Improvisations on the Hall marriage theorem: Completing Latin squares and Sudokus
The key concept of our discussion is that of a perfect matching (PM) in a bipartite graph. The expansion condition in Hall's marriage theorem can be extended to an unbiased 2-sided one. This enables an alternative (and simpler) proof of Evans' (proven) Conjecture:
A partial nxn Latin square with n-1 dictated entries admits a completion to a full Latin square.
PMs are used to successively fill the square by rows, columns or diagonals. Latin square tables correspond to quasi-groups; the ones corresponding to groups are only a tiny fraction of them, as n grows. However, for Sudoku tables of order mnxmn, the completion (say by diagonals) usually fails, even if there are no dictated entries, unless they are conjugates of a twisted product of two groups, of orders n and m.
An open problem for sudoku lovers: Is there a sudoku square (of any order) which is not a conjugate of a twisted product of groups?
No prior knowledge needed.