  A $d$-dimensional generalization of a Latin square of order $n$ can be
considered as a chess-board of size $n\times n\times \ldots\times n$ ($d$
times), containing $n^d$ cells with $n^{d-1}$ non-attacking rooks. Each cell is
identified by a $d$-tuple $(e_1,e_2,\ldots ,e_d)$ where $e_i \in \{1,2,\ldots
,n\}$. For $d = 3$ we prove that such a chess-board represents precisely one
main class.
  A subsystem $T$ induced by a family of sets $<E_1,E_2,\ldots ,E_d>$ over
$\{1,2,\ldots ,n\}$ is real if $E_i \subset \{1,2,\ldots ,n\}$ for each $i \in
\{1,2,\ldots ,d\}$. The density of $T$ is the ratio of contained rooks to the
number of cells in $T$. The distance between two subsystems is the minimum
Hamming distance between cell pairs. Replacing $k$ sets of $<E_1,E_2,\ldots
,E_d>$ by their complements, a subsystem $U$ is obtained with distance $k$
between $T$ and $U$. All these subsystems, including $T$, form a partition of
the chess-board. We prove that in such a partition, the number of rooks in a
$U$ and the density of $U$ can be determined from the number of rooks in $T$
and the number of cells in $T$ and $U$ and the value of $(-1)^k$. We examine
the subsystem couple $(T,U)$ in the $2$- and $3$-dimensional cases, where $U$
is the most distant unique subsystem from a real $T$. On the fly, a new
identity of binomial coefficients is proved.
