## XY-Loops

Denis Berthier in his Hidden Logic of Sudoku,  makes the statement that XY-loops are worthless.  Since we had such productive fun drawing curves through the bv cells – including XY-loops – back in the posts of December 2011, I should explain why sysudoku solvers  keep on drawing them, loops and all. It’s also time for an update on differences between XY-loops and the X-loops of recent posts.

One notable difference is that confirming loops, the almost nice loops yielding a clue,  cannot be formed by XY chains.  The links between bv cells must function as winks.  If you have thought about the possibility of using the internal slink of the bv twice to confirm the starting candidate, as in the diagram below, forget it.

Here is a starting  bv cell, where “o” stands for the starting candidate, and “x” stands for the candidate of the outgoing link.  By XY-chain logic, if “o” is false then “x” and every succeeding outgoing wink candidate is true, going out to last cell before the starting one.  The bv partners receiving incoming winks,  including  “x” in the starting cell, are false. Fortunately, quantum mechanics does not extend to Sudoku logic, and one candidate cannot be true and false at the same time.

In AIC chains, there is the possibility that an incoming slink connects to “o” combining with the internal slink for a confirmation.  This occurs, but not in an XY chain.

Berthier classifies the forbidden XY- loop above as a pseudo xy-loop,  and states, though without proof,  that “a pseudo xy-loop allows no conclusion”.   Denis does not cite his pseudo xy-loop for the violation above.   He does say that a pseudo xy-loop is not a “full xy-chain”, which he defines as every cell  having a left(incoming link) candidate and a different right(outgoing link) candidate.

The curves we draw to tame XY-chains contain Berthier’s full XY-chains exclusively, with all cells configured as shown at left.  They can form loops, and candidates within them do form toxic sets.  There is no cause for worry about such loops that form in the bv map as you draw connecting chains.  We never go all the way around a loop in search of toxic sets, which Berthier does prove to be “logically useless”.  Our curves, branching in and out of loops, creates more chains with toxic ends.  There is no need to examine loop properties of XY-loops for candidate removals, as we did with slink loops.  Every XY loop removal is justified by an XY-chain within the loop.

You might have wondered why each branch and merge in out XY-webs goes in a single direction.  The diagram above accounts for it, and this is what maintains full xy-chains throughout.

Introducing his analysis, Berthier does make the scary statement that “xy-loops are useless”, but he is careful, as always in Hidden Logic of Sudoku, to define “useless” to mean generating no resolution (solving) results unavailable by simpler means. We agree, in our case by regarding XY-chain eliminations as simpler than XY-loops.  Berthier makes clear that the purpose of his work is better understanding of the logical beauty of Sudoku, and more efficient and penetrating computer based solving. He is not specifically seeking human engineered manual solving methods, as we are.   Berthier has made excellent contributions in Hidden Logic of Sudoku,  and continues to instruct the community on the logical realities of the most purely intriguing of puzzles.

Sometime back, I  promised the XY network produced by the Washington Post 6-star’s bv’s.  The connections between chains create new toxic sets, and several loops.

Now let’s compare solutions to Andrew Stuart’s little gems. First, the one we were exploring together. If you checked with Andrew’s scanraid site, you might have had this variation, with the same victims.

On the 6-panel, you actually may have found these answers to the questions posed by Andrew on his site, and me in the last post: