How do we show that is $P$ if a $n\times n$ Latin square with $\le n-1$ filled cells, then $P$ can be completed to a proper Latin square?
Here is the definition of a Latin square.
(WHAT I HAVE DONE SO FAR: I have worked examples for $n=2,3,4$ but am unable to find a pattern that is rigorous, or any pattern for that pattern. A step-by step proof would be appreciated- or if you could tell me what facets here are of paramount importance when it comes to my observation.)
Could someone explain the inductive step in the link provided by Gerry Myerson in Theorem 5? I don't get it.