## Finding vulnerable cells in a solution pattern set

Advanced methods and approaches for solving Sudoku puzzles

### Finding vulnerable cells in a solution pattern set

Often when people begin applying some of the more advanced solving techniques such as colors, forcing chains, or POM, they are at a loss for a good starting point to try. In applying POM, I have noticed some relationships between the solution set of all possible patterns for a number, and the cells where conflicts are most likely to show up. In the following diagrams, a '.' represents a cell where a number cannot be placed, and letters represent all of the possible patterns that can complete the solution for that number, otherwise termed the solution pattern set for the number. Below is an example of the simplest solution pattern set.
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`...........A....B...B....A.......................................................`

Vulnerable cells in a solution pattern set are pairings of cells which have a certain relationship to each other that may be easily exploited.

Vulnerable points in the solution pattern set must fulfill criteria 1 and either criteria 2 or 3, or both.

Criteria 1: The requirement, a pair of vulnerable cells must not lie in the same house; that is, they must not have either a row, column, or box in common. Simpler methods have already reduced things as much as possible for cells in the same house, so you are not likely to learn anything more there.

Criteria 2: (the strong criteria) A pair of vulnerable cells can be two cells that, taken together, posess all of the patterns of a given number's solution pattern set. The vulnerability here is if you can find an assertion that implies that neither of the cells contains the given number, then that assertion must be false (POM's general pattern exclusion rule)

Criteria 3: (the weak criteria) A pair of vulnerable cells can be two cells that have no two solution patterns for a number in common. The vulnerability here is if you can find an assertion which implies that the given number is placed in both of these cells, then that assertion must be false.

Pairings which satisfy both the strong and the weak criteria are doubly vulnerable. For these cases, in addition to the above, any assertion which merely implies that the pair has the same value must be false.

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`.  .  .  .  .  .  .  .  ..  .  .  A  B  C  .  .  ..  .  .  .  .  .  .  .  ..  .  .  .  .  .  .  .  ..  .  .  BC A  .  .  .  ..  .  .  .  .  .  .  .  ..  .  .  .  .  .  .  .  ..  .  .  .  C  .  .  AB ..  .  .  .  .  AB .  C  .`

In the diagram above, cells r5c4 and r9c6 would be strongly vulnerable. Cells r2c4 and r9c8 would be weakly vulnerable, and the r2c6-r8c8 pairing would be doubly vulnerable.

The advantage of these vulnerability points is that you are half way there. You just need to find the assertions that get you back to the start using whatever method you like best.

All is not skittles and beer. Many solution pattern sets do not have vulnerabilities. The simplest doesn't, and neither does the following.

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`.  .  .  AB .  CD .  .  ..  .  .  .  .  .  .  .  ..  .  .  .  .  .  .  .  ..  .  .  C  .  A  .  DB ..  .  .  .  .  .  .  .  ..  .  .  D  .  B  .  AC ..  .  .  .  .  .  .  .  ..  .  .  .  .  .  .  .  ..  .  .  .  .  .  .  .  .`

Nevertheless, here's hoping that these points lead to a few good starting places in solving your puzzles.
Myth Jellies

Posts: 593
Joined: 19 September 2005

Quite frankly I'd say that anyone who can understand that deserves a skittles and beer!
emm

Posts: 987
Joined: 02 July 2005