I have been reading the paper: "The Dynamics of Successive Differences Over $\mathbb{Z}$ and $\mathbb{R}$", describing the n-value game.
The n-value game works as follows. A polygon with $n$ sides is formed, with numbers assigned to each vertex. This is the initial state of the game. The next state of the game is achieved through applying a transition to the previous. Here, the previous state's midpoints are used to form a new $n$ sided polygon. The numerical values assigned to the new verticies of this state are determined as the absolute value of the differences of the two adjacent corners (forming the new vertex).
The paper above worked on properties of this game for $n = 3, n = 4$ over $\mathbb{Z^+}$, finding when the game enters a cycle (repeating a pattern of states) and when the game converges onto a certain state.
I wish to find a way to generalize for which $n$ will the game converge, and for which values will it enter a cycle. How could I go about approaching this problem?