Lecture 10

Recreational Problem Solving

Here's a tip for you, young people: problem solving is everything!

Your whole life will depend on it - whether it be personal affairs, professional career, or any other thing you might do. Your success will be determined by how much you are able to identify, handle and solve problems.

Well, truth is, not all problems are mathematical in nature. In fact, different kinds of problems require different kinds of solutions. But one thing is certain: in some way or another, we need to solve them. Einstein is known to have once said, "It's not that I'm so smart; it's just that I stay with problems longer."

In this lecture we shall work on a general principle of solving problems - especially mathematical ones - in order to develop an approach. Then, we shall apply this approach to solve recreational problems and puzzles.

Polya's Four Step Approach

George Pólya was a 20th century Hungarian mathematics professor who mainly worked on discrete mathematics, numerical analysis, and probability. He wrote dozens of books and published even more highly cited articles. His contributions in mathematics are still relevant today.

Aside from being a noteworthy mathematician, Pólya is also remembered as a fine mathematics educator. He taught at ETH Zurich (Switzerland) and at Stanford University (California, USA). Perhaps, what math students should be thankful of him for is for organizing what is now called Polya's Four-Step Approach to Problem Solving.

This method proposes a very systematic way of handling a problem. Although different problems require different solutions, Polya's approach puts a system to the process.

According to Pólya, problems (both mathematical and non-mathematical) can be solved efficiently if one follows the following steps:

Understand the problem
Make a plan
Execute the plan
Look back and reflect

Even experienced mathematicians who find themselves lost in a problem, would only need to be reminded of Pólya's approach and that fixes everything to put them back on the right track. Note however that this approach cannot stand alone. One would still need an understanding of the context of the problem.

Understanding (Preparation)

"The first step to solving a problem is recognizing there is one," says Will McAvoy from HBO's The NewsRoom (click here for the clip). The first challenge for anyone encountering a problem is identifying what the problem really is.

Might be hard to believe but many people fail in this very first step. Car owners spend so much money bringing their broken vehicles to mechanics who seem to fix the problem only to find out days later the same problem occurring. Parents go to different heights to fix their children's declining scholastic performance from hiring tutors to scolding the child but still the child continues to fail. Managers do everything to save their companies but still bankruptcy is inevitable. All these may have been caused by one reason - failing to specifically identify what the problem is.

Understanding the problem is the step which tells us to identify what the problem really is. What is needed to be found? What is being asked?

This might be a little easier for mathematical problems since math problems are very specific in what they ask. With a little practice, students can easily identify what they need to solve.

Planning (Thinking Time)

Once the problem is identified, a plan on how to solve the problem has to be developed. Planning includes identifying what resources are available or accessible or what are given. It also includes what method to use. Since there might be a number of methods available, one may choose the most appropriate one, the easiest or fastest one, or the one he is an expert of.

For example, if a car is broken and the owner or the mechanic points out that a piston is malfunctioning, then the next step would be identifying what are available to them such as tools, skills, or a spare part. Then, they decide if they could fix it now or should they call another mechanic or bring the vehicle to another shop.

In mathematics, planning involves identifying what values are given and deciding how to solve the problem. Choices include graphing or illustrating, using a formula, application of theorems, using calculators, or a combination of some of these, and others.

Executing (Insight)

When things are set, we are now ready to execute the plan. This is the part that requires the most effort. Here we apply the technical skills we learned from our math classes and any other applicable strategy.

This step is the reason why we worked the last 10 years of our lives learning about mathematics -- so that we could apply all of them here in solving problems! Now, problem solving is not just about knowledge or skills, it's also about organization, accuracy, carefulness, patience, and many other scientific virtues.

On a final note in execution, when the plan does not work, just go back to Step 2 and try again.

Looking Back (Verification)

References

[1] OPEPP (2022). Tool for the Field: Polya’s Problem-Solving Method. From OPEPP.org [Click here]
[2] Veritasium (22 May 2021). Math's Fundamental Flaw [Video]. YouTube [Click here.]
[3] Stefanowicz, A. (2014). Proofs and Mathematical Reasoning. University of Birmingham. [Click here]

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