Red Ed wrote:I'm starting to wonder if this isn't completely the wrong approach though, as Guenter is doing some pretty amazing things over on the grid enumeration thread ...

I think that the two problems (enumerating complete grids, and finding the minimum number of clues) are quite different in nature, so that as far as I can tell progress in the former problem will not help the latter.

Of course, it is entirely possible that Guenter, or someone else, will come up with some new ideas for the minimum clue problem, but it will require something pretty spectacular for a proof.

I firmly believe that there are 16s out there somewhere, but they are incredibly sparse... I now have over 350000 distinct 18-clue uniquely completable puzzles and about 2500 17-clue ones. A ratio of 100:1 from 18 to 17, but of course this is likely to be an artefact of my construction procedures and not a legitimate value.

Here is a question: take a valid grid G, and consider the proportion of m-hint subgrids that are uniquely completable.... Let p_G(m) be this function (so in other words p_G(0) = 0, and p_G(81) = 1). Can we say anything at all about this function? Is it significantly different for different grids?

By randomly selecting a largish number, say 10^7, random 27-hint subgrids and checking how many are uniquely completable (usually very few), we can get an estimate for p_G(27). If we have two grids G and H where p_G(27) > p_H(27), then can we conclude that G is a "more fertile" source of uniquely completable subgrids of other sizes, and thus focus more attention on that grid?

Cheers

Gordon