In 2009, I posted a calculational proof of the handshaking lemma, a well-known elementary result on undirected graphs. I was very pleased about my proof because the amount of guessing involved was very small (especially when compared with conventional proofs). However, one of the steps was too complicated and I did not know how to improve it.
In June, Jeremy Weissmann read my proof and he proposed a different development. His argument was well structured, but it wasn’t as goal-oriented as I’d hoped for.
I am currently in Salamanca (Spain), attending the conference Tools for Teaching Logic III. My talk was on teaching logic through algorithmic problem solving and it went quite well, I think. In particular, it seems that the audience enjoyed the examples that I have used and the teaching scenarios that I have shown. As a result, I have promised that I would upload my PhD thesis into this website. Since the thesis can also be useful for other people, I have decided to write a new blog post.
Suppose you write a program and you send the source code to two of your friends, ${\cal A}$ and ${\cal B}$. Your two friends read the code and when they finish, $A$ errors are detected by ${\cal A}$, $B$ errors are detected by ${\cal B}$, and $C$ errors are detected by both. So, in total, $A+B−C$ errors are detected and can now be eliminated. We wish to estimate the number of errors that remain unnoticed and uncorrected.
I found the following problem on K. Rustan M. Leino’s puzzles page:
[Carroll Morgan told me this puzzle.]
Prove that for any positive K, every Kth number in the Fibonacci sequence is a multiple of the Kth number in the Fibonacci sequence.
More formally, for any natural number n, let F(n) denote Fibonacci number n. That is, F(0) = 0, F(1) = 1, and F(n+2) = F(n+1) + F(n).
Last November I solved Problem 15 of Project Euler (a counting problem involving paths in square grids), and, although the problem admits a simple solution, some of the solutions presented in their forums are very complicated. Thus, I thought it would be a good idea to present my solution, as I consider it very simple.
Problem statement Starting in the top left corner of a $2{\times}2$ grid, there are 6 routes (without backtracking) to the bottom right corner.