How To Solve It: A New Aspect of Mathematical Method

My book of this week is How to Solve It: A New Aspect of Mathematical Method by eminent mathematician George Polya. Polya was one of the most influential mathematicians of the 20th century. O’Connor and E F Robertson wrote a short biography of Polya, giving us a glimpse of this extraordinary scientist and teacher.

In the book Mathematical People: Profiles and Interviews, there is a quote from Polya: “I came very late to mathematics. … as I came to mathematics and learned something of it, I thought: Well it is so, I see, the proof seems to be conclusive, but how can people find such results? My difficulty in understanding mathematics: How was it discovered?” This might be one motivation behind Polya’s writing of a book like How to Solve It. I like this short review of the book:

How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be “reasoned” out – from building a bridge to winning a game of anagrams. Generations of readers have relished Polya’s deft – indeed, brilliant – instructions on stripping away irrelevancies and going straight to the heart of the problem.

How to Solve It was first published in 1945. While reading this book, I could not help questioning myself: how did I not come across and read it in my early teens? Damn me for the hefty loss of not having done so! It would have enlightened and saved me from many struggles of searching for ways to solve problems not only in maths, but also other subjects. As I had not read it back then, I cannot tell what a young student would think of this book. I am indeed curious to know your thoughts, if you had. As a trained computer scientist with a lot of passion for mathematics, many articles in this book brought me moments of epiphany. Polya’s words brought me the realisation of what we have learned to do through our scientific education and research, with the shortcoming that we have done so with less reflection and less articulation of what that set of thought processes are.  

How did I come across How to Solve It eventually? I owe this discovery to two gentlemen: Nuwan Jayasena and Charles Simonyi. Many thanks to Nuwan that I attended a Stanford Engineering Hero Lecture, given by Charles Simonyi in 2015. In this lecture, Charles noted how much he valued the book How to Solve It and recommended everyone to read it. How fascinating that it took one Stanford PhD (Nuwan) to invite me to a talk given by another Stanford PhD (Charles), in which I was connected with a book by a Stanford Professor (George Polya)! I am very grateful.  

You might start to wonder: woman, are you ever going to tell me what this book is about?

There are four parts: In the Classroom; How to Solve It: A Dialogue; Short Dictionary of Heuristics; Problems, Hints and Solutions. Here I focus on the “How to Solve It” list and some of the heuristics discussed in the book.

In the “How to Solve It” list, Polya distilled the problem solving processes to the following four steps. These suggestions and questions are quoted directly from the book.

  1. Understanding the problem
    • You have to understand the problem.
    • What is the unknown? What are the data? What is the condition?
    • Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
    • Draw a figure. Introduce suitable notation.
    • Separate the various parts of the condition. Can you write them down?
  2. Devising a plan
    • Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
    • Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
    • Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
    • Could you re-state the problem? Could you restate it still differently? Go back to definitions.
    • If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? 
    • Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary?
    • Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown?
    • Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?
    • Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
  3. Carrying out the plan
    • Carrying out your plan to get to the  solution: check each step.
    • Can you see clearly that the step is correct? Can you prove that it is correct?
  4. Looking back
    • Examine the solution obtained.
    • Can you check the result? Can you check the argument? Can you derive the result differently? Can you see it at a glance?
    • Can you use the result, or the method, for some other problem?

Are you excited and motivated enough to read the book yourself by these questions now? I love reading this book, if not for anything else other than the questions. Polya’s writing style of this book reassured me that I am not too crazy of asking many questions. But then, I never care at all whether others think less of my intellectual capability, because I would admit I am an idiot forthright. I am an idiot who is trying very hard to shed off each layer of stupidity every day through learning and working. Is it not wonderful to imagine that you are losing weight that way metaphorically, without sacrificing your awesome appetite for a giant neapolitan pizza?

A final point: although many examples in this book are maths related, the book is much more fundamental than prescribing recipes for solving maths problems. To me, many suggestions for investigation, thought processes, heuristics, planning and evaluation are very applicable for any domain, such as litigation, economics, social science and civil engineering.

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