Subfield: Combinatorics


Question: At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \geq 3$, the chairs are labelled $1,2,3, \ldots, n-1, n$ in order around the table. A table is considered full if no more people can be seated without having two people sit in neighbouring chairs. For example, when $n=6$, full tables occur when people are seated in chairs labelled $\{1,4\}$ or $\{2,5\}$ or $\{3,6\}$ or $\{1,3,5\}$ or $\{2,4,6\}$. Thus, there are 5 different full tables when $n=6$.

![](https://cdn.mathpix.com/cropped/2023_12_21_ed73f7f90f51d294b52fg-1.jpg?height=401&width=331&top_left_y=1304&top_left_x=1469)
Determine all ways in which people can be seated around a table with 8 chairs so that the table is full, in each case giving the labels on the chairs in which people are sitting.

Solution:
If there are 5 or more people seated around a table with 8 chairs, then there are at most 3 empty chairs. But there must be an empty chair between each pair of people, and this is not possible with 5 people and 3 empty chairs.

Therefore, there are at most 4 people seated.

If there were only 2 people seated, then there would be 6 empty chairs which would mean that at least one of the two "gaps" around the circular table had at least 3 empty chairs, and so another person could be seated, meaning that the table wasn't full.

Therefore, there are at least 3 people seated.

This means that a full table with 8 chairs has either 3 or 4 people.

If there are 4 people, there are 4 empty chairs, and so there is exactly 1 empty chair between each pair of people.

Thus, people are seated in chairs $\{1,3,5,7\}$ or in chairs $\{2,4,6,8\}$.

If there are 3 people, there are 5 empty chairs.

With 3 people, there are 3 gaps totalling 5 chairs, and each gap has at most 2 chairs in it. Therefore, the gaps must be 1, 2, 2 in some order. This is the only list of three positive integers, each equal to 1 or 2 , that adds to 5 .

The gap of 1 can be between any pair of seats. In other words, the gap of 1 could be between $\{1,3\},\{2,4\}$, and so on. In each case, the position of the third person is completely determined because the remaining two gaps have 2 chairs each.

Thus, with 3 people, they are seated in chairs

$$
\{1,3,6\},\{2,4,7\},\{3,5,8\},\{4,6,1\},\{5,7,2\},\{6,8,3\},\{7,1,4\},\{8,2,5\}
$$

In total, there are thus 10 ways to seat people at a table with 8 chairs:

$\{1,3,5,7\},\{2,4,6,8\},\{1,3,6\},\{2,4,7\},\{3,5,8\},\{4,6,1\},\{5,7,2\},\{6,8,3\},\{7,1,4\},\{8,2,5\}$

Final answer: If there are 5 or more people seated around a table with 8 chairs, then there are at most 3 empty chairs. But there must be an empty chair between each pair of people, and this is not possible with 5 people and 3 empty chairs.

Therefore, there are at most 4 people seated.

If there were only 2 people seated, then there would be 6 empty chairs which would mean that at least one of the two "gaps" around the circular table had at least 3 empty chairs, and so another person could be seated, meaning that the table wasn't full.

Therefore, there are at least 3 people seated.

This means that a full table with 8 chairs has either 3 or 4 people.

If there are 4 people, there are 4 empty chairs, and so there is exactly 1 empty chair between each pair of people.

Thus, people are seated in chairs $\{1,3,5,7\}$ or in chairs $\{2,4,6,8\}$.

If there are 3 people, there are 5 empty chairs.

With 3 people, there are 3 gaps totalling 5 chairs, and each gap has at most 2 chairs in it. Therefore, the gaps must be 1, 2, 2 in some order. This is the only list of three positive integers, each equal to 1 or 2 , that adds to 5 .

The gap of 1 can be between any pair of seats. In other words, the gap of 1 could be between $\{1,3\},\{2,4\}$, and so on. In each case, the position of the third person is completely determined because the remaining two gaps have 2 chairs each.

Thus, with 3 people, they are seated in chairs

$$
\{1,3,6\},\{2,4,7\},\{3,5,8\},\{4,6,1\},\{5,7,2\},\{6,8,3\},\{7,1,4\},\{8,2,5\}
$$

In total, there are thus 10 ways to seat people at a table with 8 chairs:

$\{1,3,5,7\},\{2,4,6,8\},\{1,3,6\},\{2,4,7\},\{3,5,8\},\{4,6,1\},\{5,7,2\},\{6,8,3\},\{7,1,4\},\{8,2,5\}$