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| The '''Franken Swordfish''' is a [[fish]] pattern of the [[Franken Fish]] type. This type allows either the [[defining set]] or the [[secondary set]] to contain [[box]]es, as well as [[line]]s. | | The '''Franken Swordfish''' is a [[fish]] pattern of the [[Franken Fish]] type. This type allows either the [[defining set]] or the [[secondary set]] to contain [[box]]es, as well as [[line]]s. |
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| This is a [[project:fish diagram|fish diagram]] of a Franken Swordfish:
| | === Maximal Franken Swordfish === |
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| [[image:franken swordfish.png]] | | This is a [[project:fish diagram|fish diagram]] of a Franken Swordfish with all 12 possible candidates present. Note that this rarely happens: |
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| | {{grid|5=/|7='''n'''|8='''n'''|9=e|14=/|16='''n'''|17='''n'''||18=e|23=/|25='''n'''|26='''n'''|27=e|32=/|34=/|35=/|36=a|37=e|38=e|39=e|40=e|41='''n'''|42=e|43='''n'''|44='''n'''|45=e|50=/|52=/|53=/|54=a|59=/|61=/|62=/|63=b|64=e|65=e|66=e|67=e|68='''n'''|69=e|70='''n'''|71='''n'''|72=e|77=/|79=/|80=/|81=b}} |
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| | ;Legend: |
| | * '''n''': Franken Swordfish pattern for digit n |
| | * /: May not contain candidate for digit n |
| | * a: one or both of these must have candidate n, or the pattern collapses to a [[Pointing Pair]] |
| | * b: one or both of these must have candidate n, or the pattern collapses to a [[Pointing Pair]] |
| | * e: candidate n can be eliminated from each of these cells |
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| In this example, the defining set contains columns 5, 7 & 8. The secondary set contains rows 5 & 8 and box 3. All the [[candidate]]s in the defining set (marked with '''X''') are located inside the secondary set. The remaining candidates from these 3 [[constraint]]s can be eliminated. | | In this example, the defining set contains columns 5, 7 & 8. The secondary set contains rows 5 & 8 and box 3. All the [[candidate]]s in the defining set (marked with '''X''') are located inside the secondary set. The remaining candidates from these 3 [[constraint]]s can be eliminated. |
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| | | === The 5 Patterns for the Franken Swordfish === |
| '''The 5 Patterns for the Franken Swordfish'''
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| The previous example is the maximum pattern for the franken swordfish. It contains 12 cells and because of this its occurence is much rarer than the any of the other 4 patterns. Logical analysis of the minimum pattern shows that 4 of the cells in box 3 of the example are redundant. The other 4 patterns occur when 1, 2, 3, 4 of these redundant cells are removed. The only restriction on this is there must be at least 1 cell in each of the box 3 columns in the pattern. The minimum pattern occurs when there is only 1 cell in each of columns 7 and 8 of box 3. Since there are 9 different ways these cells can occur. The probability of occurence is 9 times that of the maximum pattern. There is nothing finnish or sashimish about any of the 5 patterns. In a finned or sashimi row or column swordfish cell eliminations occur only in the box with the fin. In contrast the 5 franken swordfish patterns with 8 through 12 cells all give exactly the same cell eliminations. They are analogous to the 4 patterns of the row or column swordfish with 6, 7, 8 and 9 cells. Because of the 5 patterns of the franken swordfish, the probobility of occurence may be comparable to that of the row or column swordfish. | | The previous example is the maximum pattern for the franken swordfish. It contains 12 cells and because of this its occurence is much rarer than the any of the other 4 patterns. Logical analysis of the minimum pattern shows that 4 of the cells in box 3 of the example are redundant. The other 4 patterns occur when 1, 2, 3, 4 of these redundant cells are removed. The only restriction on this is there must be at least 1 cell in each of the box 3 columns in the pattern. The minimum pattern occurs when there is only 1 cell in each of columns 7 and 8 of box 3. Since there are 9 different ways these cells can occur. The probability of occurence is 9 times that of the maximum pattern. There is nothing finnish or sashimish about any of the 5 patterns. In a finned or sashimi row or column swordfish cell eliminations occur only in the box with the fin. In contrast the 5 franken swordfish patterns with 8 through 12 cells all give exactly the same cell eliminations. They are analogous to the 4 patterns of the row or column swordfish with 6, 7, 8 and 9 cells. Because of the 5 patterns of the franken swordfish, the probobility of occurence may be comparable to that of the row or column swordfish. |
Revision as of 23:38, 3 November 2021
The Franken Swordfish is a fish pattern of the Franken Fish type. This type allows either the defining set or the secondary set to contain boxes, as well as lines.
Maximal Franken Swordfish
This is a fish diagram of a Franken Swordfish with all 12 possible candidates present. Note that this rarely happens:
- Legend
- n: Franken Swordfish pattern for digit n
- /: May not contain candidate for digit n
- a: one or both of these must have candidate n, or the pattern collapses to a Pointing Pair
- b: one or both of these must have candidate n, or the pattern collapses to a Pointing Pair
- e: candidate n can be eliminated from each of these cells
In this example, the defining set contains columns 5, 7 & 8. The secondary set contains rows 5 & 8 and box 3. All the candidates in the defining set (marked with X) are located inside the secondary set. The remaining candidates from these 3 constraints can be eliminated.
The 5 Patterns for the Franken Swordfish
The previous example is the maximum pattern for the franken swordfish. It contains 12 cells and because of this its occurence is much rarer than the any of the other 4 patterns. Logical analysis of the minimum pattern shows that 4 of the cells in box 3 of the example are redundant. The other 4 patterns occur when 1, 2, 3, 4 of these redundant cells are removed. The only restriction on this is there must be at least 1 cell in each of the box 3 columns in the pattern. The minimum pattern occurs when there is only 1 cell in each of columns 7 and 8 of box 3. Since there are 9 different ways these cells can occur. The probability of occurence is 9 times that of the maximum pattern. There is nothing finnish or sashimish about any of the 5 patterns. In a finned or sashimi row or column swordfish cell eliminations occur only in the box with the fin. In contrast the 5 franken swordfish patterns with 8 through 12 cells all give exactly the same cell eliminations. They are analogous to the 4 patterns of the row or column swordfish with 6, 7, 8 and 9 cells. Because of the 5 patterns of the franken swordfish, the probobility of occurence may be comparable to that of the row or column swordfish.
The diagram of a minimum franken swordfish pattern is shown below.
How It Works.
Since A and B are conjugate pairs either A exclusive or B is X. If A is X none of the peer cells in row 5 can be X. This creates 2 conditional conjugate pairs (EG and FH) in columns 7 and 8. Since G and H are peers and E and F are peers, one and only 1 of each of the peer groups must be X. Therefore X can be no additional X's in row 7 are box 3. If B is X none of the peer cells in row 7 can be X. This creates 2 conditional conjugate pairs (CG and DH) in columns 7 and 8. Since G and H are peers and C and D are peers, one and only 1 of each of the peer groups must be X. Therefore X can be no additional X's in row 5 are box 3. QED
The logic for the other 4 patterns is similar except that one or both of the conditional conjugate pairs become conditional group conjugates. The logic is still valid for this case.
Combined Franken/Column Swordfish
It is possible to have an 8 cell pattern which is both a franken swordfish and a column(row) swordfish. In this case additional cell eliminations can be made. An example of this is shown in the diagram below.
Minimal Franken Swordfish
Similar to a regular Swordfish pattern, it is possible to have empty spots in the pattern, but these are limited to the rows or columns running through the box of the pattern. This is the smallest possible (6 cell) pattern.
- Legend
- n: Franken Swordfish pattern for digit n
- /: May not contain candidate for digit n
- .: part of the pattern without a candidate n
- a: one or both of these must have candidate n, or the pattern collapses to a Pointing Pair
- b: one or both of these must have candidate n, or the pattern collapses to a Pointing Pair
- e: candidate n can be eliminated from each of these cells
Finned Franken Swordfish (ineffective)
It is also possible to have a 9 cell pattern which is both a franken swordfish and a finned column(row) swordfish. In this case the finned swordfish pattern does not have any effect on eliminations.
The fin is a regular part of the fish pattern, not an addition.
Finned Franken Swordfish (effective)
Here is example of a finned franken swordfish. The fin in this example occurs in box 5 and the only cell eliminations for this pattern are also in box 5.
See Also
