dailysudoku.com

[Ben]

So I consider myself an intermediate (maybe advanced intermediate?) sodokuite based on how I solve regular and squiggly sudokus and what I read in these excellent dailysudoku threads.

However, I need some intermediate/advanced level help with the Law of Leftovers. I've searched, read, and understand the discussion group threads about the Law. I've Googled/Yahooed everything I can about the Law. But, there appears to be another advanced level of the Law that I'm missing. I sometimes think there may be a part of the Law that involves overlapping 2 different areas of "innies" and "outties" that I'm not aware of. Or maybe it's just how the line is drawn to make the innies and outties. What really irks me is when using the hints in the Sumocue program. For example, the program will provide a hint of "Law of Leftovers in Columns 1-3" and the program can then implement the hint. Even after the hint step is performed by the program, I can't even reverse engineer how the Law was invoked.

Anyway, if you know of an advanced Law of Leftovers primer/aide or tips somewhere, I'd appreciate it. Thanks in advance and thanks for the great discussions!

Ben

[Steve R]

I think http://www.sudopedia.org/wiki/Law_of_Leftovers sets out pretty much all there is to the logic.

Steve

[Pat]

[wapati]

I'm with Pat, show us a stage were you are stuck.

[Myth Jellies]

The Law of Leftovers is really just a way of looking at and using particular type of constraint set very similar to a fish group (such as an x-wing, swordfish, or jellyfish)

Consider a simple x-wing group formed at the intersection of 2 rows (r1 & r9) with two columns (c2 & c8 ). This x-wing fish group consists of the four intersection cells r19c28. I usually represent this x-wing group as r19/c28 This group is considered "true" for a digit if it contains exactly 2 true candidates. The complementary fish group to this particular x-wing is the 7-fish in the 49 cells r2345678/c1345679. The 7-fish is true if it contains exactly 7 true candidates of a particular digit. Whenever a fish group is "true" for a particular digit, the complementary fish group is also "true" for that digit.

Every fish group also has two sets of cells where fins may lie. One set is associated with the rows of the fish group, and the other set is associated with the columns. For our particular r19/c28 x-wing, the row-fin cells are r19c1345679, and the column-fin cells are r2345678c28. For any particular digit, a fin group is considered true if it contains at least one true candidate of that digit. If a row-fin group is "true" for a particular digit, then the column-fin group must also be true.

The fish groups and fin groups have a conjugate relationship to one another, i.e. either both the fish group and the complementary fish group is true while all fin groups are false, or the fin groups are both true and the fish groups are both false. Since fins and fish are conjugate, they are both strongly and weakly linked, and they can even be colored in a group fashion.

So by now you may be wondering how this all relates to the Law of Leftovers.

Well it turns out that fish group logic can work with nonets instead of columns or rows. For example, the intersection of the first three nonets with the first three rows in a squiggly puzzle forms a fish/constraint group which I label b123/r123. The LoL "innies" are nothing more than the row-fin group of b123/r123. The innies are the cells that are part of rows r123 but not part of nonets b123. Likewise, the LoL "outies" are the nonet-fin group of b123/r123, hence they are the cells that are part of b123 but are not part of r123.

Thus, for a particular digit, the outie fin and the innie fin must have the same truth state, and that state is the opposite of the fish group (actually a frankenfish group).

All of the fun you have with finned fish in normal sudoku, including using them as subpattern components in AICs, you can also have with these fishy constraint sets in squigglies--just keep in mind the difference in what it means for a fish group versus a fin group to be true.