I recently looked through my old math notes, and found some of my personal research projects. It's quite fun for me to read; in those days I had much more time to pursue my interest in math, and I actually managed to prove some interesting things. Here is an example:

If you have looked at my book log recently, you may have noticed a drastic drop in the number of books I have been reading starting mid-2012. There are two reasons for this, one of them being Project Euler.

Project Euler is a math/programming puzzle web site. I can't remember where I read about it first — perhaps it was in an article on Slashdot? (Edit: Actually, yes, I think it was in this article — which also nails down the date I joined Project Euler: The 3rd of October 2012.) Anyway, the idea is this: Every weekend (more or less) a new problem is published. The problem is usually mathematical in nature, but almost always requires some programming to solve. When you have found the correct answer to the problem (most of the time in the form of a number), you submit it to the site and gain access to a forum where you can discuss the problem and the solution.

Back in 2009 I began keeping a log of the books I read. It's a mixture of Science Fiction, Fantasy, Thrillers, Murder Mysteries and a few computer science books. Some are in danish, some in english. As you can see, I didn't include the start and finish dates at first, but I quickly realized that it would be a good idea to include that information as well. I wish I had started this back when I was still studying at the university — back then I used to buy around 20 books each summer because the local book store was selling 5 books for 100 DKK (which is less than what one book usually costs), and now I have a hard time remembering which ones I've read.

One of the important – indeed central – aspects of mathematics is the idea of abstracting out interesting structures in order to study those structures in general. The study of groups, rings, fields, vector spaces, category theory, and so on, are examples of this. Other types of abstractions are general methods and principles, such as proof by induction or the Pigeonhole Principle. One such principle caught my attention recently, namely the Inclusion-Exclusion Principle.

I became interested in making music back in the good old days of the Commodore 64 home computer. I listened to the wonderful music of Rob Hubbard, David Whittaker, Ben Daglish, Martin Galway, Fred Gray and all the other musicians who composed for the C64 (I still sometimes find myself whistling the tunes from Spellbound, Master of Magic, Street Surfer, Krakout, Parallax, Shadowfire and other games that I haven't played for at least 20 years).

Recently I have begun to use javascript whenever I need to calculate something. As an example I can mention the Queens Problem (see below). Someone asked me about the problem, and I just had to try to write a program that could find the solution(s). Javascript seemed the quickest way to get a working program, and it's easy to display the results of the program in html. Also, the recent browser wars have made javascript so much faster, so speed is no longer a reason to avoid javascript. This post is really just a place for me to put all of my small javascript projects.

The first two mathematical conjectures that I describe in this post go back to the days when I was studying math at the University of Copenhagen. At that time I always had several "mathematical pet projects" that I was thinking about whenever I could find some free time. Usually my projects were generalizations of math problems from my classes, and most of them involved number theory, group theory or combinatorics.

I have a Roland JW-50 Music Workstation, and I've been worried about what would happen to all my music (stored on old 3.5" disks) when the keyboard (or the built-in floppy drive) dies one day. The JW-50 actually lets you save your songs in MIDI format, but I have had problems with that function — I have at least one song that crashes the keyboard if I try to save it in MIDI format.