There are $30$ people at an alumni dinner, seated at $3$ round tables of $10$ seats each.
After every time interval $\Delta t$, a position change event is required where everyone changes position simultaneously, in order to have the opportunity to sit next to someone different on his/her left and right. This results in a different seating configuration.
What is the minimum number of seating configurations (i.e. number of position change events $+1$) required for everyone to have sat next to every other person just once?
NB - If this is not possible then the last condition can be modified from “just once” to “at least once”, but please specify accordingly.
I tried to work this out for a smaller number $n$ of up to $10$ people, seated only on $1$ table, and found that the minimum number of configurations is $\displaystyle\bigg\lfloor\frac n2\bigg\rfloor$.
However, it gets complicated when there are different tables.
This is an example of the Oberwolfach problem, and various papers and articles on this are available on the web, most of them dealing with generalized cases and require a fairly good understanding of graph theory.
It would be appreciated if anyone could derive a user-friendly solution to the question in this particular case.