Hi Bill!
Firstly, I apologise for non-standard terminology - I have a mathematical/computational perspective, from which viewpoints the most convenient definition of the
grid includes all cells, so a square grid is of size
N x N, in which rows and columns are numbered from 0, so the effective area in which digits can appear is
(N-1)^2.
Smythe Dakota wrote:There cannot be two adjacent rows, each of which has no black cells (other than the "forced" black cell in the leftmost column). If there were, you could simply interchange the contents of those two rows, cell by cell, and thus produce a second solution.
Consider the example above ("N=9, NB = 56, Unique solution"). The bottom two rows have no interior black cells, yet this solution is the only one that matches the given
hints, or "word sums". It's true that swapping the rows makes no difference to the sums, either horizontal or vertical, but you have to take into account the
entire grid, whose sums in this case are enough to force the values in these two rows to be just what they are!
Smythe Dakota wrote:More generally, there cannot be two "words" (runs) in adjacent rows, one directly beneath the other, which have the same length and which start in the same column.
Again, this is not true, and for the same reason.
Smythe Dakota wrote:Still more generally, there cannot be two words, even in non-adjacent rows, which have the same length and start in the same column, and in which each cell in one word is rook-connected to the corresponding cell in the other.
I think that all your observations are essentially correct, but only if the "words" in question have the same combination of symbols (what I would call a
cycle). If two 6-digit words appear together in a 2x6 pattern, that's only a problem if both of them contained a
permutation of the same values, eg {
1,2,3,4,5,6}.
In the example I mentioned above, the two 8-digit "words" have different symbol sets, one is {
1,2,3,4,5,7,8,9} and the other is {
1,2,3,4,6,7,8,9}.
Cycles certainly are an indicator of non-uniqueness of solution, and your observation on rook-connectivity of the pairwise symbols is suely relevant, but only if the digits match.
In fact there can be even more exotic cases of cycles, e.g. 2 matching digits spread over 3 or 4 rows/cols, or even some more complicated than that!
That's why the "Acyclic cycle" example I show above is of particular interest. The 4 symbols are all different, yet they are still interchangeable!
The consequence of cycles is this: say that I were to generate a grid with a given set of values, then produce the corresponding
puzzle by adding up the sums, and the feed those sums (hints) into a solver (man or machine).
If there were any cycles in the original grid, I would certainly find that the puzzle had more than one solution.
Cheers
Jim White