Solving Problems: Georg Polya and Beyond

Okay, this is going to be just a shell for a post.

In the nineteen forties, a Cambridge scholar called George Polya gave his research students a checklist of strategies that could help them get started on solving a problem (usually a research program), or get past some sticky obstacle.  Since Polya, lots of people have extended and elaborated this checklist, and little kids in elementary school are taught some of these strategies (which they probably proceed to forget right away!), and I felt I should read up on the latest thinking on it.  But it is such a useful list of strategies that getting it written down ASAP, even if it is a little premature, is probably going to be useful.  OK, straight onto the list.

01.  Understand the problem.  State the problem clearly.  This is less useful for those of us with simple problems that are easily stated, but often doing this clarifies a misunderstanding that has been an obstacle.

02.  Define your terms.  (This is a little technical, but examining the words used in stating the problem is often surprisingly helpful.)

03.  Study similar problems that actually have been successfully solved.

04.  Simplify your problem.  If a simplified version of your problem can be solved, the original problem might have a solution obtained by slightly tweaking the simpler solution.

05.  Break your problem into parts, and solve them separately.

06.  Draw a picture, or make a graph, or build a chart.  (Often problems that have to do with dealing with numbers sometimes suggest a solution when you study the numbers visually.)

07.  Make a list, or a table.  (Online sales points offer tables for comparing different models of electronic equipment.)

08.  Look for a pattern.  This makes most sense when trying to find a property of a number.

09.  Work backwards.  This strategy is in the context of trying to find out why something is the way it is (which comes up in various sorts of research).  Often trying to understand a chain starting from one end of it leads to a major obstacle.  Starting at the other end often does the trick.

A formal description of George Polya's methods is found in Wikipedia.