Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be an even positive integer. Show that there is a permutation
![\left(x_{1},x_{2},\ldots,x_{n}\right)](/media/m/5/6/d/56dd85b48c30d84aec8f848385017305.png)
of
![\left(1,\,2,\,\ldots,n\right)](/media/m/3/0/4/304e1944cec541049f468e227d494111.png)
such that for every
![i\in\left\{1,\ 2,\ ...,\ n\right\}](/media/m/5/b/e/5bed8b07c5171c95c24dd859ffeed6d1.png)
, the number
![x_{i+1}](/media/m/0/a/3/0a3d074a8ffa40f9b99c8995140f780d.png)
is one of the numbers
![2x_{i}](/media/m/c/c/d/ccde43d9dd8322025e5356cff29164e0.png)
,
![2x_{i}-1](/media/m/d/2/5/d25810004c40ee121b392c1566f4d55e.png)
,
![2x_{i}-n](/media/m/2/e/f/2ef560e6c8f05471f10b18dfa3ab7acc.png)
,
![2x_{i}-n-1](/media/m/3/3/e/33edceaa8fb3ffea45137b9dab2cc4a8.png)
. Hereby, we use the cyclic subscript convention, so that
![x_{n+1}](/media/m/7/5/2/75214f45c163a8c4377e5d658c1dd4c4.png)
means
![x_{1}](/media/m/2/6/d/26debca594a117cd7ba8f81eb1a27d2c.png)
.
%V0
Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.