TITLE: Prove that the 25 people can be seated in this way
QUESTION [34 upvotes]: 5 mathematicians, 5 biologists, 5 chemists, 5 physicists, and 5 economists sit around a large round table.
Prove that the 25 people can be seated such that, if A and B are two different people with the same specialty (for example, two mathematicians), then the people sitting to the immediate left of A and to the immediate left of B are of different specialties (for example, a biologist and a chemist).
I thought to use induction, since $n=1$ and $n=2$ are easy to do. However, I don't know how to continue.

REPLY [3 votes]: Given that there are 5 types of person, each of whom sits to the left (or right) of ONE other person, this relationship can be shown as a square (person by neighbour). Also, the seating is going to be a continuous sequence where the 'left' person of the current pair will be the 'right' person of the next pair.
Each neighbour relationship can be numbered 1 to 25. Provided a relationship has a unique number and points to another valid relationship (exception being made for the last relationship of course), then the system is valid.

M   M   B   C   P   E   B   P   M   E   C   B   B   E   E   M   C   C   E   P   P   C   M   P   B   M