## Rank’s Little Gotcha

Here we encounter the taxing complication in rank theory which occurs when base sets are allowed to intersect.  This further limits rank theory as a basis for human solving methods.  This blog opts out of rank field theory.

As I finished my review of the extreme puzzles on KrazyDad.com and begin to think about the puzzles of the forum fascination class, my label for them was “super extreme”.  Exploring the ways they have been solved, I’ve been coming across a more appropriate label, namely “monster”.  I like it because it’s not that the constraints on candidates are more complicated.  Rather, it’s that the deluge of candidates drowns out all logical perception.  I’ve used the term “fog” to describe it. A monster overwhelms all reason and hope, hiding its vulnerability behind a screen of terror.  Small victories (eliminations) don’t seem to count.  It keeps on coming. Yeah, that fits.  Let’s go with monster.

So what is a computer free defense against monsters?  The last two posts reviewed the SudokuOne introduction to basic rank theory, which is where I started looking.  I tried to read further, but couldn’t concentrate.  I needed motivation.

Here’s a  truckload of motivation, the first in a series of 67 Set Link Graphs (SLG)  that SudokuOne’s program used to unlock Golden Nugget.  Just search for SudokuOne Golden Nugget.

In the sysudoku grid version of the SLG, truths are solid curves; covers, dotted ones.  Four colors link numbers 1, 2, 4 and 7.

Shockingly different from SudokuOnes’s first two examples, isn’t it?  No slink truths. Monsters don’t give you many slinks or bv.

The base count is 4+3+3+3 in rows  plus 1 column(6)+3 in the NE box(247) = 17, and there are 3+3+3+3 column + one box (NW) +5 cell =18 cover sets.  SudokuOne says the “raw rank”  is 1, but the removal calls for a rank of zero, because the 3-candidate victim is in one cell cover set, and is not  a base candidate.

The explanation for the discrepancy is somewhere in SudokuOne’s “General Logic” treatise.  Part 2 starts with the effects on rank when base sets intersect.  This is disconcerting, because the proofs for rank eliminations of the last two posts clearly require that base sets do not intersect.  In the SudokuOne SLG above,  truths in row 1 and row 3 intersect with box truths of the same numbers.    In Part 2, “General Logic”  explains how rank varies candidate by candidate when base sets intersect with base sets and cover set (link) intersections.  There are two types of intersection, which divide the plumbing into regions of differing rank.  The complex details are summarized in  “A basic guide to general logic” on the SudokuOne site.

So let’s think about this.  First of all, there is no additional infrastructure such as slinks, winks, AIC, and patterns to guide the human solver in the selection of base truths and cover sets from among a large number of possibilities.  Those possibilities expand when intersections are permitted.  Further, it turns out that the number of base and cover sets is indefinite, and the rank field of every prospective SLG of monster class must be carefully calculated to find and justify eliminations.

My conclusion is that the set link graph isn’t a humanly tractable approach.  Rather, it’s an alternative way to program computers to solve Sudoku puzzles.  Since my focus is human solving, I see no advantage in going there. Reading SLG solutions is therefore a waste of my time. I will keep the “no base intersection”  base and cover rank theory in mind as a possible basis for human solving methods, but I’ll have to have some evidence that the set link graph approach is humanly tractable and goes beyond  the tools we have.

Can a human find the removal of 3r7c9  above?  Yes.  The remaining December posts will examine a forum refined approach, the exocet.  Can a human conquer a monster unaided by computer search?  Yes.  Easily?  No, but that’s why humans generate Sudoku monsters.

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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