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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.Based on this representation, we apply a uniform method to prove the M. Hall's,Ryser's and Cruse's theorems for completion of partial Latin squares. With thehelp of this proof, we extend the scope of Cruse's theorem to compact bricks,which appear to be independent of their environment.  Without losing any completion you can replace a dot by a rook if the dot mustbecome rook, or you can eliminate the dots that are known not to become rooks.Therefore, we introduce primary and secondary extension procedures that arerepeated as many times as possible. If the procedures do not decide whether aPLSC can be completed or not, a new necessary condition for completion can beformulated for the dot structure of the resulting PLSC, the BUG condition.

Another proof of Cruse's theorem and a new necessary condition for completion of partial Latin squares (Part 3.)