George Pólya: How to Solve It

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Ya’akov

Joined Jan 27, 2019
8,505
I was reminded of a very important little book by George Pólya, a Hungarian mathematician who devised a set of heuristics for problem solving. Most people who have studied math at the university level will have encountered this book. I came across it as I was working on my own heuristics to explain problem solving to university students who worked for me.

Pólya is focused on mathematical problems but his method strongly overlapped with mine and leaving out the maths-specific content it is entirely applicable to any engineering problem.

He offers four steps:

1. Understand the Problem
2. Devise a Plan
3. Carry out the Plan
4. Review/Extend

For each step, he offers techniques. Even the Wikipedia article has enough detail to the useful. Obviously his book is even more so.

I learned that people often simply don't know how to go about solving a problem. Having a system is very helpful. Most successful engineers will automatically use some form or the Pólya system, so you will probably recognize your own methods, and maybe learn some new ones.

What heuristics have you found useful in problem solving? What tricks do you use when things bog down and progress isn't being made?

SamR

Joined Mar 19, 2019
4,911
Most people can't get past step 1. Understand (and be able to describe) the problem... Simply stating "the car won't run" is not enough to get to step 2. So, jumping to step 3 with "dispose of it" is not a solution...

Thread Starter

Ya’akov

Joined Jan 27, 2019
8,505
Most people can't get past step 1. Understand (and be able to describe) the problem... Simply stating "the car won't run" is not enough to get to step 2. So, jumping to step 3 with "dispose of it" is not a solution...
Yes, that's why Pólya's heuristics are so important. Not rules, heuristics and various methods to accomplish them.

wayneh

Joined Sep 9, 2010
17,469
What tricks do you use when things bog down and progress isn't being made?
Distraction. Recognize you're stuck, and go do something else to let your brain work on it "in the background". Like Archimedes in his bathtub. With wisdom you realize that the harder you're trying, the more likely you're doing it wrong.

Thread Starter

Ya’akov

Joined Jan 27, 2019
8,505
Distraction. Recognize you're stuck, and go do something else to let your brain work on it "in the background". Like Archimedes in his bathtub. With wisdom you realize that the harder you're trying, the more likely you're doing it wrong.
I make it a point to shout "eureka!" when I am in the shower just in case it provokes some brilliant brainwave.

SamR

Joined Mar 19, 2019
4,911
It works better when ensconced upon your porcelain throne!

visionofast

Joined Oct 17, 2018
106
any problem is simply able to be broken into "causes" and "effects",
ignorant peoples mostly devote their time and resources for dealing with "effects",but wise ones focus on "causes".

jgessling

Joined Jul 31, 2009
82
I’ve had good success in my software career by “sleeping on it”. It’s been important to me to purposely stop working on a problem and then think about it again the next morning. Often an unexpected and better solution presents itself. And a good nights sleep is a good idea anyway.

Papabravo

Joined Feb 24, 2006
20,600
The book which I first encountered in the late fall of 1966 was the key to turning around a semi-disastrous start to an engineering education and career.

Thread Starter

Ya’akov

Joined Jan 27, 2019
8,505
The book which I first encountered in the late fall of 1966 was the key to turning around a semi-disastrous start to an engineering education and career.
The book helped several budding engineers of my acquaintance as well. The math majors were exposed to it but the engineers were largely unaware of it. If you really try the method, it really works.

Papabravo

Joined Feb 24, 2006
20,600
It was like somebody turned on a light bulb. In one semester I went from a 2.2 to a 3.74 and never looked back. The two courses that I crushed in the spring of 1967 were Differential Equations, and Strength of Materials. In both cases I was able to use the heuristics to visualize the solutions before I ever put pencil to paper. The following semester I crushed Fluid Dynamics and the applications of the Navier-Stokes equations for the same reasons. I surprised even myself.

Thread Starter

Ya’akov

Joined Jan 27, 2019
8,505
It was like somebody turned on a light bulb. In one semester I went from a 2.2 to a 3.74 and never looked back. The two courses that I crushed in the spring of 1967 were Differential Equations, and Strength of Materials. In both cases I was able to use the heuristics to visualize the solutions before I ever put pencil to paper. The following semester I crushed Fluid Dynamics and the applications of the Navier-Stokes equations for the same reasons. I surprised even myself.
Very cool, it's a great testimony to doing things right.

jgessling

Joined Jul 31, 2009
82
There are a couple other classic books that I would highly recommend.

“The Mythical Man Month”. F. Brooks. Written by a manager from the IBM OS/360 project it includes the all time most important conclusion for large projects. That is if a project is running behind schedule then adding more staff will delay it even further

“Are your lights on?” R. Austin. The problem presented is what sign should be on the highway where it exits a tunnel? After considering all the typical engineering answers, like: if your lights are off and it’s daylight then leave them off, if your lights are off and it’s night time then turn them on, etc. That kind of a flowchart sign would make drivers crazy and accomplish nothing. I guess it’s an argument for using some common sense, and paying attention to what your goal really is instead of getting lost in your process.

Delta Prime

Joined Nov 15, 2019
1,311
had good success in my software career by “sleeping on it”. It’s been important to me to purposely stop working on a problem and then think about it again the next morning.
I would ponder a problem while falling asleep and awake with a solution. Alas, fleeting was the moment I awoke amidst the aroma of coffee dissipating my solution. To this day! Pen and paper in reach upon a sturdy solid oak nightstand!

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