  Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$,
define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of
$\mathbb{Z}_n$ are written in a circle. So \[ ||i,j||_n=\min\{(i-j)\bmod
n,(j-i)\bmod n\}. \] A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$
is \emph{$(s,k)$-clash-free} if $||\pi(i),\pi(j)||_n\geq k$ whenever
$||i,j||_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close
elements (at distance less than $s$) to a pair of elements at large distance
(at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can
be reformulated in terms of certain packings of $s\times k$ rectangles on an
$n\times n$ torus.
  For integers $n$ and $k$ with $1\leq k<n$, let $\sigma(n,k)$ be the largest
value of $s$ such that an $(s,k)$-clash-free permutation of $\mathbb{Z}_n$
exists. Strengthening a recent paper of Blackburn, which proved a conjecture of
Mammoliti and Simpson, we determine the value of $\sigma(n,k)$ in all cases.
