Problem solving

A problem of the form What can I figure out about this new idea? focuses on a mathematical object, abstracted from the trappings of the real world. For example, the problem may be about understanding the notion of graph, tree, geometric shape, permutation, or function.

Getting to know an idea

You begin work on this kind of problem by brainstorming. To brainstorm, you list as many observations and questions as you can without evaluating them. While still brainstorming, you look at lots of examples, examining them for patterns, looking for properties (characteristics) of the mathematical object being examined.

Once you have a conjecture, your mission is more focused: you want to prove or disprove the conjecture. To prove a conjecture, you write a proof, which is a logical, convincing arugment. Ironically, failed attempts to prove a conjecture often lead to the feeling that the conjecture is false, and lead you to work to find an example of where the conjecture fails. Such an example is called a counterexample. Even if your work to find a counterexample fails, it might lead you to insights about the mathematical object—insights that contribute to a proof that conjecture is true after all. If proved, a conjecture is elevated to the status of a theorem.

Solving all problems

Solving a problem of the form How can I solve all problems of this type? requires finding a general method, called an algorithm. As you might expect, working on several problems of the same type can give you ideas about how to work such problems in general.

Sometimes algorithms are expressed in a formal structured language so that their correctness can be checked easily. Then they can be used in proofs. Often algorithms are expressed less formally, however.

However you express the algorithm, you need to test it on as many examples of problems as you can.

In solving any kind of problem, you might want to monitor your work as you go. Here is an idea from John Mason, Leone Burton, and Kaye Stacey, in their book Thinking Mathematically.

While working on a problem, make these marks in your notes to keep tabs on how you're doing:
STUCK
AHA!
TRY
CHECK
REFLECT
KNOW
WANT
QUESTION
THINKING
WARMING UP
ASSUME
Stuck?

All mathematicians get stuck while trying to solve problems. Galovich, in the book Doing Mathematics: An Introduction to Proofs and Problem Solving, compiles ideas about being stuck from Polya and Schoenfeld:

Write down what you know
Write down what you want
Draw a picture
Label some variables
Try one or more special cases (specialize) and look for patterns
Consider extreme cases
Consider using symmetry properties
Restate conditions in equivalent form
Recombine elements of the problem
Introduce auxiliary elements
Change perspective or notation
Assume you have a solution and determine its properties
Try a simpler problem
Try an analogous problem
Generalize the problem
Isolate a variable

Look up stuff
Make sure you're aware of relevant results by others.
Think backward
Imagine that you have the information that is asked for in the problem but donŐt have the information in the question. Does this give you any ideas about the relationship between the question and the solution?

When you're not stuck and are going fine, keep asking

Is this taking too long?
Should I keep going on this track or try something else?

When you've been working on one track but haven't made progress for awhile, consider yourself stuck and ask those questions again.