## Andrew Stuart’s Guardians

Andrew Stuart is the well regarded author of The Logic of Sudoku, and a Sudoku programmer whose solver and website are valuable aids to puzzle makers and solvers alike. In  www.scanraid.com , also linked as www.sudokuwiki.org ,  Andrew had a page on Rod Hagglund’s “guardians”.  In his Gallery of Strategies,  Andrew now says

“As of March 2010 this strategy has been deprecated. It is a complicated way of looking at what is ultimately a nice loop with off-chain eliminations. It has been removed from the solver. The documentation will remain on this site.”

I’m glad that the text and the colorful examples of Andrew’s original description remain on the site. Hagglund’s guardian is a candidate that cannot be removed because its presence prevents the closure of a slink loop of odd length.  The details are a good follow up to my posts on X-chains.  And also, the guardian story is a good example of how Sudoku theory progresses by means of examples.

By Stuart’s guardian rules, a single guardian, such as r4c7 in his example shown here, is a clue.  My last post cited  Phillip McCollum’s “Sudoku for Everyone” treatment of this pattern as a slink chain of even length ending in a unit, with removals of both ends of the chain. Stuart’s guardian conclusion is the identical result when the chain ending unit has three candidates.

A second guardian rule states “ If there is more than one guardian, any cell that is seen by all of the guardian cells cannot contain the candidate number.”  Here we must translate into sysudoku speak, where candidates, not cells,  see each other. Doing that, we can take Andrew’s assertion to mean that a slink chain can create a toxic set.  Of course, the remaining candidates in the chain ending unit, where the  forced wink of my last post closes the loop, is a toxic set we already know about.  Andrew has something else in mind.

Stuart’s second example, labeled “Double Guardians”,  I am presenting here as two almost nice removals.  For the purpose of illustrating Andrew’s second rule, it can be pictured as two guardians at r7c1 and r8c7, acting together to prevent the wink paths from becoming slinks – creating a slink loop of length 5.  The two guardians  become a toxic set, removing the same two candidates as the X-chains,  r7c7 and r8c3.

So in this instance the guardians did not prove to be uniquely powerful.  In fact, add a candidate in r1 and we have a case in which the X-chains succeed and guardians don’t apply.  But that doesn’t mean Andrew’s assertion is not true.  In fact, it can be proved, and Andrew did so informally in his introduction to the guardian rules.  What seems impossible is being able to add “and no simpler technique can make the same removals”.  That’s kinda my problem with the workbook.  It may not get done in workbook form, for lack of puzzles that get to the state requiring the advanced technique I want to illustrate. Help!

To round out comments on the Stuart’s original guardian page, here is his Guardian 3 example, with three different X-chain removals, each being a way for the two guardians to prevent a slink loop of length 5. The guardian 3 panel appeared on the web page.  My Guardian 4 gives two X-chains duplicating both guardian removals.

To get technical about it, Andrew errs a little bit in his 2010 guardian depreciation announcement.

The single guardian is confirmed by  a slink chain of even length closed by a wink and removing three X-candidates.  In the second example, the double guardian removals are duplicated by almost nice loops.  In Guardian 4, the almost nice loops remove both second guardian rule victims.  In Guardian 3, the almost nice loop removes one of the guardian victims. After that removal, the second victim is removed by . . . an almost nice loop.