If nothing else, Sudoku solvers have imagination when they name techniques!
This idea is completely analogous, logically, to pairs, triplets, etc. Forbidding matrices
expose this analogy.
Here is the idea as an X wing:
- Consider any one candidate
- Consider any two distinct rows
- Let the candidate be limited to no more than two cells in each of these two rows
- Let this group of no more than 4 cells share exactly two columns
- The candidate is forbidden from all the cells in those two columns outside of those 4 cells
Because of symmetry, one can exchange columns for rows in this
rule.
X wing example
Note the following:
- All possible locations for 3's in row 9 are f9,h9.
- All possible locations for 3's in row 5 are f5,h5.
- Forbids 3 from f4,f8,h4,h6.
This step could be presented in a puzzle proof as follows:
- xwing on 3's at fh59 forbids f48,h46=3.
It should be clear that, for example, f4=3 is impossible. If f4=3, then in order for rows 5,9 to have
any threes, both h5=3 and h9=3. Since h5,h9 are both in column h, one concludes
f4=3 is forbidden by the rules.
This idea extends easily:
- Consider any one candidate
- Consider any N distinct rows, N>0.
- Let the candidate be limited to no more than N cells in each of these N rows
- Let this group of no more than N^2 cells share exactly N columns
- The candidate is forbidden from all the cells in those N columns outside of those N^2 cells
Because of symmetry, one can exchange columns for rows in this
theorem.
With N=1, we have the trivial case of a Unique Possibility. With N=2, we have a standard
X wing. With N=3, we have a swordfish. With N=4, we have a jellyfish.
with N=5, the creative name squirmbag is used.
To prove this technique, I prefer to use induction.
This technique merely crosses the possible locations of a candidate in N rows with N columns.
It is analogous to an Ntuple, which crosses the possible locations within a large container of N candidates with N cells.
Got all that? Now you are ready for the Swordfish Technique