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A partial Latin square of order $n$ can be represented by a $3$-dimensionalchess-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 manyrooks as their capacity. That implies a simple capacity condition for thecompletion of partial Latin squares which is in fact the Cruse's necessarycondition for characteristic matrices. Andersen-Hilton proved that, except forcertain 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 beexcluded. Identifying the structures of the chess-board that can be overloadedwith $n$ or $n+1$ rooks, it follows that a PLS derived from a chess-board withat most $n+1$ non-attacking rooks can be completed exactly if it satisfies thecapacity condition. In a layer of a Latin square, two subsystems of a remotecouple are in balance. Thus, a necessary condition for completion of a layercan be formulated, the balance condition. For an LSC, each 1-dimensionalsubspace of the chess-board contains exactly one rook. Consequently, for thePLSCs derived from partial Latin squares, we examine certain sets of1-dimensional subspaces because they indicate the number of missing rooks.

Analysis of subsystems with rooks on a chess-board representing a partial Latin square (Part 2.)