Michael Kleber's post led me to wonder about a related issue: Suppose a sudoku has *exactly* 2 solutions. Then what is the maximum number of boxes that can differ in the two solutions? I'd like this question a bit more if it were imo "purer": The whole sudoku idea could be played based instead on finishing merely a Latin square (each row and column of an NxN square must contain all N of a set of N distinct tokens). So here's a question: What is the set comp(N) of numbers C such that there exists a partially completed NxN Latin square having exactly C completions? I strongly suspect that if N >= 4, then 2 belongs to comp(N). In which case we can go back to the original question for each N and ask, Given an NxN partially completed Latin square having exactly 2 completions, what is the maximum number of squares that can differ between its completions? --Dan