## Higher Rank Theory

This post extends the proof of the rank theorem to the rank = r >0 case, and finds a sysudoku interpretation of a rank 3 removal from the General Logic page of www.SudokuOne.com .  The method is Andrew Stuart’s cell forcing chains, judged earlier to be impractical for human solving.

A rank rule for rank >0, say rank = m, can be stated this way:

As proof for rank = r > 0, we can assert that

1. Each base set contains one true candidate.
2. Every true base candidate is contained in at least one cover set.
3. Since all cover set candidates see each other, a cover set can contain no more than one true candidate, so there are n cover sets containing a base candidate.
4. At rank = r > 0, a candidate in m+1 cover sets must be in at least one of the n cover sets containing a base candidate.
5. Any such candidate not in the base is not a true candidate, but sees one.

What this means in the plumbing example of last post, is that the orange candidate 5 is in no base set, but is in four cover sets.  To check this, we start  with  SudokuOne’s 2-D representation of the rank 3 removal example.

Despite the keypad marking, we can tell from the base and cover sets chosen which sets are slinks, and which cells are bv, although other candidates are not identified.

There are 7 base sets and 10 cover sets.  The red 5-candidate is in four cover sets,  including the 35 bv of r2c2, and is therefore eliminated.  In considering the difficulty of finding this removal, we need to recognize that many other cells could have been chosen as base sets, other slinks could have qualified other units as base sets, and r2c2 could have been chosen as a base set.  Also, the scan could have involved other sets of numbers.

Actually, the array of slinks one wink away from the r2c2 5-candidate suggests another reason for its removal.  It is a beautifully simple case of a quadruple cell forcing chain.  Left in, it forces out all four candidates of r5c5.

I rejected searching for such cell or unit forcing chains as a human solving method in earlier posts.  The example doesn’t increase my confidence for a human’s finding this removal by the rank  method.

The next post shares  my observations on how an extended form of rank theory called link  set theory is applied to the solution of the Golden Nugget.  As a preview, you might tackle Part 2 of SudokuOne’s “General Theory” page.