Applying Polya’s principles to problem solving

In this post I will talk a little about  problem solving. To be more precise, I am gonna apply Polya’s four step method (adding two extra steps of my own but which are implied by Polya as well) to a mathematical problem, explaining each step and why it’s important.

First of all, I would like to emphasize that if you can’t solve a particular problem, that may not be because you are not smart enough and it would better to let it to a more able person. You probably are as smart as needed but lack sufficient motivation, a good toolbox of mathematical ideas and the idea that even if you do a couple of mistakes,  it’s the big picture that counts. You can learn more about this at this awesome post by professor Tao.

From a quick look around, I have seen that there are numerous sites that mention the four step method but not that many that actually apply it to a problem. So we are gonna do just that! We will try to hit two birds with one stone by tackling a problem from undergraduate real analysis. But let’s begin with the method. All of the steps are equally important.

I should state here that following the steps that I propose won’t magically solve every problem. But by using them every time, you will develop the mindset of a good problem solver and gradually build your skills.

First step : Know as much as you can of the theory.

That includes having a suitable reference close to you or, even better, try to understand it before you attack any problems. Most of the time, you will have to combine different ideas from the things you know in order to come up with a plan. It doesn’t make any sense to try to solve a problem in mathematics if we don’t have a sufficient knowledge of as many things as we can. For example, let’s say that the problem is  : 

Let f be a continuous function in \mathbb{R}, which decreases outside of an interval I as fast as \frac{1}{e^{|x|}}\ . Show that \int_{-\infty}^{+\infty}f(x)dx exists.

So, we know how this function is like outside of the interval but what about on the inside? Is the integral on I bounded or is it not? A student that has prepared by reading a little bit of his calculus before seeing this exercise should have no problem finding the core idea. But for us to really see the power of the four step method, we need a better problem. So let’s us suppose that we studied all that we could or that we have a book or Wikipedia close to us and let’s try to solve the following problem.

Let f be a continuous function on an interval [a,b], f(x)\geq 0 for all x in that interval and \int_{a}^{b} f(x)dx=0. Prove that then f(x)=0 for all x in [a,b].

OK! We have a problem, so let’s start.

Second step : Know your enemy!

Start by trying to see what is given by the problem and what you need to prove or find. Make absolutely sure that you understand all the statements and if you don’t, take a pause, open a book and revise. Don’t try to think of an idea yet, unless you have one that you are very sure about. It really does pay off to know what exactly you are trying to prove and what is given by the problem. In this case, I will color them accordingly. 

Let f be a continuous function on an interval [a,b], f(x)\geq 0 for all x in that interval and \int_{a}^{b} f(x)dx=0. Prove that then f(x)=0 for all x in [a,b].

We have our hypothesis in green. Take a good long look at it. Your first reaction should be to bring in your mind statements of it that you can un-wrap. You may say “Hey, f is a continuous function. Let’s keep that in mind cause I know a couple of theorems about them”. There is no reason to try to recall any of them right now, just keep in mind that this is a key property. That means, that this property gives us a bunch of nice theorems that we can use. Keep moving to the rest of the hypothesis.

 f(x)\geq 0 for all x in that interval. This is pretty straightforward. Our function doesn’t get negative anywhere on the interval given.

\int_{a}^{b} f(x)dx=0. This could also be a key property. We do know (or can find) a couple of theorems about integrals too. Again, there is no need to start brainstorming right now. We just want to get acquainted with everything, making sure that we don’t miss a vital piece of information. Lastly, here comes what we want to prove.

 Prove that then f(x)=0 for all x in [a,b].

OK. So, now we have read and made sure that we understand every part of the statement. Let’s go to the next step.

Step 3 : Get that idea & devise a plan

I am sure you will agree that this is the hard part of the job. But if you know the general theory, you are already a step closer to the solution. Let’s see how we can get you closer. Here’s a bit of general advice that you can use in every problem. In every one of the following suggestions, you should always try to connect what you know and what you hope to prove/find.

Write it down in suitable notation : Remember the old “How many apples do I have at the end of the day?” problems. They got so easy because we humans are good when we work with the right notation. So try to write down the problem in the best way you can, if it’s not already stated that way.

Visualize the problem : Visualizing your problem in your mind, in a piece of  paper or with the aid of software can help a lot towards finding a solution. Connections between the different parts of the hypothesis will come easier this way, so in every case where some kind of visualization is possible, do it.  Let’s see how can we visualize our problem. The hypothesis states that we have a continuous and nowhere negative function on an interval. It doesn’t really matter which function we will draw as long as it satisfies those two requirements. We can also choose whichever interval we like as long as our function continues to behave. Let’s choose f(x)=\sin(x) and [0,\pi].

OK. So we have a continuous function which is nowhere negative over our interval. We are just following the statements in the hypothesis and constructing the function to see what will happen.

Ask yourself (possibly dumb) questions : Remember that some of the greater problems in mathematics came from seemingly very basic questions. Your questions may appear dumb to you but they could actually be very deep, so don’t be afraid to ask them and then to try to answer them. For example, we visualized our function. Have we used all our hypothesis? No. We haven’t use the fact that the integral has to be zero over the interval. What does that mean? Does that integral has some connection with the graph of the function? What functions do we know that have such an integral? Why do they have it like that? Can our function be one of those? Why or why not? Asking and then answering those questions will surely ignite an idea in your head.

If the problem seems unapproachable, change it : But not a lot! What this basically means is that this is your problem and your final goal should be to solve it and learn something that you didn’t know before. If that means that you have to solve an easier problem first, so be it. Just keep in mind that the easier problem should have a connection with your first one or else you are just losing your time. For example, here we want to prove something but we can’t see the idea yet. Why should a function which is continuous and non-negative, with and integral which is zero, be equal to zero everywhere? Constant functions g(x)=c, c\geq 0 could also fit the statements in the hypothesis. What would it mean if they had zero as an integral? Can we see how they relate to a continuous function?

Think what happens if you surrender the game : That means that you should take some of the assumptions and turn them on their heads. Some combinations will not lead to anything useful. For example, what if my function is not continuous but is non-negative and has an integral of zero?  Does that mean that it’s zero everywhere on the interval? What if it’s negative at some points?

What if my function is continuous and has an integral of zero but there is a point that f(x)>0? What would that mean on the graph? Could my integral be zero now?

You got an idea? OK. Try now to think what you can do to make it work. Do you need to add something new to the problem? Will this idea get you close to your solution or will you need something more? Remember that as long as you move to the right direction, every step is important.

Step 4 : You found the plan! Carry it out now.

So, after all this tampering with the problem, you got an idea which you think that should lead to a solution. Not only that but you also have a general idea how to make it work. Proceed then to carry out your plan very carefully, using the correct notation. Check each step using your logic and the facts that you know are true. Do not rely on your intuition for this!

Did you hit a wall? That’s perfectly fine, you just have to see how big is the damage.

Keep asking yourself dumb questions : Did the argument seem to be work but now you are stuck with another problem? Hopefully, it will be smaller, so attack it in the same way as before, keeping in mind what new information you have! You are one step closer!

Did you follow your idea but it didn’t work? Again, this is new information. Ask yourself again. Did I do a mistake somewhere? Did I forgot to use a part of the hypothesis? Can I change something to make it work? Can I use some other part of the hypothesis in a different way?

If the answer is no to all those questions, then just note why your argument didn’t work and return to the previous step for another idea (if you don’t already have one). Do not fall into the “my idea must work” hole! It’s pretty easy to get passionate with an idea we think should work when we already found out that there are good reasons that it shouldn’t. Again, note what went wrong and then try with another one.

With a little luck and  patience, you will get that idea and finally solve it! Congratulations!

Step 5 : Where are you going? It’s not over yet! 

Now this is a very important step. Just because you solved the problem that doesn’t mean that it’s over!  Every time you solve a problem, you have an amazing opportunity to become a better problem solver/mathematician. Look back at your solution! Check your steps, see your train of thought. Try to think of the whole argument as whole. Is there something here that you can re-use in another problem? Can you use the result of your proof in another problem? Is there a better, more natural proof? Perhaps we can change the hypothesis a little bit, make it weaker and see if the result still holds.

In truth, every interesting problem can lead to some that are even more interesting. Don’t be afraid to give it time!

And the final step is :

Step 6 : Solve other problems. 

Practice makes perfect and that’s also true for mathematics.

I hope this post helps.

Until next time, when I will comment on the awesomeness of the fourier transform.

“Wait, you didn’t tell us the solution of the problem!”

I leave it for practice. 😀


4 thoughts on “Applying Polya’s principles to problem solving

  1. i find more merit in finding a way than applying principles, learning how to find your way can be an unmatchable experience, focus on experience rather than focusing on knowledge, intuisionism,… no doubt i m french. but i ve been through both points of view

    1. I agree but sometimes you have to walk before you can fly and you can’t really rely only on experience, because there is always a problem where standard methods break down. That’s part of the joy of science. And don’t forget that this advice is generic and it totally is about finding your own way! I am just proposing to experiment with a problem until you come up with the right idea.

  2. Experience is built, we can loose it though, about the fact that standard methods dont work in special cases i think thats exactly why we should not rely on knowledge but come on, we are Ms students, we should not ignore knowledge or we’ re doomed…

    i meant that randomness has a part of determinism, if that word exist on the previous post, but we all understood what I meant.

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