If $n>0$ is an integer, let $[n]=\{1,\ldots,n\}$. Let $S_n$ denote the set of all permutations (bijections) $\pi:[n]\to[n]$.

For which positive integers $n$ is there a bijection $\Phi:[n!]\to S_n$ such that for all $x\in[n!-1]$ we have $ (\Phi(x))(k) \neq (\Phi(x+1))(k) \text{ for all }k\in [n]$?

**Note.** This can be formulated in the language of Hamiltonian paths: Put a graph structure on $S_n$ saying that $\pi_1,\pi_2\in S_n$ form an edge if $\pi_1(k)\neq \pi_2(k)$ for all $k\in[n]$, and find a Hamiltonian path in this graph.