Final answer:
The problem is a variation of the combinatorial round table or social golfer problem, which, for 31 people, remains unsolved in mathematics. It seeks a seating arrangement where each person has different neighbors each day, a complex task without a known solution for this specific number of participants.
Step-by-step explanation:
The question relates to a combinatorial problem in mathematics, specifically to seating arrangements around a round table. When considering different neighbors for each person every day, we are dealing with a variation of the famous round table problem, also known as the social golfer problem.
Firstly, let's set one person as a reference point, since the table is round there is no distinct 'head' of the table. That reduces the problem to arranging the remaining 30 people.
Now, the key is to understand that each person can have at most 29 different neighbors to one side and 29 to the other side since the first and last day their neighbors would be the same if we managed to give them a new neighbor every single day in between.
However, arranging 31 people so that each has different neighbors every day is in fact an unsolved problem in mathematics. This falls under a larger set of problems known as combinatorial design.
For smaller numbers of people, it might be possible to determine a specific number of days. But with 31 people, the problem becomes very complex and as of the current state of mathematical research, there is no known solution that maximizes the number of days for this number of people without repeating neighbors.