Beyond the Rectangle


The alternative deadly uniqueness formations of the Sudocue Guide are detailed in this post.  Also, there is a UR checkpoint on Sheldon’s Master Class # 3 that goes beyond the UR cases surveyed in recent posts.

I can’t leave the UR without mentioning  that, besides the rectangle, there are other bv patterns that are subject to deadly number interchange.  Of course, its going to be more than a mention.  I have to be me.  In addition to the rectangle, the following deadly interchange formations are reported in the Sudocue guide.

unique pats 1First is a deadly formation of 6 cells in three aligned boxes.  It’s represented here  as a deadly coloring cluster of 6 bv, an inner BUG, that could spoil the puzzle maker’s reputation for uniqueness, were it not for candidates somewhere in these cells that prevent this coloring.

Note that I’m avoiding the use of the word pattern, as in POM or LPO.  There are two patterns  covering the three boxes of this formation.  Five more formations are obtained by exchanging the middle box with left or right boxes, and by  interchanging left and right boxes for three more.

There are no candidates outside of the formation that are disturbed by interchanging the blue and green candidates. The slinks of the coloring have wiped them out.  Your focus is now on the removal of candidates in the cells of the formation, which enable the full coloring, and the number interchange, to occur.

unique pats 2A second formation of six cells in three boxes is possible.  Of course, six cells sharing two bv numbers is pretty rare in itself. You don’t need to think about it until a cloud of identical bv descends on you.  There’s no need to worry  about the many variations of this formation.  It will be very easy to recognize.

unique pats 3The inner mathematician in you might be buried deep, but I bet it is hankering to know if these mildly threatening formations extend beyond three boxes.  Sudocue’s answer is yes. Combining the two three box formations gives us an obvious four box formation, with  many positional variations.

unique pats 4We can cover one more box, as seen at left, but then the 69 bv in additional boxes will not connect slinkwise with the ones we have. Five boxes five, weirdly enough, seems to be the limit.

Well, I got carried away, didn’t  I?  It’s very doubtful that you will ever encounter these deadly formations in a single solution puzzle.  But it’s good to be aware that the rectangle is not the only possibility.

How did it go with Sheldon’s Apprentice Puzzle #3?  Most of the solving occurs in a long, but pretty straightforward line marking, taking you to the UR.  This really calls for a 2D trace.  In this one, some effect lists are run vertically to keep the trace on the page.

Sheldon 3 LM trace

After the monumental marking of the naked single on c3, we get  a 5-wing on r6, which we keep marked on the grid as line marking continues.  Then on the last row, a 4f, the unnamed UR appears, in r23c24.  What are we going to do with that? Are we going to be fired?

Sheldon 3 UR 1Wait a minute.  There are 2-candidate slinks bordering 4 in one of the bv corners.  Taking the other bv as the single corner, Stuart’s hidden corner analysis applies! You can now verify that this 4 will drive the rectangle deadly.

Removing the 4 doesn’t collapse the puzzle,  but the N2 sets up a avoidable rectangle!  Now it’s clear we have to get 5 into r2c4, or 3 into r3c2.

Sheldon 3 UR 2The first deadly rectangle 5-pill is prescribed by doctor 5 in r2c1 and the alternative 3-pill, by doctor 3  in r3c3. Fortunately,  these two candidates confirm each other.  If one is true, they both are.

This means that both need to be removed to avoid the deadly rectangle.  The puzzles collapses quickly with these removals and we lowly apprentices are saved!  Trump us not.

Sheldon 3 UR 3You may want to compare the UR analysis required in this Sheldon puzzle  with what Master Class has to say about the unique rectangle methods.  Did I miss an easier one somewhere?

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About Sudent

My real name is John Welch. I'm a happily married, retired professor (computer engineering), timeshare traveling, marathon running father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. The blog is about Sudoku solving. It covers how to start, basic solving to find candidates efficiently, and advanced solving methods in an efficient order of battle. It is about human solving methods, not computer solving.
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