Now consider Death Blossom. In this case, all candidates of a cell are linked to the CEC via bivalues or ALS. Elimination is performed if existance of the candidate in the CEC reduces the number of candidates in the cell to 0. I hold that Death Blossom and Kraken fish are actually examples of the same technique. In the first case the RS is a fish; in the second, a cell. In fact most eliminations performed in Sudoku can be described by this model: a candidate in a CEC is linked to a RS via elements and existance of the candidate in the CEC creates a contradiction in the RS.
Consider ALS techniques. In this case the RS is an ALS and the contradiction is reduction of the number of candidates in the ALS to n-1. In the xz-rule, the CEC is linked to the ALS directly and via another ALS. In the xy-rule, the CEC is linked to the ALS via two other ALSs.
Consider a naked set. This case is actually a special case of death blossom. The RS is a cell which links to the CEC via a single ALS.
Consider a hidden set. In this case the RS is a row, line, or box. The candidate in the CEC (which can be part of the RS) reduces the number of possible candidates in the house to 8.
Unique rectangles, BUG, and BUG-lite techniques also fall into this model. Here the RS is the UR, etc. and the contradiction the existance of the "deadly pattern". Many of the advanced techniques utilize elements to connect one cell of the pattern to the remaining non-bivalued cell.
In many coloring and strong link eliminations the RS is a house containing a conjugate pair and so is similar to a hidden set. In this case the RS can be considered to be either a chain of elements directly linked to the CEC or a conjugate pair linked to the CEC via a chain of elements.
In nice loops the RS is the CEC with the contradiction that the RS must both contain and not contain the labelled value(s). Similarly, forcing chains can be considered cases where the RS is a cell linked to the CEC via chains of elements.
There is more work required to complete the list especially with more exotic coloring and continuous loops, but the model appears to work in most cases. Of course, the usefulness of a model is its ability to predict other possibilities and that is true in this case. Consider Death Blossom. The current techique links a cell to the CEC via bivalues and ALS. Given the model, then it should be possible to utilize strong links and GSL as well. Since the RS in this case is a cell, the resulting elimination can be refered to as a Kraken Cell. A little thought shows that this is indeed the case. Here is an example:
- Code: Select all
Kraken Cell (r13c9=2=r13c7, -4-r6c9, -9-r8c7): r9c9 => r4c7<>2
+----------------+-------------------+--------------------+
| 9 47 678 | 237 2468 12368 | 247# 5 12467# |
| 3 2 567 | 579 456 169 | 8 14 14679 |
| 45 1 5678 | 279 2568 2689 | 2479# 3 2679# |
+----------------+-------------------+--------------------+
| 45 458 9 | 235 256 236 | 247- 1248 1247 |
| 67 67 2 | 8 1 4 | 3 9 5 |
| 1 458 3 | 259 7 29 | 6 28 24@ |
+----------------+-------------------+--------------------+
| 2 357 57 | 4 9 78 | 1 6 38 |
| 68 369 4 | 1 28 5 | 29% 7 38 |
| 78 79 1 | 6 3 278 | 5 24 249* |
+----------------+-------------------+--------------------+
Note: I haven't tried to find an example where this Kraken Cell exclusion is the only possible one. The example is only to demonstrate the theory.
The extension of the model to other RS's is obvious. With an ALS as the RS, xy-rule type eliminations can be made with ALSs and/or a strong links. Granted this is just a grouped nice loop, however, this demonstrates the extension of the model to the ALS approach which is a little more palatable than the much broader scope of grouped nice loops. Similar extensions exist for URs, naked sets, etc.
Not all of this is new. Ruud published a Great Unified Theory, but I was intrigued at the fact that the same model could describe Kraken fish and Death Blossom, could be extended to other techniques, and then used to predict other approaches.
[Edited to change constraint set to restricted subset, add direct replacement, and refer to the Kraken Cell approach and to add this comment]
