There are 2015 people from universities across the country, all sitting around a massive table. Each person belongs to one of four groups: students, professors, deans and presidents. Students hate professors, and will never sit next to them. Likewise for presidents and deans. Explain why, in every possible seating, there must be two people from each group sitting next to each other who are from the same group.

Question

There are 2015 people from universities across the country, all sitting around a massive table. Each person belongs to one of four groups: students, professors, deans and presidents. Students hate professors, and will never sit next to them. Likewise for presidents and deans. Explain why, in every possible seating, there must be two people from each group sitting next to each other who are from the same group.

Answer 1

To explain why in every possible seating arrangement of the 2015 people around the massive table, there must be two people from each group sitting next to each other who are from the same group, we can use the pigeonhole principle.

Let's consider each group of people as a "pigeonhole" and the seating arrangement as "pigeons" that need to be placed into the pigeonholes.

Since there are four different groups (students, professors, deans, and presidents) but only three possible pairs of groups that dislike each other (students-professors, professors-deans, and deans-presidents), we can divide the possible seating arrangements into three categories:

1. Students sitting next to professors.
2. Professors sitting next to deans.
3. Deans sitting next to presidents.

Now, let's analyze each of these categories separately:

1. If there are no students sitting next to professors, it means that all the students must be sitting next to either deans or presidents. Since there are only two types of people students can sit next to, there will be at least two students sitting next to each other. Therefore, we have two students from the same group sitting next to each other.

2. Similarly, if there are no professors sitting next to deans, all the professors must be sitting next to either students or presidents. Hence, there will be at least two professors from the same group sitting next to each other.

3. If there are no deans sitting next to presidents, all the deans must be sitting next to either students or professors. Therefore, there will be at least two deans from the same group sitting next to each other.

In any of the three cases, we can see that there will always be at least two people from the same group sitting next to each other.

By considering all possible seating arrangements, we can conclude that in every seating configuration of the 2015 people around the massive table, there will always be two people from each group sitting next to each other who are from the same group, as per the pigeonhole principle.