Difference between revisions of "Franken Swordfish"

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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.


This is a [[project:fish diagram|fish diagram]] of a Franken Swordfish:
For the Franken Swordfish, 1 of the defining (or secondary) rows (or columns) is replaced by a box, but 2 of the 3 secondary (or defining) columns (or rows) must intersect with this box.


[[image:franken swordfish.png]]
This gives us 4 varieties:
* rrr-ccb
* rrb-ccc
* ccc-rrb
* ccb-rrr


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.
If only 1 row or column would intersect with the box, the other rows or columns are an[[X-Wing]] pattern.


;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]], [[Hidden Single]] and [[Locked Candidates]]
* b: one or both of these must have candidate n, or the pattern collapses to a [[Pointing Pair]], [[Hidden Single]] and [[Locked Candidates]]
* e: candidate n can be eliminated from each of these cells
=== Maximal Franken Swordfish cccrrb ===
This is a [[project:fish diagram|fish diagram]] of a Franken Swordfish with all 12 possible [[candidate]]s present. Note that this rarely happens.
{{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}}
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 '''n''') are located inside the [[secondary set]]. The remaining candidates from these 3 [[constraint]]s can be eliminated.
When a computer solver tries the [[Finned Swordfish]] before the '''Franken Swordfish''', it will find it 3 times, with the base [[Swordfish]] in rows 1, 2, 3 respectively, eliminating just a single candidate in box 3 each time. For this reason the '''Franken Swordfish''' should be tried ''before'' the '''Finned Swordfish'''.
Rotate the diagram 90 degrees for a '''rrrccb''' pattern.


'''The 5 Patterns for the Franken Swordfish'''
=== Same with both rows in 1 band ===


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.
This pattern, although it has the appearance of a Franken Swordfish, degenerates into a series of locked candidates.


The diagram of a minimum franken swordfish pattern is shown below.
{{grid|5=/|7='''n'''|8='''n'''|9=e|14=/|16='''n'''|17='''n'''||18=e|23=/|25='''n'''|26='''n'''|27=e|31=e|32=/|33=e|34=/|35=/|36=e|37=e|38=e|39=e|40=e|41='''n'''|42=e|43='''n'''|44='''n'''|45=e|46=e|47=e|48=e|49=e|50='''n'''|51=e|52='''n'''|53='''n'''|54=e|59=/|61=/|62=/|63=a|68=/|70=/|71=/|72=a|77=/|79=/|80=/|81=a}}


{{grid|5=--|7=--|8=G|9=*|14=--|16=H|17=--|18=*|23=--|25=--|26=--|27=*|32=--|34=--|35=--|37=*|38=*|39=*|40=*|41=A|42=*|43=C|44=D|45=*|50=--|52=--|53=--|59=--|61=--|62=--|64=*|65=*|66=*|67=*|68=B|69=*|70=E|71=F|72=*|77=--|79=--|80=--}}
Before being recognized as a '''Franken Swordfish''', this pattern will be processed in 3 easy steps:


'''How It Works.'''
# [[Locked Candidates]] (labeled '''a''') in the intersection of box 9 and column 9. This eliminates the candidates in the intersection of boxes 3 and 6 with column 9.
# Locked Candidates in the intersection of box 5 and column 5. This eliminates the candidates in the intersection of box 5 with columns 4 and 6.
# Locked Candidates in the intersection of box 4 and row 4. This eliminates the candidates in the intersection of box 4 with rows 5 + 6.


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
=== Maximal Franken Swordfish ccbrrr ===


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.  
{{grid|2=/|5=/|7=/|8=/|9=/|10=e|11='''n'''|12=e|13=e|14='''n'''|15=e|16='''n'''|17='''n'''|18='''n'''|19=e|20='''n'''|21=e|22=e|23='''n'''|24=e|25='''n'''|26='''n'''|27='''n'''|29=/|32=/|37=e|38='''n'''|39=e|40=e|41='''n'''|42=e|43=e|44=e|45=e|47=/|50=/|56=/|59=/|65=/|68=/|74=/|77=/}}
 
The defining set consists of columns 2 and 5 with box 3. The secondary set are rows 2, 3 and 5.
 
Rotate the diagram 90 degrees for a '''rrbccc''' pattern.
 
=== Same with both columns in 1 band ===
 
This pattern, although it has the appearance of a Franken Swordfish, degenerates into a series of locked candidates.
 
{{grid|1=e|2=/|3=/|7=/|8=/|9=/|10=e|11='''n'''|12='''n'''|13=e|14=e|15=e|16='''n'''|17='''n'''|18='''n'''|19=e|20='''n'''|21='''n'''|22=e|23=e|24=e|25='''n'''|26='''n'''|27='''n'''|28=e|29=/|30=/|37=e|38='''n'''|39='''n'''|40=e|41=e|42=e|43=e|44=e|45=e|46=e|47=/|48=/|55=a|56=/|57=/|64=a|65=/|66=/|73=a|74=/|75=/}}
 
This pattern will be processed in 3 separate steps:
 
# '''Locked Candidates 1''' (labeled '''a''') in box 7 and column 1. This eliminates the candidates for '''n''' in the intersection of column 1 with boxes 1 and 4.
# '''Locked Candidates 1''' in box 4 and row 5. This eliminates the candidates for '''n''' in the intersection of row 5 with boxes 5 and 6.
# '''Locked Candidates 2''' in box 1 and 3. This eliminates the remaining candidates for '''n''' in the intersection of box 2 with rows 2 and 3.
 
=== Reduced Patterns for the Franken Swordfish ===
 
The examples above are the maximum patterns for the '''Franken Swordfish'''. They contain 12 cells, and as such will not be spotted a lot in the wild.
Logical analysis of the minimum pattern shows that 4 of the candidates in the defining box are redundant. Reduced patterns occur when 1, 2, 3 or 4 of these redundant cells are removed. There must be at least 1 candidate in each of the intersections between the  defining rows or columns and the box in the pattern.
 
Further reduction is possible by removing one of the pattern candidates in the rows or columns which run through the box which belongs to the pattern. The candidates cannot both be removed from the same row or column. Like a regular [[Swordfish]] each of the defining constaints must have 2 candidates.
 
Below are some effects for reduced patterns.


=== Combined Franken/Column Swordfish ===
=== 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.
It is possible to have a 6 to 8 cell pattern which is both a franken swordfish and a column(row) swordfish.  In this case additional cell eliminations can be made.
 
{{grid|1=*|2=*|3=*|4=*|5=-|6=*|7=X|8=X|9=*|14=-|16=-|17=-|18=*|23=-|25=-|26=-|27=*|32=-|34=-|35=-|36=o|37=*|38=*|39=*|40=*|41=X|42=*|43=X|44=X|45=*|50=-|52=-|53=-|54=o|59=-|61=-|62=-|63=o|64=*|65=*|66=*|67=*|68=X|69=*|70=X|71=X|72=*|77=-|79=-|80=-|81=o}}
{{grid|1=e|2=e|3=e|4=e|5=/|6=e|7='''n'''|8='''n'''|9=e|14=/|16=/|17=/|18=e|23=/|25=/|26=/|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}}
 
This pattern can eliminate up to 20 candidates for digit n.


=== Minimal Franken Swordfish ===
=== Minimal Franken Swordfish ===
Line 34: Line 85:
{{grid|5=/|7=.|8='''n'''|9=e|14=/|16='''n'''|17=.||18=e|23=/|25=.|26=.|27=e|32=/|34=/|35=/|36=a|37=e|38=e|39=e|40=e|41='''n'''|42=e|43=.|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=.|72=e|77=/|79=/|80=/|81=b}}
{{grid|5=/|7=.|8='''n'''|9=e|14=/|16='''n'''|17=.||18=e|23=/|25=.|26=.|27=e|32=/|34=/|35=/|36=a|37=e|38=e|39=e|40=e|41='''n'''|42=e|43=.|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=.|72=e|77=/|79=/|80=/|81=b}}


;Legend:
=== Franken Swordfish with a Fake Fin ===
* '''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 Swordfish'''.


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.
{{grid|5=/|7='''n'''|8='''n'''|9=e|14=/|16='''n'''|17=/|18=e|23=/|25=/|26=/|27=e|32=/|34=/|35=/|37=e|38=e|39=e|40=e|41='''n'''|42=e|43='''n'''|44='''n'''|45=e|50=/|52=/|53=/|59=/|61=/|62=/|64=e|65=e|66=e|67=e|68='''n'''|69=e|70='''n'''|71='''n'''|72=e|77=/|79=/|80=/}}


{{grid|5=--|7=X|8=X|9=*|14=--|16=X|17=--|18=*|23=--|25=--|26=--|27=*|32=--|34=--|35=--|37=*|38=*|39=*|40=*|41=X|42=*|43=X|44=X|45=*|50=--|52=--|53=--|59=--|61=--|62=--|64=*|65=*|66=*|67=*|68=X|69=*|70=X|71=X|72=*|77=--|79=--|80=--}}
In this case the finned swordfish pattern does not have any effect on eliminations, because the fin is not real. The fin is a regular part of the '''Franken Swordfish''' pattern, not an addition.


The fin is a regular part of the fish pattern, not an addition.
=== Finned Franken Swordfish ===


=== Finned Franken Swordfish (effective) ===
Here is example of a genuine '''Finned Franken Swordfish'''. The fin here does not occur in the defining box 3.


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.
{{grid|5=/|7=/|8='''n'''|14=/|16='''n'''|17=/|23=/|25=/|26=/|32='''f'''|34=/|35=/|40=e|41='''n'''|42=e|43='''n'''|44='''n'''|50=/|52=/|53=/|59=/|61=/|62=/|68='''n'''|70='''n'''|71='''n'''|77=/|79=/|80=/}}


{{grid|5=--|7=--|8=X|14=--|16=X|17=--|23=--|25=--|26=--|32=X|34=--|35=--|40=*|41=X|42=*|43=X|44=X|50=--|52=--|53=--|59=--|61=--|62=--|68=X|70=X|71=X|77=--|79=--|80=--}}
The fin in this example occurs in box 5 (marked '''f''') and the only cell eliminations for this pattern are also in box 5.


== See Also ==
== See Also ==

Latest revision as of 18:52, 7 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.

For the Franken Swordfish, 1 of the defining (or secondary) rows (or columns) is replaced by a box, but 2 of the 3 secondary (or defining) columns (or rows) must intersect with this box.

This gives us 4 varieties:

  • rrr-ccb
  • rrb-ccc
  • ccc-rrb
  • ccb-rrr

If only 1 row or column would intersect with the box, the other rows or columns are anX-Wing pattern.

Legend

Maximal Franken Swordfish cccrrb

This is a fish diagram of a Franken Swordfish with all 12 possible candidates present. Note that this rarely happens.

   
     
     
  /  
  /  
  /  
n n e
n n e
n n e
     
e e e
     
  /  
e n e
  /  
/ / a
n n e
/ / a
     
e e e
     
  /  
e n e
  /  
/ / b
n n e
/ / b

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 n) are located inside the secondary set. The remaining candidates from these 3 constraints can be eliminated.

When a computer solver tries the Finned Swordfish before the Franken Swordfish, it will find it 3 times, with the base Swordfish in rows 1, 2, 3 respectively, eliminating just a single candidate in box 3 each time. For this reason the Franken Swordfish should be tried before the Finned Swordfish.

Rotate the diagram 90 degrees for a rrrccb pattern.

Same with both rows in 1 band

This pattern, although it has the appearance of a Franken Swordfish, degenerates into a series of locked candidates.

   
     
     
  /  
  /  
  /  
n n e
n n e
n n e
     
e e e
e e e
e / e
e n e
e n e
/ / e
n n e
n n e
     
     
     
  /  
  /  
  /  
/ / a
/ / a
/ / a

Before being recognized as a Franken Swordfish, this pattern will be processed in 3 easy steps:

  1. Locked Candidates (labeled a) in the intersection of box 9 and column 9. This eliminates the candidates in the intersection of boxes 3 and 6 with column 9.
  2. Locked Candidates in the intersection of box 5 and column 5. This eliminates the candidates in the intersection of box 5 with columns 4 and 6.
  3. Locked Candidates in the intersection of box 4 and row 4. This eliminates the candidates in the intersection of box 4 with rows 5 + 6.

Maximal Franken Swordfish ccbrrr

  /  
e n e
e n e
  /  
e n e
e n e
/ / /
n n n
n n n
  /  
e n e
  /  
  /  
e n e
  /  
     
e e e
     
  /  
  /  
  /  
  /  
  /  
  /  
     
     
     

The defining set consists of columns 2 and 5 with box 3. The secondary set are rows 2, 3 and 5.

Rotate the diagram 90 degrees for a rrbccc pattern.

Same with both columns in 1 band

This pattern, although it has the appearance of a Franken Swordfish, degenerates into a series of locked candidates.

e / /
e n n
e n n
     
e e e
e e e
/ / /
n n n
n n n
e / /
e n n
e / /
     
e e e
     
     
e e e
     
a / /
a / /
a / /
     
     
     
     
     
     

This pattern will be processed in 3 separate steps:

  1. Locked Candidates 1 (labeled a) in box 7 and column 1. This eliminates the candidates for n in the intersection of column 1 with boxes 1 and 4.
  2. Locked Candidates 1 in box 4 and row 5. This eliminates the candidates for n in the intersection of row 5 with boxes 5 and 6.
  3. Locked Candidates 2 in box 1 and 3. This eliminates the remaining candidates for n in the intersection of box 2 with rows 2 and 3.

Reduced Patterns for the Franken Swordfish

The examples above are the maximum patterns for the Franken Swordfish. They contain 12 cells, and as such will not be spotted a lot in the wild. Logical analysis of the minimum pattern shows that 4 of the candidates in the defining box are redundant. Reduced patterns occur when 1, 2, 3 or 4 of these redundant cells are removed. There must be at least 1 candidate in each of the intersections between the defining rows or columns and the box in the pattern.

Further reduction is possible by removing one of the pattern candidates in the rows or columns which run through the box which belongs to the pattern. The candidates cannot both be removed from the same row or column. Like a regular Swordfish each of the defining constaints must have 2 candidates.

Below are some effects for reduced patterns.

Combined Franken/Column Swordfish

It is possible to have a 6 to 8 cell pattern which is both a franken swordfish and a column(row) swordfish. In this case additional cell eliminations can be made.

e e e
     
     
e / e
  /  
  /  
n n e
/ / e
/ / e
     
e e e
     
  /  
e n e
  /  
/ / a
n n e
/ / a
     
e e e
     
  /  
e n e
  /  
/ / b
n n e
/ / b

This pattern can eliminate up to 20 candidates for digit n.

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.

   
     
     
  /  
  /  
  /  
. n e
n . e
. . e
     
e e e
     
  /  
e n e
  /  
/ / a
. n e
/ / a
     
e e e
     
  /  
e n e
  /  
/ / b
n . e
/ / b

Franken Swordfish with a Fake Fin

It is also possible to have a 9 cell pattern which is both a Franken Swordfish and a Finned Swordfish.

     
     
     
  /  
  /  
  /  
n n e
n / e
/ / e
     
e e e
     
  /  
e n e
  /  
/ /  
n n e
/ /  
     
e e e
     
  /  
e n e
  /  
/ /  
n n e
/ /  

In this case the finned swordfish pattern does not have any effect on eliminations, because the fin is not real. The fin is a regular part of the Franken Swordfish pattern, not an addition.

Finned Franken Swordfish

Here is example of a genuine Finned Franken Swordfish. The fin here does not occur in the defining box 3.

     
     
     
  /  
  /  
  /  
/ n  
n /  
/ /  
     
     
     
  f  
e n e
  /  
/ /  
n n  
/ /  
     
     
     
  /  
  n  
  /  
/ /  
n n  
/ /  

The fin in this example occurs in box 5 (marked f) and the only cell eliminations for this pattern are also in box 5.

See Also

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