This post brings up an interesting question about the multiple solution puzzle: Does its ambiguity prevent strictly logical solving from reaching a solution?
Introducing the first selected review puzzle of the A.D. Ardson Very Hard v.2 collection on April 11, I accidentally omitted a given. Ardv2 38 should have a given 7r7c6. The post of April 18, traced the resolution, with missing given, to a match of the Ardson solution, with given.
A diligent reader, Gerald Asp, who checks his transcription by running the puzzle through Andrew Stuart’s solver, got the unexpected report that my version of Ardv2 38, missing 7r7c6, has 63 solutions! In my trace, 7r7c6 is determined on the final color wrap. So either my resolution transgressed strict logic at some earlier point, making an unwarranted assumption about the truth of a candidate, or I was mistaken in the review of Denis Berthier’s The Hidden Logic of Sudoku when I said, in the post THLS vs.Unique Rectangle of 12/13/16, that uniqueness is proved when a solution is reached with no assumption having been made about the truth of any candidate.
Inherent in that statement is the belief that resolution between the multiple solutions cannot be achieved without such an assumption. But in the traced resolution of April 18, where is that assumption? A coloring cluster does not make such an assumption. It relies on the fact that in the slink network of the cluster, candidates of one color are true, and those of the other are false. Hopefully, one color will be found to violate the rules of Sudoku, a color wrap.
Is such an assumption made when I use an AIC to constitute a strong link, extending the cluster? This form of slink fulfills the definition of a strong link between candidates.
The question in play here has nothing to do with my disputing Denis Berthier’s claim that UR methods assume uniqueness. These methods do make a much more limited assumption. They assume only that multiple placements of the simple rectangular form will not be allowed by a professional puzzle composer. This has nothing to do with the composer’s willingness to run his solver long enough to be sure that no multiple solutions of any form exist.
Well, help me out here. Next post, we will move on to resolve Ardv2 278, but do look for comments on this post in the coming weeks. Perhaps one of them will be yours.