@user21820 My apologies for the delay in getting back. I have not been able to make much progress on the exercises since uni started.

@user21820 I understand. I was just worried that I wouldn't be able to apply the technique on my own. For example, I think you had hinted here that the technique might be useful for (Q10), but I have no idea how. As someone of artistic bent, I do see the face lol, but I'm hoping knowing how to prove things will keep me safe from pareidolia when it comes to maths.

@user21820 I can't wait to get to induction. It seems like all the proofs I come across use it.

@user51462 It's true that the canonicalization technique is useful almost everywhere, including for (Q10). Just like for (Q9), there are many terms that are superficially different but can be considered equal because of one or more of the given conditions. Similarly, for (Q10) I gave a vague remark:

To be more precise, you need to grasp fully that the vertices are all essentially the same (symmetric), so you would want to prove suitable lemmas that can be applied to any subset of the vertices, and definitely not try to apply the systematic procedure for proving FOL theorems... It's not that it won't work, but that the resulting proof would be too many times as long, and completely inscrutable.

Another way to look at it is that canonicalization tells you to choose canonical cases. Either at most 5 vertices or at least 6, right? The only case that matters is at least 6. Since you are only guaranteed at least 6, there is no reason to consider more, so the 'canonical' case is to literally have 6 vertices. Say a,b,c,d,e,f. If a,b,c don't form a triangle, then pick 'canonical' cases (that cover all the cases by symmetry), and continue...

@user51462 When you get to PA, you will get to a simple form of induction. I want to also remark that induction has nothing to do with set theory, so any textbook that describes induction using sets in any way is bad (and unfortunately there are lots of bad textbooks that bad teachers use).