Thesis
A personal research topic which has a connection to Graph Theory and the Lights Out puzzle. What's the motivation
behind it? It all happened while i was playing Star Wars: The Old Republic on my computer when i encountered a
quest named Traken-4's Legacy. The incentive was to Power the Gun Emplacement to Shoot Down and Search a
Ship. to do so, i had to activate 9 pylons which were connected to each other. But there was a catch:
Activating/Deactivating a pylon would affect all connecting pylons. So, my group and i attempted to complete the
mission for approximately 1 hour with no success. I had to complete it! So, i found a cliff in the game so i could get a
better visual on the area and started thinking. "There is a pattern there" i thought...I took pen and paper and started
drawing diagrams on how the towers were connected to each other. It looked like this:
I started trying to solve the paper on puzzle for several hours, long after my character was logged out from the game
due to inactivity (My team mates had already disbanded the group long before). During the next day i discussed my
findings with my co-workers at the university and brainstormed for a few hours. I repeated that the day after and
suddenly discovered a pattern. I started applying some steps to the problem and i could see that some rules would
always have to present in order to find a solution. My intuition was correct. I was able to solve the puzzle by inferring
a set of rules:
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One of the Nodes(pylons) that i have to visit in order to solve the problem has to have as many connections
to other Nodes as there are n-1 Nodes to be visited(Solution Nodes).
This Node will be called the Centroid. The Solution Nodes have to have the same number of connections and
every 2 Solution Nodes have to be connected to one same Node.
The Solution Nodes cannot be consecutive and cannot be linked to the Node with the most Connections to
other Nodes.
Order of Visit is irrelevant but each Node should be visited only once. Multiple Multi-Connection solution
Nodes may exist. In that case, there must exist (C-1)xM additional Solution Nodes where C: Main nodes, M:
Connections of Main Nodes.
The list above is not formalized, however and there may be additional rules that apply. After using those rules i was
able to come up with a solution to the problem which looked like this:
The red stars are the visited Nodes. But that was not the end. I quickly discovered various other structures shown
below which followed the same five rules shown above:
And
Unfortunately, my findings would not lead me to the Nobel Prize for a discovery of a new graph structure. After
searching for more answers, i discovered that the basic idea of this structure already exists as a puzzle called Lights
Out. None the Less i am currently involved in extending the “Lights out” puzzle into Graph Theory (set of vertices
connected by edges.) adding set of rules and implementing an algorithm to detect and efficiently traverse such a
graph. The successful proof will explain the nature of several symmetric structures, of which some exist in our
environment in various forms.