problem solver, you must practice, practice, practice.  Hence, the more problem solving you do, the better problem solver you become.

Some tools for solving problems…..

PROBLEM SOLVING:

A Student’s Guide

Rule 1

If at all possible, avoid reading the problem.  Reading the problem only consumes time and causes confusion.

Rule 2

Extract the numbers from the problem in the order in which they appear. Pay no attention for numbers written in words.

Rule 3

If rule 2 yields three or more numbers, the best bet for getting the answer is adding them together.

Rule 4

If there are only two numbers which are approximately the same size, then subtraction should give the best results.

Rule 5

If there are only two numbers in the problem and one is much smaller than the other, then divide if it goes evenly-otherwise, multiply.

Rule 6

If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.

Rule 7

Never, never spend too much time solving problems.

This set of rules will get you through even the longest assignment in the minimum time with little or no thinking.

## Tools That May Really Help

Problem Solving Tools You May Use

1. Rephrasing:

Often a problem seems complex or hard to understand simply because the words used are complicated, vague, or confusing.  By rephrasing the problem in your own words, you can get it organized in your mind.  Put the problem in your own words until you feel comfortable with your understanding of the problem.

Try stating the goal in your own words and as completely as you possibly can.

2. Possibility listing:

One of the easiest and most effective ways to get control of a confused situation is simply to itemize the variables and possibilities involved.  This involves making a list of the key factors involved. In this case the further analysis of the puzzle can be transformed into a list of factors that make the puzzle a problem.

Try listing the variables and factors of the problem.

3. Identify sub goals:

When a problem is complex, breaking it down into sub problems and solving each part is helpful.  By analyzing the problem carefully and not being distracted by the first thing that comes to mind, you may be able to discover the one key factor that lies at the heart.

Try simplifying the problem or the puzzle by breaking it down into sub-problems and then solving the parts.

4. Trial and error:

This is the weakest and often the most inefficient method.  It is randomly trying one possibility, then another, and then another. This method is also called guess and check. The correct solution is discovered by chance. This method is testing all the possibilities at random. (It is very probable you will use other methods instead of making a completely exhaustive search)

Try guessing and checking your solution

5.     Estimate, predict or project

Get an idea what the solution would be close to. Predict the range of where the answer might be.

Try estimating what the answer would be close to

6.  Best first analysis:

This searching strategy involves testing the most probable or most desirable (or promising) possibility first. This method can also be used on sub goals.  If the first method attempted fails to produce a solution the second best choice is tried.

Try the most desirable choice first.

7. Worst first analysis:

This searching strategy involves testing the least probable or desirable (or promising) possibility first. This method can also be used on sub goals.  If the first method attempted fails to produce a solution the second least desirable choice is tried.

Try the least desirable choice first.

8. Process of Elimination:

This method is organizing the possibilities by eliminating what does not work. This process may be used to solve sub goals and categorizing trial and error testing.

Try eliminating the possibilities that do not work.

9. Jump the Track:

Often problem solvers get stuck in a mental rut and do the same process over and over.  Stopping to reconsider the whole course of your attack on the problem may help.  Start again with a completely different approach or a different point of view. Enlarge the range of options to include unusual ones.

Try a totally different approach.

10. Look for patterns:

By examining the puzzle carefully, a pattern for arranging the pieces or in the solution may be observed. This may be patterns in shapes, color, size, process of steps or a hidden code.

Try looking for a hidden pattern.

11. Draw or use a diagram, table, or model:

Problems are often approached by sketching out the process on paper.  Often Athinking with a pencil@ helps clarify the thinking process.

Try looking using a pencil to sketch or keep track of your thinking process.

12. Work backwards:

When the goal is clear, you can begin there and work backwards.  Taking a completed puzzle apart piece by piece, or working a maze backwards or completing describing the finished puzzle may help in the process.

Try working backwards by understanding what the solved problem must look like.

13. Simplify

Do a simpler problem of the same kind to understand the method.  Apply that method to the present problem.

Try doing a simpler problem of the same kind and apply that method.

14. Logic

When there are steps that depend on each other, decide which step goes first. After that, decide the steps that follow in a reasonable order.  Discover how the steps fit together with phrases such as: If I do this, then this will happen.

Try breaking the steps of problem into a reasonable order.

15. Act it Out

Often it helps to play act the problem by demonstrating the situation physically.

Try play acting the problem by demonstrating the situation.

16.  Create an equation

Practice some algebra by using letters as variables to represent unknown quantities. Solving the equation leads to the solution of the problem

Try using algebra as a mathematical “shortcut”.