Interesting, MJ, although the case you presented is incomplete. That 23 cell has to be a 123 because of the two other uniqueness patterns in there; there's no way you can eliminate that 1 without T&E, which presumably you'd use uniqueness to
avoid. (And in any case, uniqueness tests for a rectangular form can still be run if you're missing a candidate from one cell; the rules change only slightly.) So really, you have this situation:
- Code: Select all
12 |123 13
12 |12 .
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. |13 13
But to use this pattern, we need extra candidates in some of the cells.
All things considered, I'd call this a warped 3x3. In a normal 3x3 you have 3 rows, 3 columns, 3 boxes, and 3 candidates per box for 9 cells total. Here the latter restriction (3 candidates per box) is gone, but the rest of it holds up. This suggests that for higher-order uniqueness tests, the N candidates per box/column/row rule is not strictly necessary.
That being the case, you can apply all four known uniqueness tests using modified rules. I present here the generalized NxN uniqueness test:
Generalized NxN unique grid test:Requires N candidates in exactly N columns, N rows, and N boxes. The N candidates will be called set S. Any unfilled cell in the NxN grid which contains members of S is a pattern cell. Roof cells are any cells in the pattern which contain only members of S, at least 2 candidates. Floor cells are pattern cells that contain at least one member of S, along with other candidates. For the uniqueness test to apply, all of the floor cells (F) must occupy the same box and/or line, which we'll call G (their common group). It is not necessary for all members of S to appear in F, as Myth Jellies' pattern demonstrates in the very first box and column, so the candidates from S that do appear in F are set T; Tn is its size, which is also the size of F. It is also possible for two G's to exist; run tests 2-4 separately for each.
Test 1: Only one floor cell exists. Eliminate S from its candidates.
Test 2: All floor cells share the same candidate X, but no others except members of S. Eliminate X from all cells in G-F.
Test 3: The set of all floor cell candidates combined is T+E, where E represents the extra candidates. If a partial naked subset size M is found with all the members of E, in M-1 cells in G-F, or a complementary hidden subset size M is found with all the members of T, in M-Tn+1 cells in G-F, perform eliminations within G-F as if a full subset was found. (Subset solvers: Pretend all F are T+E, except one cell which is just E. Then look for subsets accordingly.)
Test 4: Within G, all members of T except for one candidate, X, are constrained to F. Eliminate X in F.
Note that the loop form currently under discussion can also be generalized in this way. In the loop form, each row or column or box in the pattern will have N or fewer pattern cells. It's a lot trickier to spot such pattern cells, however.
