This page describes a family of elimination methods based on n-sets in a bent region, the union of a box and a line. An n-set is a set of n cells containing candidates of exactly n values. By the numbers, the cells of an n-set contain exactly n true candidates, but along with other candidates of the same values. But n-set cells are allowed in remainders, so an n-set is not a naked subset.
In the sysudokie order of battle, we search for bent n-sets as we look for Sue de Coq chutes. When other advanced methods run dry, however, we might repeat this scan on the altered grid, before bringing deferred trials into court. We’re looking to find a Bent Naked n-Set 0 (BSN0), a Bent Naked n-set 1 (BNS1), or a Bent Almost Restricted n-Set (BARN), with n above 3.
To spot them, scan along lines for small numbers of the same values, then for additional cells in remainders of the line. Then test the set for the following:
Bent Naked n-Set 0 (BNS0)
For what: An n-set with no common value in the remainders.
Do what: Remove any candidate that sees all candidates of its value in the n-set.
Why it works: The n-set is locked. With no candidates in common, all candidates of every value see each other. Every set of values is a toxic set.
Bent Naked N-set 1 (BNS1)
For what: An n-set with exactly one value in both remainders.
Do what: Remove any candidate seeing all of the “bent” n-set’s candidates.
Why it works: If there are two or more common n-set values, not every value need be present in the n-set, because another common value can take it’s place. The candidates of a single common value must include a true value.
Bent Almost Restricted N-set (BARN)
For what: An n-set in which only one value “bends”, with candidates in both remainders.
Do what: Same as the BNS1.
Actually, the BARN is another way of spotting a BNS1. It’s just that this characteristic was used in Andrew Stuart’s ‘Bent’ Sets, on the popular Sudowiki Strategies page, where the BARN is taken to be another form of WXYZ-wing. It’s a fact that the BARN can be interpreted as a particular and peculiar form of WXYZ-wing, but you would never look for the wing by identifying n-sets. Andrew includes a link to my protesting post.
Above descriptions of the BNS0, BNS1 and BARN were arrayed together for comparison and reference. Now for some examples:
I met the BNS0 and BNS1 in a review of Robert Hanson’s report on his Sudoku Assistant solver. My post, Hanson’s Bent Naked n-Set, includes his example here of a regular XYZ-wing interpreted as a BNS1 with n = 3. The bent region is defined by chute Wc3, and remainder n-set values are 14 and 45, with 4’s in both remainder cells. The victim 4r6c3 sees all 4’s of the 3-set. As a BARN, 1’s are restricted to c3, 5’s to the W box, and the bent 4’s are toxic.
In his Sudoku Assistant report, Hanson has a striking BNS0 example. The bent region is Wr4, and the 5-set remainder values are 17 and 589. No common value, and this makes the sets of 1, 5, 7, 8 and 9 toxic.
Three BNS1/BARN examples are currently featured in the aforementioned Andrew Stuart’s Bent section, under “WXYZ-Wing”.
Here is his Example 4, transcribed into sysudokie. The bent region is Wr6, and 8 is the single value in both remainders. Three 8 candidates see all of the 5-set 8’s.
Also one of the 5-set 9’s must be true, and all of them are in r6, a bonus elimination.
Finally, let’s note that the generalized Sue de Coq is a close relative to the bent family. It’s based on the bent region, with a chute and two remainders. The disjoint cells mimic the BNS0 requirement. A minimum number of disjoint cells sets up a particular form of BNS0, while extra disjoint cells expand the size of the n-set.
I admit to neglecting the bent family in the reviews, and will be looking to update some solutions with a closer look. Please comment when you find instances that I can correct to your credit.
Next in the order of battle will be the XYZ-wing, and its bigger but rarer variation the WXYZ-wing mentioned above. Also this begins the use of solving templates with the bv and xyz maps.