  A partial Latin square of order $n$ can be represented by a $3$-dimensional
chess-board of size $n\times n\times n$ with at most $n^2$ non-attacking rooks.
In Latin squares, a subsystem and its most distant mate together have as many
rooks as their capacity. That implies a simple capacity condition for the
completion of partial Latin squares which is in fact the Cruse's necessary
condition for characteristic matrices. Andersen-Hilton proved that, except for
certain listed cases, a PLS of order $n$ can be completed if it contains only
$n$ symbols. Andersen proved it for $n+1$ symbols, listing the cases to be
excluded. Identifying the structures of the chess-board that can be overloaded
with $n$ or $n+1$ rooks, it follows that a PLS derived from a chess-board with
at most $n+1$ non-attacking rooks can be completed exactly if it satisfies the
capacity condition. In a layer of a Latin square, two subsystems of a remote
couple are in balance. Thus, a necessary condition for completion of a layer
can be formulated, the balance condition. For an LSC, each 1-dimensional
subspace of the chess-board contains exactly one rook. Consequently, for the
PLSCs derived from partial Latin squares, we examine certain sets of
1-dimensional subspaces because they indicate the number of missing rooks.
