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

- 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?

- 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?

- 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!*)