No THLS Case for Hidden XY Chains

This post and the next demonstrate that the examples presented in The Hidden Logic of Sudoku as requiring “Hidden” XY Chains actually have alternative, and much less demanding, resolution paths in the Sysudoku repertoire.  The burdensome maintenance of symmetry grids is not justified by this book.

In Chapter XV of THLS, Denis Berthier presents a series of increasingly long hidden XY chain puzzles, with reports on the significant removals found by his rule based solver SudoRules. The implication is that hidden XY chains are required for the solution of these puzzles. These examples are presented in support of Berthier’s contention that hidden chains are “an inescapable tool for the advanced player”.

Having gone through the Chapter XV hidden chains puzzles, I am relieved to report that THLS  fails to show that hidden chains are necessary at all.  These puzzles make a nice collection illustrating a variety of advanced techniques encountered in the Sysudoku  Order of Battle (menu above).  I invite you to compare the difficulty of these solutions with just the preparation and maintenance required to try out hidden XY chains. If you would have a program for that, then just think about searching the extra grids it maintains for you.

I’ll not include the usual basic solving traces in this report, but if you want to work the puzzles to the  point of illustration, you will be able to recover the givens and the THLS elaborations from the grids presented.

17-211-aicIn the first hXY example of THLS XV, Royle 17-211,  I was disappointed to see that the neat little red XY wing I found along the XY railway had no victim. Actually, that’s a good excuse to look for a forcing chain between two given points, a candidate seeing one terminal and the other terminal.

There’s an advantage to having a destination, and being willing to take any AIC route to get there.

THLS reports an hXY chain of four cells in rcn space, but three cells of XY chain and a look at the 7-panel gets you this removal.

17-211-xy-chainThe collapse is immediate, but there is a less decisive XY hugging the same rail. It buys two more bv on the way to a less advanced solution.








17-619-xy-anlThe next example, Royle 17-619 was reported as a four cell XY chain, followed by a four cell hidden rcn chain. The THLS XY chain is shown in red. I happened upon the black one, with an second victim. You could combine them to eliminate 9r5c2, but there’s a shorter route through r5c3. It’s a good example of life along the railway.

So now what is coming in place of the hidden XY chain?

 Would you believe a Death Blossom Lite? 

17-619-db-liteThe generous patch of bv, festooned with slink webbing, is typical of THLS examples, but you never see them in the book. They beg to be colored, and that is an apt finish for 17-619


17-619-coloring-1I add a second red/orange cluster for the uncovered 6 and 9 candidates. In classic bridge fashion, blue and red 6 in the same box means green or orange is true, which merges orange and blue.

Maybe you’d like to verify that the merger dooms blue and green(red) wins.

I know its unfair, but unlike Denis, I just can’t hold anything back.


17-520-i148-wingIn the next example, Royle 17-520, another 4-cell XY and hXY pair of chains is waiting, but I get to an irregular i148-wing first. The 18 wing is attached by a forcing chain wink. The removal of 1r4c2 leaves a box/line on 1r1c3.






17-520-coloringNow I have to give in to the bv patch and add coloring. Row 5 prohibits orange and green, so red or blue is true, or both. Anyway, an unlucky 8 sees both. Now it’s possible for a “shortcut” wink based on color to tie together an XY wing (three cell CXY chain) to trap a green soldier, condemning the whole green army. The blue army overruns 8r4c5 and its over.




17-11212Finally, here is the coloring of the elaboration of Royle 17-11212, with its 5-cell hidden XY chain threading through this near-BUG elaboration. It’s a lot harder to interpret in rcn space, where this hidden chain lurks. Here, two green candidates are forced into r3c8 for a blue solution. I added freeforms to indicate the order of coloring. There is little need to construct any chains for this.

Next time, three more puzzles with even more impressive, but completely hidden, and equally unnecessary, XY chains.

About Sudent

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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