An odd integer
![n \ge 3](/media/m/e/5/0/e50da5b5bfc67c0a42c000bd5d2ca5a4.png)
is said to be nice if and only if there is at least one permutation
![a_{1}, \cdots, a_{n}](/media/m/8/2/4/82469cb02e1bd0b9a8954760a6256eac.png)
of
![1, \cdots, n](/media/m/a/9/b/a9b2c9072aecef66565c31f582061359.png)
such that the
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
sums
![a_{1} - a_{2} + a_{3} - \cdots - a_{n - 1} + a_{n}](/media/m/6/b/0/6b09ca0c22a47d203f504d05b53cb2a3.png)
,
![a_{2} - a_{3} + a_{3} - \cdots - a_{n} + a_{1}](/media/m/7/f/d/7fdbada9d07ca48c74129e736166564e.png)
,
![a_{3} - a_{4} + a_{5} - \cdots - a_{1} + a_{2}](/media/m/7/3/d/73dbb80ee8a3978ce9340f489663f88b.png)
,
![\cdots](/media/m/3/b/3/3b3f59fde5e3bfd745da44e4db64f7e5.png)
,
![a_{n} - a_{1} + a_{2} - \cdots - a_{n - 2} + a_{n - 1}](/media/m/a/b/f/abf4b47713a290f148f62c221b79beae.png)
are all positive. Determine the set of all `nice' integers.
%V0
An odd integer $n \ge 3$ is said to be nice if and only if there is at least one permutation $a_{1}, \cdots, a_{n}$ of $1, \cdots, n$ such that the $n$ sums $a_{1} - a_{2} + a_{3} - \cdots - a_{n - 1} + a_{n}$, $a_{2} - a_{3} + a_{3} - \cdots - a_{n} + a_{1}$, $a_{3} - a_{4} + a_{5} - \cdots - a_{1} + a_{2}$, $\cdots$, $a_{n} - a_{1} + a_{2} - \cdots - a_{n - 2} + a_{n - 1}$ are all positive. Determine the set of all `nice' integers.