Question 6
A mathematics conference has $6n+4$ mathematicians and exactly $2n+1$ meetings, where $n\ge1$. At each meeting they sit around one four-seat table and $n$ six-seat tables; the seats are equally spaced around each table. Any two mathematicians who sit next to or opposite each other at one meeting must not sit next to or opposite each other at any other meeting.
(a) Prove that a seating arrangement is possible when $n=1$.
(b) Can a seating arrangement be made for every $n\gt 1$?
(a) Prove that a seating arrangement is possible when $n=1$.
(b) Can a seating arrangement be made for every $n\gt 1$?