Alternating X-Loops


X-loops of significance are alternating wink and slink, and pure slink loops over candidates of the same number, or such loops almost perfectly alternating.  They produce removals and clues.  Today we go around alternating X-Loops.

First, solutions for diabolical 239.  Hey, isn’t that a 4-chain a skyscraper?  Anyway, the x cells in this panel are actual removals, which  leave a new bv in r5c8, connecting the XY chain to reach both toxic 9-candidates. Good bye, diabolical.

And what about the W wing?  Is it anything more than an X chain?  Well, yes and no. 

You could say it is an X-chain with bv providing the ending slinks.  Certainly the x-candidates being in bv cells is of no significance.  You could say that the W wing is an AIC ending in bv cells.  Do we need to know about it? Nope.  If we know about X-chains and the internal slink in bv, we’re good to go.

Now about X-loops:  X-chains form the three types of loop illustrated below.  Types a and b are called almost nice loops, because they have a single flaw in the pattern of alternating links.  Type c is a nice loop, having perfect alternation throughout.

Actually, we’ll just forget about type a, and look at it as a victim seeing the toxic ends (some call them pincers) of an X-chain.  Been there, and continuing to do that.

Loop b is one X-chain with the same candidate at both ends.  By AIC logic, if that candidate is false as the chain begins, it is also true as the chain ends.  That is to say, it cannot be false, it’s a new clue!  I call type b a confirming X- loop.  We’re accustomed to type “a” removals in this blog, but I don’t have a good straightforward example of a type “b” confirmation at hand.  Maybe you have one for me.  Don’t be shy.  Credits and applause await! Please send it to me in any form imaginable.

There is always an odd number of links in “almost nice” loops.  In the “nice” loop of type c, every adjacent pair of candidates is a toxic set.  In fact, every pair separated by an odd number of links is a toxic set, because all such pairs are end candidates of an X-chain going around one side of the loop. Very nice. 

In an X-loop of odd length, as long as the winks of a loop are an odd number of links away from each other, candidates for elimination and/or removal can be located.  The odd length guarantees that the outer candidates of the winks must be false. You can locate confirmations and removals by tracing around the loop in both directions from the winks.

An almost almost nice loop, with two winks together, and elsewhere, two links between winks, produces nothing.  The inference cannot progress through the double wink.

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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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