## To Nishio or Not to Nishio

This post examines Paul  Stephens’ favored form of Nishio, from his two current books on advanced Sudoku solving.  The analysis of the Addicts 140 determines the relationship between Stephens’Nishio and a simple test for grouped almost nice eliminating loops on the Sysudoku X-panel..

In the introduction to his year of instruction in Mastering Sudoku Week by Week, Paul  states his number one golden guideline of Sudoku solving.  It is “1. Never guess.”  Yet, even though standard advanced solving techniques are omitted, he spells out in Weeks a favorite trial and error method, saying that it is the method named after the Japanese puzzle master Tetsuya Nishio.  Paul characterizes his Nishio as “controlled trial and error”, pointing out that it can only prove the guess wrong, never right.  But then he recommends guessing the value of an arbitrarily chosen bi-value  cell, often proving the bv partner right.

Paul’s Nishio procedure is widely known among trial and error methods, but I am reporting here that it has a better use as an X-panel analysis mechanism for finding grouped x-chain ANL removals.

I suggest giving a goal directed name for it in this context,  the ANL Test.  If conducted in the controlled way that Stephens prescribes, the procedure either reaches no conclusion, or it establishes guideposts for an almost nice loop removing the test candidate in the position of the Nishio candidate guessed to be false.   It is a humanly practical way to find very obscure eliminating almost nice loops on the X-panel.  The ones constructed from X-chains with grouped nodes.  The many  options in these constructions  try  human patience.  If used to find such extreme ALN’s after advanced methods are exhausted,  Nishio procedure can be welcomed as a tool for human solving.   Proof by pudding on each individual case.

In Weeks, Paul’s Nishio is demonstrated on a full grid, but we need only the left panels below.  Start by choosing the candidate to be eliminated. In his demonstration, Paul chooses the bv r1c2 with numbers  2 and 7, choosing 7 as the number, to be proved false.

The left 7-panel shows Paul’s “red pin” candidate shaded red.  Now mark out all candidates the chosen candidate sees.  These are “x”ed  in the right panel.  If any unit then has only marked out candidates, a contradiction has been found, and the chosen candidate is removed.  If, as in the case here, units remain with a single unmarked candidate, then mark out all remaining candidates these single candidates see.

In the right panel  above the letter “o” marks these.  A unit, r8, is completely marked out.  The test has succeeded in removing the chosen candidate (but not the single remaining candidate in r3c6).  Repeat the marking as long as single candidate units remain each time.  If no units have been emptied, the test has failed, but this does not prove that the chosen candidate is true.  If the process continues isolating single unmarked candidates until every unit has an single unmarked candidate and a pattern has been formed, the Nishio has failed, and no conclusion is made.

Looking at the left panel again, did you notice the finned 7-wing that removes Paul’s Nishio victim, but without any guesswork?  The unfortunate red 7 finds itself in the box with the fin. The other would be victims escape.

A similarly inconvenient truth emerges  in the Week 32 puzzle that is intended for Weeks readers to test their Nishio skills.  In the puzzle notes Paul says, “This one hits a brick wall, with hardly any squares solved by the time you need the Nishio.  If you can solve it, then you’re well on the way to becoming a true Sudoku expert.”

Well, yes and no.  Box marking brings no clues, but 20 slinks.  Line marking is tough, with a naked single, but four 5f lines, four 6f lines and four 7f lines. No bv scan results. Then on the 1-panel, you get this, a finned 1-wing immediately collapsing Week 32.  No brick wall requiring Nishio here.

But wait, there is more to the Nishio demo example.  The demo panel is reproduced below on the left.  On the right is the almost nice loop constructed directly from the demo panel.

Start forcing chains from the chosen candidate to the rejecting unit and to single candidates that the chosen candidate identified.  Each single candidate sees a next single candidate it identifies,  and a slink can be drawn.  The forcing chains arrive in the rejecting unit on a wink, and  there are two of them, so a connecting slink in the rejecting unit can always be drawn.

Did we just prove that every successful  Stephens Nishio creates an x-chain almost nice loop? Almost.   Ironically, Paul Stephens writes elsewhere that a successful Nishio provides an X-chain ANL.  He actually makes this assertion in Addicts in the notes to his most difficult puzzles.  But without showing how it happens, and obscuring it by his interpretation of x-chains as links between “squares” and his inexplicable lumping of almost nice loops(ANL)  into “Nice loops”.

In my post of November 20, 2012, Nishio Forcing Chains?,  I deal with a similar misinterpretation of almost nice loops as Nishio tests by Andrew Stuart.  Andrew turns it around by taking an ordinary ANL, even one formed by an AIC chain, and claims it to be a Nishio on some bv candidate on the chain.  Andrew claims a distinct solving technique called a Nishio forcing chain, consisting of starting at a candidate guessed to be false, and extending two forcing chains from that candidate until they meet.  For X-chains, Paul’s Nishio procedure is perhaps a quick way to determine if the “Nishio forcing chain” is going to work, but my answer to both Stephens and Stuart  is: FIND THE ANL!  In this context, you   may be interested in my answer to a reader who read into that post that my aversion to trial and error is ideological. No, ideology is not logic.

Whew! That felt good.

Along with the finned X-wings, we might well have found the simple ANL in Weeks 32 without recourse to a Nishio on one of its candidates.  A better case for the ANL test is made by puzzle 140 in The Sudoku Addict’s Workbook , for which Paul Stephens strongly recommends the Nishio.

Next post,  we will continue with Addicts 140 and the ANL Test.  If you are extremely ambitious, you might like to do the basic solving on Addicts 140. To participate in the hunt, you can skip the unproductive bv scan and go directly to a full x-panel.