Professor Plum has a dinner party with n² guests. Her guests are seated at n tables, with n guests at each table. After each dish, a new seating arrangement is made so that every guest sits down at a table with guests they have not already shared a table with.
Can every guest share a table with every other guest during the dinner? If not, what requirements do we have on n to allow all the guests to share tables with each other?
(The poster is the original author of this problem.)