# Elfenland and Graph Theory

Quite a while back I supervised a class project on graph theory. One inspiration was the boardgame Elfenland. Back then I made quite an elaborate series of lessons involving graph theory and that game. I remembered this because of a tweet involving this post on math strategies for games. Unfortunately I couldn’t find all the documents. On a more modest basis I have recreated some of the activities.

Elfenland is a game that involves using cards to travel from one point to the other. The board is below.

The first step is to transform this board to a graph. There are some difficulties and/or assumption:

- If one can travel from A to B I have added a vertices and an edge.
- I did not model any weights or “one way” edges (like 17 to 16, 16 to 15, for labels see below).

I then inputted the graph in Graph Magic (http://www.graph-magics.com/).

Now, assuming that one has to start and end in the capital at vertex 17, It is clear that because of vertex 7, a Hamiltonian circuit cannot be found, as vertex 9 is passed at least twice. So, I excluded vertex 7. Calculating the circuit for the remainder of the graph would suffice, just as long as the player would visit vertex 7 when arriving in vertex 9. The result was:

In this optimal route edges between 17,16 and 15 are not used so the problem of “one way” edges does not have a direct consequence for my strategy. In the next steps weights were added according to the Elfenland rules. In the real game, chance plays a role as travel is done by using playing cards. I have no time to improve it in this occasion, I’ll leave that to you. I think it is a nice introduction to some Graph Theory concepts.

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