Question 48
Find the number of ways that we can write $1, 2, \dots, n$ on the vertices of a regular $n$-gon ($n \ge 4$) such that if the diagonals connecting $(a, b)$ and $(c, d)$ cut each other then
$$
a + b \not\equiv c + d \pmod{n}.
$$
(Two arrangements of numbers matching after a rotation are considered the same.)
$$
a + b \not\equiv c + d \pmod{n}.
$$
(Two arrangements of numbers matching after a rotation are considered the same.)