## Bent Naked What?

Here I object to Bob Hanson’s use of the XY-wing and the XYZ-wing, in Sudoku Assistant – Solving Techniques, to introduce his more complicated concept of “bent naked subsets”. This treatment limits the wings to unit based links.  Also I’m calling him out for bending Sudoku terminology in confusing ways.

What attracted me to examine Bob Hanson’s Sudoku Assistant write up in detail, was the apparent abandonment of convention in some of his ideas. Realizing that he inhabits a different Sudoku world, I was after fresh ideas, and a point of reference from which to judge Sysudoku  infrastructure. I’m not disappointed in that, but on the bent naked subset, I have to draw some lines.

It’s good to start with a picture. Bob begins his report section Bent Naked Subsets with this representation of an XY-wing:

Here Bob makes the startling assertion that an XY-wing is a naked triple, a bent one. Wait a minute, let me deal with that. To me, naked triples are locked sets, a.k.a. subsets. Sets of cells and candidates are locked with respect to single units, i.e. lines or boxes. Now “locked” means something else? To Bob, yes. The bent naked whatever turns out to be something much more complicated than an XY wing. I’ll get to that, but right now I simply object to explaining something simple as a special case of something complicated, to be explained later.

Bob does make a proper argument that one of the wing 2-candidates must be true, else they are 1 and 3, leaving the hinge with no candidate. In my neighborhood, that makes the XY-wing’s 2-candidates a toxic set, because seeing both of them is fatal. But that does not make the hinge and wings a locked set. Another 3-candidate in r7 is not removed because it sees both 3’s of the XY-wing. If it sees all 3’s of a locked set, it must be removed, because you cannot remove a number from a locked set. So Bob, you must stay after school and write 25 times (first offense):

XY-wings are not naked triples, and bent naked sets are not subsets.

Also, Bob is getting more red marks for order of battle anomalies in his examples. The problem crops up again in Bob’s example of the XY-wing as bent naked whatever:

In the full grid of the XY-wing example shown above, now transcribed from keypad to slink marking, there are removals for three naked pairs and a slink mark. Does SA find XY-wings first? That’s strange.

My switch to the slinks and winks reflects the Sysudoku view of the XY-wing as a short XY-chain. Any form of alternating chain has a slink on each end, and the end candidates are a toxic pair. If one end candidate is false, the alternating links make the other end candidate true. The outstanding payoff is that any pair of candidates of the same number along the chain, with slinks pointing toward each other, are a toxic pair. A chain can become several almost nice loops. Student Assistant manages without all that, according to Bob.

The payoff for human solvers is well demonstrated by Bob’s example above. The extension of the XY-wing in both directions, into an XY-chain, gains two more ANL toxic sets, and three more removals.

Finding such XY-chains is made easy by a simple technique, using the bv map. In a table of bi-value cells, curves are drawn connecting candidates in the path as they are ordered in a XY-chains. The curves connect like rails, limiting the direction of travel from each connection.

These XY railways can be searched for matching candidates, and corresponding victims. Search for a matching number starts with the bv slink, setting the direction. It ends on a bv slink, making every other matching number along the path eligible to be the matching toxic set partner.

The relevance of XY-chains in this example, where the bv railways reach 19 of the 21 bv, suggests that Sudoku Assistant does not use alternating chains as a solving resource. That may be all right for a computer based machine, but definitely not for a carbon based solvers.

In my next post, I’ll take up Bob’s actual definition of the new entity he inaptly names the bent naked subset. I can’t go along with his treatment of the XY-wing and XYZ-wing as special cases of it. I’ll stand by my earlier definitions of these wings, with a reminder that both wings are limited to unit based links in Bob’s definition of them. That hurts. These wings are standbys of Sysudoku advanced solving, and this blog has extended their use by introducing ER and forcing chain versions of them.