Question 8
An integer is written at each vertex of a regular $n$-gon. A move is to find four adjacent vertices with numbers $a$, $b$, $c$, $d$ (in that order), so that $(a - d)(b - c) \lt 0$, and then to interchange $b$ and $c$. Show that only finitely many moves are possible. For example, a possible sequence of moves is shown below:
| 1 | 7 | 2 | 3 | 5 | 4 |
|---|---|---|---|---|---|
| 1 | 2 | 7 | 3 | 5 | 4 |
| 1 | 2 | 3 | 7 | 5 | 4 |
| 1 | 2 | 3 | 5 | 7 | 4 |
| 2 | 1 | 3 | 5 | 7 | 4 |
| 1 | 7 | 2 | 3 | 5 | 4 |
|---|---|---|---|---|---|
| 1 | 2 | 7 | 3 | 5 | 4 |
| 1 | 2 | 3 | 7 | 5 | 4 |
| 1 | 2 | 3 | 5 | 7 | 4 |
| 2 | 1 | 3 | 5 | 7 | 4 |