I implemented the k=1 & k=8 rules, and this is "level 1".

Interestingly, these 2 rules are somewhat dual to one another -- perhaps someone else has better insight on this?

i think this is well understood. consider the 9 x 9 "possibility matrix" whose rows are indexed by the nine cells in the row [column, box] and whose columns are indexed by the nine possible values, and whose entries are either 0 ("not possible") or 1 ("possible"). to fill in the nine cells, we need to find a "transversal" of the matrix, which is a set of nine non-zero entries, one from each row and each column. suppose the matrix has the property that the rows and columns can be permuted so that the matrix becomes "upper triangular in block form", i.e. looks like / \ | A B | | 0 D | \ / where A and D are square. then it is clear that any transversal must arise from a transversal of A and a transversal of D . the duality is between the sizes of A and D . alternatively, we can interchange the roles of "cell" and "value" which corresponds to transposing the matrix. then we seek a "lower triangular block form" of the matrix, but this is equivalent to upper triangular by reversing the order of rows and the order of columns. mike