Third Week of Instruction (1/19, 1/21)

Wednesday (1/19)I want the students to work on the friends in class problem. This problem is not difficult if it is turned into a problem about graphs. This requires me to give my second mini-lecture (the induction was the first) on elementary graph theory.

The main result I derive is the following:

Let (V,E) be a graph then the sum of the degrees of the vertices is twice the number of edges.

This mini-lecture takes about half the class time and since I want the students to have a reasonable time in groups I decide to let them spend the balance of the period working on problems.

I return the homework. I have used TeX to include a written comment for each solution submitted. The response is very favorable. One student said they had never, ever gotten back feedback or comments from any professor they have had at UCSC. I hope they like it as much when they actually read the comments. I want the students to redraft their work and I have indicated what is of value in the work they have submitted but also whether it is correct and complete (and then there may be some suggestions for generalization), on the right track but incomplete (with some suggestions for how to finish), or off on the wrong track and ideas for backtracking and finding the right direction. Quite a few students turned in induction proofs and for the most part these were correct but I did not particularly like the way many would complete the induction. These students treat the equality they are trying to establish as an identity and make changes to both sides until they are equal whereas I would prefer them to transform the term that involves the use of the induction hypothesis exclusively until it has the correct form for the succeeding case. Perhaps with my comments and more opportunities they will get this.

Friday (1/21)Today I talked about a new type of method: Coloring. An example of a problem amendable to this argument is the chessboard with opposite corners removed problem. We also talked about parity. Students worked in groups on problems drawn from the first set I gave them.

The students are not as animated as previously, probably as they adjust to their new groups. I receive new submissions from 10 students. We spend some time discussing previous problems that the students have worked on. One such problem is the lattice points problem which, it appears, nearly everyone has finally figured out. One student goes to the board and gives a fairly good account: makes boxes labeled by pairs of o (for odd) and e (for even). There are four boxes, five points, so two points get in the same box and for that pair the midpoint is a lattice point. When I ask about generalizing, more than one student says that 9 points in three space work. For n space this would be 2^n + 1 points. They definitely got this one. The generalization to three dimensions was a Putnam problem some years ago!