Problem: Given $4$ circles, we define the following set of rules:
i) Any circle which contains $\ge 3 $ elements transfers exactly one of its elements to each of other $3$ circles.
ii) Circles which contain $<3$ number of element do not transfer any of its elements to the other three circles.
In one operation all rules are applied simultaneously.
There are two states which can be achieved under these rules:
This is a stable state reached.
This is an oscillating state.
Question: I am curious to find what is the maximum value of $k$ such that some initial configuration cycles through $k$ distinct configurations and returns to the original configuration (for the first time) after $k$ operations? $k$ is also called the least period of the transformation.