The no-tridagon-in-solution pattern
For any valid solution grid and any triplet of digits, say {1 2 3}, the analysis of the first post of this thread remains valid: no valid solution grid can have the tridagon pattern without guardians in any rectangle of blocks.
As a result, for any triplet of digits {1 2 3}, the following no-tridagon-in-solution pattern must be present (modulo isomorphisms) in any solution grid and in any rectangle of blocks. After all, this is just expressing the Sudoku constraints and this is just how a partial 3-digit template on a rectangle of 4 blocks can be instantiated.
There are two isomorphism classes for a potential 3-template pattern in 4 blocks: tridagon and no-tridagon, and the first is impossible.
- Code: Select all
+-------------------------------+-------------------------------+
! 123 . . ! 123 . . !
! . 123 . ! . 123 . !
! . . 123 ! . . 123 !
+-------------------------------+-------------------------------+
! 123 . . ! 123 . . !
! . 123 . ! . 123 . !
! . . 123 ! . . 123 !
+-------------------------------+-------------------------------+
Note that this pattern is identical to the tridagon pattern, except for moving two cells in block b22.
Now, what does this tell us about the possibility of a direct search for tridagons starting from "tridagon" properties of solution grids, when one is certain the no-tridagon pattern is always there? What kind of a pre-filter of solution grids can this be?
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