Hex(agons), lies, and ternary trees

A handful of interesting trends emerged this week, and the overhead projector reminded me of its vulnerability to overheating (and, thus, under-lighting).

At this point in the course, I use the technique of simple induction to lie to my students in three different ways. My motivation is to vaccinate them against false proofs by letting them work through a few. Both the evening and day section seemed to spot the sleight-of-hand fairly readily. Although it's not meaningful to generalize, perhaps this cohort of students has its BS-detector cranked up higher than previous cohorts, or I'm a worse liar than I once was.

In another thread, a challenging (I think) counting problem that isn't due for a month has generated enough controversy about the number of distinct ternary trees with five nodes that I've had to resort to writing a short program to convince myself of the correct answer. I'm curious to see whether my students begin to use their programming language of choice to explore questions, and whether this hides or obscures the connection between programs and proofs.

In lecture, one student brought out the subtle point of whether we can really assume we have a full binary tree with n+1 nodes if such a configuration is impossible. She noted that in some cases (when n+1 is even) there will be no such FBTs. What makes this situation interesting is thiat it corresponds to vacuous truth --- of there are no FBTs of size n+1, then any claim about them is true (they have n edges), so the only case where there's work to do is in the case where there [b]is[/b] a FBT of size n+1.