  In this paper we study a tour problem that we came cross while studying
biembeddings and Heffter arrays, see [D.S. Archdeacon, Heffter arrays and
biembedding graphs on surfaces, Electron. J. Combin. 22 (2015) #P1.74]. Let $A$
be an $n\times m$ toroidal array consisting of filled cells and empty cells.
Assume that an orientation $R=(r_1,\dots,r_n)$ of each row and
$C=(c_1,\dots,c_m)$ of each column of $A$ is fixed. Given an initial filled
cell $(i_1,j_1)$ consider the list $
L_{R,C}=((i_1,j_1),(i_2,j_2),\ldots,(i_k,j_k),$ $(i_{k+1},j_{k+1}),\ldots)$
where $j_{k+1}$ is the column index of the filled cell $(i_k,j_{k+1})$ of the
row $R_{i_k}$ next to $(i_k,j_k)$ in the orientation $r_{i_k}$, and where
$i_{k+1}$ is the row index of the filled cell of the column $C_{j_{k+1}}$ next
to $(i_k,j_{k+1})$ in the orientation $c_{j_{k+1}}$. We propose the following
"Crazy Knight's Tour Problem": Do there exist $R$ and $C$ such that the list
$L_{R,C}$ covers all the filled cells of $A$?
  Here we provide a complete solution for the case with no empty cells and we
obtain partial results for square arrays where the filled cells follow some
specific regular patterns.
