Fellow Bulbmaniacs!

First of all, Bulbmania isn't just any old run-of-the-mill computer game: depending on how you want to see it, it can either be a strategy game involving math or a math game that requires strategy! You could theoretically play Bulbmania simply with a piece of paper and a pencil, but we tend to think it's a lot more fun on a computer. And while playing, don't forget how important math is for this game!

The goal is simple: get all the light bulbs to turn off!

Clicking on a light bulb changes its state, meaning that light bulbs that are on are turned off and vice versa. The hard part is that light bulbs on either side of the one you click on will be turned on or off, too (Bulbmania Classic). In the Stefan Problem, the two light bulbs on the left and the two light bulbs on the right even change!

For skilled Bulbmaniacs, the challenge is getting all the bulbs to go out as quickly and with as few clicks as possible.

## Origination

The first "Bulbmania" game was developed and named by Jörg Sautter for the open house day ("Tag der Forschung") at the Heinrich-Heine-University in Düsseldorf. The program was written in MatLab and required either a MatLab license or a huge standalone binary file. The files are still available at www.bulbmania.de.

There the game variant "Advanced" is called "Stefan Problem". It is the name of another employee (Stefan Henn) who asked if the same game would be possible if not only the nearest but additionally the second nearest neighbors were toggled. And the answer is, yes, but with an additional restriction: The number of bulbs is not allowed be a multiple of five. However, the same problem arises in Bulbmania Classic at multiples of three.

In spring 2010, a major part of Prof. Hochbruck's group moved from Heinrich-Heine-University Düsseldorf to Karlsruhe Institute of Technology (KIT). We thought, Bulbmania might be a nice contribution to the "day of Energy" (September the 25th 2010) — a public event where the KIT presents its research areas. This was the inducement for Dominik Löchel to set up a new Bulbmania code, which should run on any web browser.

The program code was written piece by piece and tested, i.e. playing Bulbmania until the program chrashed. These plays gave rise to some questions: Are all initial settings of bulbs solvable? Which settings are solvable? How does the optimal strategy for a minimal number of clicks look like? What is the minimal number of clicks for each setting?

These are many questions and some go deep into the mathematical background. Some answers are: If the number of bulbs in the setting is not a multiple of three in Bulbmania Classic or five in Bulbmania Advanced, then each initial setting of switched on or off bulbs is solvable. The game can be won with a number of clicks less or equal to the total number of bulbs. The settings with a multiple of three in Bulbmania Classic or five in Bulbmania Advanced are either impossible to solve or boring. Therefore you cannot choose those settings in our Bulbmania game.

Now, enough hints are given; play Bulbmania until you understand the math behind it!