Berthier’s xyt-Chains Rated Extreme

This post suggests a limited role for Denis Berthier’s xyt chain in human solving. I found no THLS examples of longer chains not requiring prior xyzt or hidden chains.The last prospect, Royle 17-33442 proves to be both unjustified and unnecessary. In the Sysudoku Order of Battle, the xyt chain is therefore will be rated as extreme, and place among last resorts. Being aware of how they work, you might stumble on a promising one before that, but systematic search is not practical.

17-33442-basic-trAs a reminder, Sysudoku basic derives its candidates with a minimum of unnecessary additions and eliminations. A slink marking bypass uses unwritten slinks to derive clues, then adds pencil marks for box slinks, then adds line slinks, finding subsets in the process.

Berthier number scans first, then removes candidates by subsets, mostly hidden, and box/line interactions similar to slink marking.

The basic solving of 17-33442 took some effort, but not as much as number scanning. And its product is much more ready for advanced solving. That is a major theme of Sysudoku.

17-33442-lm-grid-fake-8On the line marked grid, here is the path of SudoRules first advanced move, as reported in Berthier’s THLS trace. It is supposed to be a c-chain, eliminating 8r4c3.

But it isn’t that. Although THLS displays no candidates, we know that 8r4c4 to 8r2c2 is not a strong link, because the victim 8r4c3 is there to be removed.

17-33442-grouped-anlTHLS does not acknowledge what this actually is,  but it solves the puzzle, without help from xyt chains. It is a grouped 8-chain ANL, confirming 8r2c4 or eliminating 8r1c56, depending on which slink you put in at the top. It does eliminate 8r4c3, as required for Berthier’s xyt-chain, but then you have to stop it.




17-33442-xytNow if we go along with the removal of 8r4c3 without the rest of them, we do get to the same collapse by means of a longer xyt-chain. The xyt chain logic: starting in r1c3, if 8r1c3 is false, the chain removes 8r1c56, as interfering candidates 9r2c2, 9r4c2, 3r4c3,  9r4c1, and finally, r2c4 are removed.  And if 8r1c3 is true, the same two are removed, so they are indeed eliminated, giving us the same decimation of 8’s.


17-33442-collapse-trOne conclusion we can draw is that, not only do longer xyt become tedious and difficult, as Berthier acknowledges, their ultimate success or failure becomes impossible to predict. If you think an xyt is necessary, and you don’t see its completion before starting, you’re making an arbitrary choice, and flirting with trial-and-error.

We’ll never see it happen in advocate examples, but the repairing XY chain under xyt revision may just peter out. Or worse, the chain starting assumption may be contradicted without completing the full chain. In that case, assumption then becomes a confirmed guess. If the xyt chain from this assumption is not simple enough to be readily predictable, it’s a type of trial. Defer the search and find something less nebulous if you can.

royle-17-9373This THLS account on behalf of the real human solvers winds up with Berthier’s xyzt chains, a derivative of the xyt chain with family ties to the Sysudoku iXYZ-wing. I think regular readers will understand why I’m skipping the hidden hxyt-chain.

If you’d like to do sysudokie basic on the first xyzt-chain example in the THLS chapter introducing it, here it is, Royle 17-9373. I didn’t find the xyzt-chain when I got there, but I did find an easier alternative in the regular advanced repertoire.


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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2 Responses to Berthier’s xyt-Chains Rated Extreme

  1. Strmckr says:

    The named human solving types of M, w, s, l, h : wings are all xyt chains
    short versions consisting of 3 strong (grouped or non grouped) links between 2/3 digits
    Aka continuous and discontinuous nice loops.

    Which can and do crack alot of puzzles wide open readily and very common, I wouldn’t dismiss this technique concept as extreme.

    • Sudent says:

      Yes, and an XY-chain is an xyt-chain with no extra candidates, Strmckr. But none of this makes it practical for human solvers to follow up every linked bv pair as a possible xyt-chain.

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