The Four-Colour-Problem

Yesterday I was a bit bored, and then I recalled having read the biography of a German mathematician called Heinrich Heesch several years ago, who spent most of his life on solving the so-called four-colour problem. ( The four-colour problem is about the question whether four colours are always enough to colourize a two-dimensional map in such a way that there are no adjacent countries having the same colour.

I pondered over this problem and after some time I came to the conclusion that the problem is trivial: Every two-dimensional map can be represented by a planar graph that has no crossings. (I called it a "two-dimensional graph".) In such a graph, it is possible to draw up to 4 nodes which have edges leading to each other, but it is not possible to draw more than 4 - otherwise, there would be a crossing. Therefore, four colours always suffice.

The "official" proofs of this problem, which is now called a theorem, a…

Meine Lerntheorie

Beim Lernen von Theorie lassen sich zwei Arten des Lernens unterscheiden:

1. Konzepte lernen
2. Daten lernen

Konzepte lernen heißt zu versuchen zu verstehen, wie und warum etwas funktioniert bzw. funktionieren könnte.

Daten lernen heißt zu versuchen, sich Fakten zu merken.

Studieren heißt vor allem, Theorie zu lernen. Welche Art des Lernens vorherrscht, ist vom Fach abhängig. Im Medizinstudium kommen beide Arten vor: Fächer wie Physik oder Chemie handeln vornehmlich von Konzepten, Fächer wie Anatomie oder Mikrobiologie von Daten. Im Informatikstudium dagegen geht es fast ausschließlich darum, Konzepte zu lernen; Daten lernen kommt fast nicht vor.

Das Lernen von Daten ist wahrscheinlich vornehmlich eine Frage der Merkfähigkeit und der Übung, während das Lernen von Konzepten auch von der kognitiven Begabung des Studierenden abhängig ist.

Lens effect

There is a cool new 256b intro at, called ind. It shows a rotating sphere divided into several fields, which flicker in funky colours.

This intro is basically based on the lens effect. The lens effect is a distortion that projects all pixels from a square onto a circle surrounding it. How is this done?

If we try to imagine how it works, we will perhaps start with the y coordinates of the projection of the topmost line of the square: We notice that the points in the horizontal center are the topmost ones and the elevation declines to the edges, where it becomes zero, i.e. the edges are printed at the same position as in the square. With what mathematical formula can this be achieved? Simple subtraction wouldn't make it a circle. We need a trigonometric function. Which one? Let's figure it out: We use the distance from the horizontal center point as its parameter. If it is 0, the height must have the highest value, and if it is maximal, the height must be 0. A trigonom…

Chebyshev's inequality

Somebody posted a task involving Chebyshev's inequality to ( It took me more than two hours, but I think I solved it.

Sudoku Puzzle Solver now Interactive!

The first fruit of my OpenGL learning: I've made an Interactive Sudoku Puzzle Solver. Based on my algorithms I wrote in 2005, it now displays the solution of the puzzle step by step and you can control it interactively.

XNA Game Studio

Finally I've been able to install XNA Game Studio on my PC (the required Service Pack from the website didn't work on my students' version of Visual Studio 2005 Professional, but the file I downloaded from the server of our university did). I've already drawn a moving sprite. XNA GS has an easy method of transparency, every purple pixel isn't drawn on the screen. So the main problem I've had with previous solutions has gone. Now I have to learn how to put text on screen and how to evaluate the user inputs.


Today I had a look on Savage Charts #1 and had the impression that the music reminded me of the style of Imphobia. Nostalgic feelings came up. I then had a look on Imphobia #12. It does not work correctly on modern PCs, it flickers like hell, but it's legible and the music can be played in the background using an external sound player. I've been impressed at the number of people who had contributed articles to this diskmag. Imphobia attracted more writers than any other PC diskmag ever. This is unique in diskmag history. It is unrealistic that this will ever repeat as nowadays people have Internet forums where they can be heard.

The Monty Hall Problem

Yes, this problem illustrates why I sometimes get so enraged with people who seem to be worse at logics. According to Wikipedia, even a lot of academics, including Ph.D.s in mathematics, doubted Marilyn vos Savant's correct solution to this problem and accused her of "innumeracy". Of course Marilyn was right. But it's difficult to convince somebody of one's rightness if this person is worse at logics because this person will have a hard time understanding the argumentation.

That's why there are mathematicians after all. But, as I said, in this case even a lot of Ph.D.s in mathematics were initially unable to follow Marilyn, and only after several more explanations they finally had to admit that she was right and they had been wrong.

A similar problem occurs on discussing about subjects that require a relatively high level of education, while there are many popular misconceptions. Politics is a good example, for there are many common misconceptions about terms…