Mathematical Inquiry

The following lists some good questions to ask when investigating something mathematically:

Determining what is "interesting"...

  • Is there a pattern?
  • Do these things have something in common?
  • Is there a recursive relationship? ...is there a non-recursive relationship (e.g., perhaps a formula for predicting this thing)?
  • Can these things be combined in some way? What happens?
  • Where does the behavior change?
  • When does a change have no effect?
  • Did I expect this result? If not, does it always happen? Can I prove it?

Methods of argument

  • Can I argue directly? What can I immediately conclude from what I know?
  • Can I work backwards from where I want to be? What might the next-to-last step be?
  • Can I argue indirectly (i.e., proof by contradiction)?
  • Can I appeal to the Pigeonhole Principle here?
  • Can I use the Principle of Mathematical Induction to argue this? ...or the second principle of mathematical induction (i.e. strong induction)
  • Can I use the Well-Ordering Principle to argue this?
  • Can Symmetry be exploited?
  • Have I considered the most extreme cases?
  • Can I split the set of all possibilities into a small number of cases?
  • Can I reduce the number of cases I must consider by making an assumption without loss of generality?
  • What do the smallest several cases tell me? Is there a pattern to how they are argued? Can the argument(s) used be generalized?

Otherwise Useful Questions

  • What happens with a specific example?
  • What does the definition tell me?
  • Have I named everything that might be significant?
  • Can I write one of the things in the problem in terms of another?
  • Can I "chip away" at this problem? (i.e., Can I write this problem in terms of a simpler problem?)
  • Is there a way to organize or classify what I'm looking at in a way that might be beneficial?
  • If I'm arguing by mathematical induction, have I used the inductive hypothesis? ...can I make pieces of the inductive hypothesis visible in what I am trying to prove in the inductive step?
  • Does the implication work in reverse? (i.e., does the converse hold?)
  • What divisors does this value have? Is this value prime? Do these two values share divisors or are they relatively prime? (these are particularly useful questions to ask in number theory)
  • How can I simplify what I'm looking at?
  • What is the most "ugly" part of this? How can I get rid of it?
  • Can I insert what I want, and compensate for the insertion? (...and hope the compensatory piece works out somehow)
  • How can I combine these things together?
  • What properties does this thing have?
  • Are there any special cases?
  • How does the general case behave? ...or if I am defining something in a more general context, how should it behave?
  • Does this seem to behave like something else I am familiar with?
  • Can I solve a similar (or simpler), but related problem? How does what I'm looking at compare to something similar (or simpler) that I know something about?
  • Can I make an intelligent guess? (Note: intelligent guessing, as opposed to blind guessing, implies you have some specific reason for the guess you make, or that you have narrowed down the possibilities in some way.)
  • Can I write a program to answer this question?
  • Can technology help (e.g., Calculators, Excel, Mathematica, wolframalpha.com, etc...)
  • Can I attack the problem with brute force? (i.e., Can I consider, or make an argument for, every single possibility -- despite the large number of possibilities? ...although this is almost never considered an elegant approach! )

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