This post examines Bob Hanson’s Bent Naked Subset rules in Sudoku Assistant – Solving Techniques, and provides a more relevant rationale – and a more appropriate name – for his technique. Also, XYZ-wings not covered by the BNS are identified, as well as types of BNS not covered by XYZ-wings.

Last post, I objected to the use of the term “subset” in Bob’s name for this formation. I’ll use the term *n-set* instead, the *n* standing for the number of values in the set of cells. In the name *Bent Naked N-Set* ( or *BNS*) “Bent” refers to the two intersecting units, and “Naked” further restricts the BNS to n cells containing only candidates of n numbers. The cells of the intersection of the units is the “hinge”and the naked set cells in each of the two subunits outside of the intersection are the “wings”. Bob describes two types of BNS. I’ll label them *BNS1*, having exactly one of the *n* numbers common to the two wings, and *BNS0* having no numbers common to the wings. One or none numbers in common is the requirement for toxic set eliminations

For his only example of a BNS1, Bob uses this 154-wing.

Then, where you’d expect a proof of this assertion to be, Bob explains why this simple XYZ-wing removes 4r6c3. His explanation lacks the generality of his BNS1 rule, and is a little off base even for the wing, stating that either value of r2c3 forces 4r6c3 out. Using a forcing chain argument to explain an XYZ-wing, to illustrate a BNS? All backwards.

Bob’s unproved BNS1 assertion reads:

**“If a bent naked subset contains one and only one candidate k that is present in both of its nonintersection subdomains, k can be eliminated as a candidate in any cell that sees all the possibilities for k in the subset.”**

We’re left to decide if he means one and only one candidate, or one and only one value(number) , in each wing. Bob often says “candidate” when he means “number”. I’m guessing the less restrictive “number”.

Is there a proof of Bob’s BNS1 assertion? After showing that an XYZ-wing works, Bob represents a bent naked wing by one of his generalized method diagrams. It represents the naked set candidates in each unit and both units. Bob explains that if all of the candidates of k, the number common to both wings, are removed from A and B, no remaining candidates can be duplicated to make up for the missing number in the naked set. Do you understand what “duplicate” means here? Let me explain what Bob didn’t. I went around in a puzzled state for a long time.

The “one and only” condition of the BNS1 rule is necessary because Bob’s “subset”, the naked n-set, is not a subset. It does contain *n* and only *n* numbers in *n* cells, but some of those cells are in different units, allowing solved cells to contain the same clue number. This generally allows the true k-candidate to be absent from the naked n-set.

With two (or more) common numbers, k1 and k2, the k1 candidates of the n-set are not a toxic set, because in the final solution, a duplicate of k2 true candidate could fill a k1 cell in the other wing. But with only one common number in the wings, no duplicate is available.

This kind of bent naked n-set is a generalization of the regular XYZ-wing whose wings are attached by unit induced winks, including those whose victims see toxic candidates by means of forcing chains.

It does not cover XYZ-wings whose wing-to-hinge winks are constructed from forcing chains, such as this one, found in KrazyDad Insane v.4, b.1, n.5. Here, 3r2c9 sees the hinge 3r9c4 by a grouped forcing chain, creating the 367 wing that leaves three removals when the smoke clears.

Now consider Bob’s other rule, for BNS0. If there is no common wing numbers, the naked n-set behaves like a naked subset. That is, candidates of every number form their own toxic set. That is clearly the case, because all candidates of any n-set number are in the same unit.

The BNS0 is nothing like the XYZ wing, because its wings have no common number.

Bob provides a real BNS0 example. In his example grid, three of the completion candidates are missing, namely theshaded ones 9r4c3, 7r4c8 and 2r9c8. The omissions don’t affect the example. There is a self verifying Sue de Coq Wr4 = 4(1+7)(8+9) +489 that removes 7r4c8.

The BNS numbers are 15789. The wings and intersection are marked in blue, green and orange.Notice that 89r4c8 is left out of the naked set. This in itself shows the naked n-set is really is quite far from a subset. By the BNS0 rule, 9 has to go because it sees all 9-candidates in the naked set. It was very convenient to be able to leave out the r4c8 cell.

Bob points out that this 9 would also reduce Wr4 to 471, reaching a contradiction. That’s interesting, but not actually relevant, because we are looking for logical methods of making removals, not for candidates that cause contradictions.

To summarize:

A Bent Naked n-Set, or BNS, is a set of n cells with candidates of n numbers, contained in two intersecting units. Two types of BNS produce toxic sets:
In type BNS1, the two wings contain one number in common. The BNS candidates of the common number are a toxic set. In type BNS0, the two wings have no number in common. The BNS candidates of each BNS number are a toxic set. |

I have no generalized BNS examples yet. Before analyzing Hanson’s report, I was not aware of that possibility, and thought that Hanson’s bent naked whatever was simply an XYZ-wing. That’s what out-of-the-box thinkers do for us.

But now that you’re on to it, I’d be happy to publish your examples, with full acknowledgement, of course. You can attach to sysudoku@gmail.com .

Next we explore Bob Hanson’s views on almost locked sets. Without looking, you could review the Sysudoku posts of July 2012 and anticipate what he’s going to say about the ALS in the example above.

By the way, there’s a new page, titled Find It. There, you will be able to scan titles and key words on a complete list of posts, to find the one back there somewhere, that you want to review again.