The Deranged Mathematician: Counterexamples and Contradictions

Continuing on my series on how to survive proofs, we now turn to two related questions: how to search for counterexamples and contradictions for a conjecture? The first is, I think, mostly self-explanatory, although it has a trick that I find that some students miss: your counterexamples should only ever be as complicated as they need to be. How to figure out how complicated they need to be? For that, you need an intuitive feeling for the problem, and possibly an iterative approach. The second is much richer: proof by contradiction is a very powerful tool when used well. This is a great time to give one of my favorite such proofs, which is the proof that it is impossible to draw a regular heptagon on a square grid. I have seen it presented as a proof without words---it is that visual (and quite pretty!). Read the full post (for free) on Substack: Counterexamples and Contradictions