In addition to the other answers...experiment! Write SAGE (or other) code to look at a hundred or a thousand examples of what you're trying to say something about. You'll probably see the pattern more clearly when it's sitting right in front of you in numerical form, and probably catch a class of exceptions you hadn't thought of before. If your particular intuition isn't easily calculable on a computer, then generate as many examples as you can by hand, and organize the data in such a way that your brain can see any patterns more clearly (and use SAGE or other code to simplify tedious calculations along the way).
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