Problem Solving
and the Book
How to Solve It
by George Polya
Table of Contents
…
Background
George Polya [Bibliography: http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/polya-george.pdf accessed 9-12-2020] was a Hungarian mathematician who wrote the book How To Solve It [Bibliography: George Polya, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.] which has sold over a million copies and has been translated into 17 languages [Bibliography: https://math.berkeley.edu/~gmelvin/polya.pdf accessed 9-12-2020]. Polya provides a general approach for solving a wide range of problems, both mathematical and non-mathematical. Among his mathematical work he proved that a random walk returns to the original point in one or two dimensions, but not in three or more dimensions [Bibliography: https://arxiv.org/abs/1803.00811 accessed 9-12-2020].
The Four Steps
In his book Polya identifies four basic principles of general problem solving which can be represented by the 4 letters U-P-I-R. [Footnote: To help you remember this think of it as a single word mnemonic pronounced “You-per”.] Following are these four steps:
Step One: Understand the problem
Step Two: Make a Plan
Step Three: Implement the plan
Step Four: Revise and reflect, looking back on what you’ve done
Each of these four steps can be described in more detail as follows:
Understand the problem
What are you trying to find or show? Do you understand all the words used in stating the problem? What data is given? What limiting conditions must be worked around? Can you explain the question to someone else? Try to state the problem another way.
Make a Plan
Draw a picture or diagram. Look for a pattern. Make a list. Use a table. Smart guess-and-check, keeping track of what you’ve tried. Solve a smaller similar problem. Work backwards.
Implement the plan
Using the work you’ve done so far now solve the problem and state the solution in complete sentences.
Revise and reflect, looking back on what you’ve done
If your plan doesn’t work, discard it and choose another. Recheck the result and arguments used. Does the answer make sense? If not, recheck the method and the calculations. Was there another way to solve this? Can you see that your solution is right? Can you get the result in a different way? Can you use this for another problem?
Application
Let’s consider how we might apply this approach to solving a real-world problem of finding a good restaurant to go to with a friend you haven’t seen in some time.
1. Understand the problem
Consider any dietary constraints (Vegetarian? Gluten free? Drinks?). Think about how much you can afford for a meal like this, and what transportation is available to you (Public transit? Walk? Bike? Car?) how far away the restaurant is and when you need to get there. You probably want a place that is quiet enough to enable conversation.
2. Make a Plan
Consider how you can find answers to your constraints. One plan would be to use the website Yelp [Bibliography: Yelp.com accessed 9-12-2020] to narrow down the restaurant. Google maps [Bibliography: maps.google.com accessed 9-12-20] can help you compare options for how to get there. Choose the top 3 restaurants that are not too expensive, that are open for dinner, and are between where you and your friend live. Plan on sending these to your friend to see which they prefer.
3. Implement the plan
After the data is gathered, list the top choices and send them to your friend. Together choose the one you want and go together to that restaurant.
4. Revise and reflect, looking back on what you’ve done
Once you arrive, if it is too noisy, leave and instead go to the next one on your list. After you eat there make note of the experience so you can consider going back there again or not. Remember anything you have learned about restaurant review sites and maps to help you solve a similar in the future.
Conclusion
Becoming a better problem solver can help us not only in math classes, but in every-day thinking. The discipline of creating solutions using limited computer instructions is called programming. A step-by-step set of instructions is known as an algorithm. [Footnote: The word algorithm is derived from the Latinized name al-Khwarizmi, who was the 9th-century Arabic mathematician Muḥammad ibn Musa al-Khwarizmi. Besides having the word algorithm named after him, he also introduced the idea of “al-jabr” balancing, known today as Algebra.] A recipe can be thought of as an algorithm for cooking a particular dish. The process of programming algorithms can be thought of as a “sandbox” for sharpening our general thinking skills.
Bibliography