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What are ideas and strategies on improving at discovering counterexamples?

I originally posed this as an Example Question.

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    $\begingroup$ In my opinion this is too broad. Also, it depends on the subject and any general advice would have to be extremely high-level. $\endgroup$
    – dtldarek
    Mar 15 '14 at 16:35
  • $\begingroup$ I agree. Perhaps we could edit this to be for a particular topic? Or a particular level? $\endgroup$
    – adamblan
    Mar 15 '14 at 16:49
  • $\begingroup$ I agree. The strategy for finding counterexamples to statements in introductory calculus will be very different from the strategy for finding counterexamples to statements in pure mathematics courses. Narrowing it to a level sounds like it would work well. $\endgroup$
    – Chris Cunningham
    Mar 15 '14 at 18:14
  • $\begingroup$ I think this question would be fine if the OP would add some context. What course or courses does the OP have in mind? What level of students? What kinds of counterexamples? $\endgroup$
    – Jim Belk
    Mar 15 '14 at 21:58
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The question is really broad and a bit vague, but some remarks hoping they match the intent.

  1. A key to discovering counter-examples is to understand the examples. Taken an example and try to modify to get more examples. See when/if they break. If they do not break think about why they did not break. Then 'attack' the point that prevented breaking in the next round.

  2. Dually a key to discovering counterexamples is to understand why it is hard/impossible to prove the can not exist. Try to prove that a counterexample cannot exist. Think about why you are unable to prove the nonexistence.

  3. (Depending on context.) Keep an open mind. Perhaps if you really cannot find a counterexample it is because there really is none. This happens. A famous case is Carleson's theorem, he first tried to find counterexamples to what is now his theorem (then Luzin's conjecture) but eventually got convinced there is none and proved this. I quote a part from an interview with him, on this process (replace notions you might not know by something that makes sense, just to feel the route):

I was at that time working on Blaschke products, and I said maybe one could use those to produce a counterexample. Zygmund was very positive and said "of course, you should do that." I tried for some years and then I forgot about it before it again came back to me. Then, in the beginning of the 1960s, I suddenly realized that I knew exactly why there had to be a counterexample and how one should construct one. Somehow, the trigonometric system is the type of system where it is easiest to provide counterexamples. Then I could prove that my approach was impossible. I found out that this idea would never work; I mean that it couldn’t work. If there were a counterexample for the trigonometric system, it would be an exception to the rule. Then I decided that maybe no one had really tried to prove the converse. From then on it only took two years or so. But it is an interesting example of "to prove something hard, it is extremely important to be convinced of what is right and what is wrong." You could never do it by alternating between the one and the other because the conviction somehow has to be there.

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    $\begingroup$ I would add that edge-cases are often useful, depending on the context of the problem. $\endgroup$
    – mkasberg
    Mar 15 '14 at 19:33

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