The is a round table with 60 chairs around it. 30 people come into the room and sit around the table leaving 30 chairs empty.

Question

How do you show that one can always find six chairs next to each other so that exactly 3 chairs are taken and 3 are empty? I think that the Pigeonhole principle could be used to answer this question.

Answer 1

Consider a window sliding over $6$ chairs in an arc and totalling the number of people sitting in it. When this window slides one place, there is either an outgoing person or not and there is either an incoming person or not. Thus the total changes by $1,0$ or $-1$.

If the window has at most $2$ people in it at some point and at least $4$ in it at another point, by the intermediate value theorem (and the limited variation of the window sum) it must have exactly $3$ people in it somewhere. Thus if there is no window of $3$, the window total is either at most $2$ everywhere or at least $4$ everywhere. But we can get the total number of people by summing all window totals and dividing by $6$, and these two scenarios result in too few and too many people respectively, so they are impossible.

Therefore there is at least one window with exactly $3$ people.