The Art of Problem-Solving

“If there exists a difficult problem you can’t solve, there is a simpler one you can solve”. — Polya’s Dictum.

Doing Math is like standing on the shoulders of giants, however you must see through its problems. It is these problems that make the subject mesmerizing. There is beauty in the problems of mathematics. This becomes clearer when you understand that there is always a solution to all problems.

Math is too technical, its the language of the gods, gods of some distant sola systems in the outer suburbs of very distant galaxy, these are what a huge class of people think about the subject. This is grossly unfortunate, but its not your fault, even the algebra textbook is always frowning (looks old) because it has these many problems. Truth is, Math isn’t all of these. And the Australian mathematician Terrence Tao would say, “one does not have to be a genius to do mathematics!”

Let’s get into the kitchen quickly and do some Math. How do you cook a dish of “Jollof beans”?

i. Boil beans in a pot keeping temperature regulated; wait for a while (you can read a book if you wish) and drain beans when sufficiently softened.

ii. Then transfer beans into a pot of heated oil, measuring a volume you can look up in cullinary recipe books.

iii. Add different spices and stir well enough until beans is cooked

iv. Don’t skip any step I must have missed

Ok think about this for a moment. How do you add 4 and 7? By breaking up 7 into 1’s, then 1 to 4, to 5, until we exhaust the 1,s. Simple!

The point is that, just like the kitchen example and with adding 4 and 7, Math becomes a lot easier if we know some algorithm, some step-by-step flow to approach problems with.

The Hungarian mathematician George Polya actually outdid his dictum above by writing a book titled “How To Solve It”, where he reduced the act of solving math (perhaps general class) problems to a few classical steps:

1) Understand the Problem

What is the structure of the problem? Is it of the form “Prove/Show that…”, “Evaluate/Simplify…”, “Find…”, which are the basic three categories. So knowing the problem’s shape gives the hint on what is required; whether to prove or give a counterexample. While naming and choosing variables, do well to use simple notations and connect your variables as much as possible — you would find it difficult to over-emphasize those connections on the long run.

2) Sketch A Plan

There are no definite rules as to solving problems, however intuition always shines a bright torch, especially when such problem is strange to you. Know a number of theorems related to the topic, and detect the constraints in the given problem. Link the knowns and the unknown. It fuels the process if you had solved a related problem previously. You can look up similar problems and their solutions to get a hint on what strategy would likely work; and once you solve a type of problem, solve so many of it. That way you would be quick to see it tip-toeing around if it ever comes back your way.

3) Execute the Plan

Strike the problem with the weapons of your planned strategy; sometimes the first strike does not crack the problem, in such case modify your plan slightly or grossly depending on how close you seem to be from the solution. Sometimes intuition could lead you somewhere very far, for example in solving the Basel problem, the legendary mathematician Leonhard Euler connected an infinite series to the square of the constant π; which is not obvious. Do not be deterred by the failure of a blow to break a problem at first.

4) Examine Your Solution

It does not solve the problem better to look back at the solution to the problem, but it does improve your intuition about solving other slightly related problems. You can search it if there is possibly other approaches you can take.

These steps still work, at least for amateur mathematicians, who aren’t trying to crack the Riemann Hypothesis just yet. At those levels of researching Math, you’d be more comfortable with these steps and would be prepared to start taking complex turns and twists when thinking Math.

I hope someone finds these steps useful. Keep solving problems. Cheers!