## Sudoku Unification Model

Advanced methods and approaches for solving Sudoku puzzles

### Sudoku Unification Model

I'd like to present a point of view, which is not entirely new, which unifies many of the techniques that are used in Sudoku. Consider the Kraken fish. In a Kraken fish elimination a "fish" (n rows or columns) is linked to a cell where the elimination of a candidate is performed (the candidate elimination cell or CEC) via a set of link elements - direct replacement (the CEC is a cell in the fish, but with a different candidate), direct links (cells of the fish which share a house with the CEC), strong links, grouped strong links (GSL), bivalue cells, Almost Locked Sets (ALSs), or a chain of these. The elimination is performed if existance of the candidate in the CEC reduces the size of the covering set of the fish to n-1. The fish is a Restricted Subset (RS) and elimination occurs when existance of the candidate in the CEC creates a contradiction. Note that all four basic types of fish fit into this model with a single direct link linking the CEC and RS

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:
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`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]
Last edited by Mike Barker on Mon Oct 16, 2006 11:13 pm, edited 1 time in total.
Mike Barker

Posts: 458
Joined: 22 January 2006

SUM has been used to create a number of different approaches to solving puzzles. Kraken Blossom (or Kraken Cell) links a cell as the Restricted Subset and an empty cell as the Contradiction Condition. Here's an example which solves Top1465 #2 (once considered one of the toughest Sudokus). As Daj has pointed out this can also probably be solved with forcing chains. This is an alternative.
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`Locked Row Line/Box: r3c46 => r3c789<>7Locked Row Line/Box: r3c46 => r3c789<>8Locked Column Line/Box: r46c7 => r279c7<>8Locked Row Line/Box: r5c89 => r5c2346<>5Hidden Row Pair: r3c46 => r3c4=78,r3c6=78Locked Column Line/Box: r23c5 => r479c5<>3A=3 cell ALS xz-rule: r4c456-9-r5c4789 => r5c6<>7Overlap 3-element Grouped Nice Loop: ALS:r2c1235-5-r4c5-9-r4c1=9=r2c1~9~ => r2c79<>9Multiple Overlap 3-element Grouped Nice Loop: ALS:r2c1237-7-r456c7=7=r5c89-7-ALS:r1235c2~6~ => r3c3<>3UR+3C/2SL (36): r23c25 => r3c5<>64-element Grouped Nice Loop: r5c89=7=r456c7-7-ALS:r2c12357-5-r4c5-9-r4c13=9=r5c3~7~r5c89 => r5c3<>75-valued/2-link Kraken Blossom (SUM Exclusion) (r2c5-3469-r2c1237-7-r456c7=7=r5c89-7-, r2c5-5-r4c5-9-r5c4789-7-): r2c5 => r5c2<>7+-------------------------+-----------------------+----------------------+|     7     126        8  |   459   4569    4569  |     3   1456   1259  ||   469#     36#    3469# |     2  34569*      1  |   467# 45678    578  ||     5    1236    12469  |    78    349      78  | 12469    146    129  |+-------------------------+-----------------------+----------------------+|   189       4     1579  |  3579     59@   3579  |   178@#    2      6  ||     3    1267-    1269  |   479@     8    2469  |   147@  1457@   157@ ||   268   25678     2567  |     1   2456   24567  |   478@#    9      3  |+-------------------------+-----------------------+----------------------+|   128       9    12357  |     6    125    2358  |   127   1378      4  || 12468   12368    12346  |  3489      7   23489  |     5   1368   1289  || 12468  135678  1234567  | 34589  12459  234589  | 12679  13678  12789  |+-------------------------+-----------------------+----------------------+Hidden Column Pair: r69c2 => r6c2=57,r9c2=57Locked Column Line/Box: r789c3 => r2c3<>3A=3 cell ALS xz-mer: r7c157-57-r2c12357 => r7c38<>1,r7c36<>2,r1469c5<>5,r4569c7<>7,r2c8<>4,r2c8<>6Locked Column Line/Box: r23c3 => r89c3<>4Locked Row Line/Box: r1c46 => r1c9<>9Locked Row Line/Box: r5c89 => r5c4<>7Locked Column Line/Box: r89c3 => r3c3<>1Locked Row Line/Box Pair: r1c56 => r1c2<>6Locked Column Box/Box: r35c2|r36c3 => r89c3<>6Naked Block Pair: r6c2|r4c3 => r6c3<>57Naked Row Pair: r6c35 => r6c6<>26Naked Block Pair: r9c4|r7c6 => r9c6<>58Naked Block Pair: r9c7|r8c9 => r9c9<>29Row X-Wing Fillet-o-Fish: r6c26|r7c36 => r9c2<>5The Solution is completed with singles `
Mike Barker

Posts: 458
Joined: 22 January 2006

Mike Barker wrote:5-valued/2-link Kraken Blossom (SUM Exclusion) (r2c5-3469-r2c1237-7-r456c7=7=r5c89-7-, r2c5-5-r4c5-9-r5c4789-7-): r2c5 => r5c2<>7
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`+-------------------------+-----------------------+----------------------+ |     7     126        8  |   459   4569    4569  |     3   1456   1259  | |   469#     36#    3469# |     2  34569*      1  |   467# 45678    578  | |     5    1236    12469  |    78    349      78  | 12469    146    129  | +-------------------------+-----------------------+----------------------+ |   189       4     1579  |  3579     59@   3579  |   178@#    2      6  | |     3    1267-    1269  |   479@     8    2469  |   147@  1457@   157@ | |   268   25678     2567  |     1   2456   24567  |   478@#    9      3  | +-------------------------+-----------------------+----------------------+ |   128       9    12357  |     6    125    2358  |   127   1378      4  | | 12468   12368    12346  |  3489      7   23489  |     5   1368   1289  | | 12468  135678  1234567  | 34589  12459  234589  | 12679  13678  12789  | +-------------------------+-----------------------+----------------------+ `

Even the Kraken Blossom step can be put in single-threaded AIC or Nice Loop terms if you wish:
((7&1&4&5)=9)r5c4789 - (9=5)r4c5 - (5=(3&4&6&9&7))r2c12357 - (7)r456c7 = (7)r5c89 => r5c2 <> 7

Of course, finding it without being told it is there is another thing entirely
Myth Jellies

Posts: 593
Joined: 19 September 2005

SUM derived techniques have been used to solve puzzles with the restricted subset consisting of:
1) cells (Death Blossom or Kraken Cell) with eliminations of all candidates in the cell as the contradiction condition (CC)
2) a row, column or box (Kraken House) with elimination of all occurances of a candidate in the subset as the CC
3) a fish (n rows or columns - a Kraken Fish) with reduction of the size of the covering set to n-1 as the CC
4) a UR and BUG-lite (a Kraken UR or Kraken BUG-Lite) with creation of a deadly pattern as the CC (these haven't been overly effective since existing techniques cover most of the eliminations)
5) an ALS (a Kraken ALS was used to solve Unsolvable #33) with reduction of the number of candidates in the ALS to n-1 as the CC
A next logical technique is a Kraken Almost ALS. An Almost ALS is a set of n cells in a common house with n+2 candidates. As Havard has pointed out, a Kraken cell with 3 candidates in the cell is an example of an Almost ALS. This approach can be extended to AALS with more cells. Just like a Kraken ALS the CC is reduction of the number of candidates to n-1, thus linking of the cells with 3 candidates of the Almost ALS allows for elimination of the linked candidate in candidate elimination cells common to the end nodes of all of the linking elements or chains. This technique along with other "Kraken" technqiues can be used to solve Emily's Friend's Puzzle.
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`Locked Column Line/Box: r13c4 => r59c4<>1Locked Row Line/Box: r4c78 => r4c4<>9Column X-Wing Fillet-o-Fish: r689c3|r69c9 => r9c1<>33-element Nice Loop: r1c3=4=r2c3-4-r2c6-8-r1c4=8=r1c9~4~r1c3 => r1c9<>4A=4 cell ALS xz-rule: r9c1457-1-r5c14578 => r4c7<>2B=4 cell ALS xy-rule: r1279c9-3-r9c1345-1-r5c14578 => r6c9<>24-link Grouped Advanced Colouring: r5c6=1=r5c5-1-r9c5=1=r9c9=3=r6c9=4=r6c456~4~r5c6 => r5c6<>45-valued/2-link Kraken Cell (r4c8-2-r5c78=2=r5c6=1=r5c5-1-, r4c8-3-r6c9=3=r9c9=1=r9c5-1-, r4c8-9-r4c7=9=r89c7-9-r7c2689-1-): r4c8=239 => r7c5<>1+---------------------+----------------------+--------------------+|    56     26   246  |  1458      7      9  |     3   1246  128  ||     1      9  2467  |   458      3     48  | 24578   2467  248  ||   357     37     8  |   145      2      6  |   457    147    9  |+---------------------+----------------------+--------------------+|     4   2378     1  |    78      5   2378  |    89\$   239*   6  ||   368      5     9  |   468   1468@  1238@ |   248@   234@   7  ||  3678  23678  2367  | 46789   4689  23478  |     1      5  348# |+---------------------+----------------------+--------------------+|     2    178\$    5  |     3  489-1   1478\$ |     6   1479\$  14\$ || 36789  13678   367  |     2  14689   1478  |   479\$ 13479    5  ||   679      4   367  |   679    169#     5  |   279\$     8  123# |+---------------------+----------------------+--------------------+2-link Kraken ALS (r5c4|r6c4|r4c4-8-r1c4=8=r1c9=1=r13c8-1-, r6c4-4-r6c123569-9-r7c2569-1-, r5c4-4-r6c456=4=r6c9-4-r7c9-1-): r4569c4=84 => r7c8<>1+---------------------+------------------------+---------------------+|    56     26   246  |   1458@     7       9  |     3   1246@  128@ ||     1      9  2467  |    458      3      48  | 24578   2467   248  ||   357     37     8  |    145      2       6  |   457    147@    9  |+---------------------+------------------------+---------------------+|     4   2378     1  |     78*     5    2378  |    89    239     6  ||   368      5     9  |    468*  1468    1238  |   248    234     7  ||  3678# 23678# 2367# | 46789*\$ 4689#\$ 23478#\$ |     1      5  348#\$ |+---------------------+------------------------+---------------------+|     2    178#    5  |      3    489#   1478# |     6  479-1   14#\$ || 36789  13678   367  |      2  14689    1478  |   479  13479     5  ||   679      4   367  |    679*   169       5  |   279      8   123  |+---------------------+------------------------+---------------------+2-link Kraken ALS (r5c6=1=r5c5-1-r9c1345-3-, r5c7-2-r9c7=2=r9c9=3=r9c3-3-): r5c14567=32 => r8c1<>3,r6c3<>3+-----------------------+----------------------+--------------------+|     56     26    246  |  1458      7      9  |     3   1246  128  ||      1      9   2467  |   458      3     48  | 24578   2467  248  ||    357     37      8  |   145      2      6  |   457    147    9  |+-----------------------+----------------------+--------------------+|      4   2378      1  |    78      5   2378  |    89    239    6  ||    368*     5      9  |   468* 1468*@ 1238*@ |   248*   234    7  ||   3678  23678  267-3  | 46789   4689  23478  |     1      5  348  |+-----------------------+----------------------+--------------------+|      2    178      5  |     3    489   1478  |     6    479   14  || 6789-3  13678    367  |     2  14689   1478  |   479  13479    5  ||    679@     4   367@# |   679@   169@     5  |   279#     8  123# |+-----------------------+----------------------+--------------------+Locked Column Box/Box: r356c1|r346c2 => r8c2<>32-link Kraken ALS (r5c5-1-r9c1345-3-r8c3=3=r8c8-3-, r5c7-2-r9c1347-3-r8c3=3=r8c8-3-): r5c14578=12 => r5c8<>3+--------------------+----------------------+---------------------+|   56     26   246  |  1458      7      9  |     3    1246  128  ||    1      9  2467  |   458      3     48  | 24578    2467  248  ||  357     37     8  |   145      2      6  |   457     147    9  |+--------------------+----------------------+---------------------+|    4   2378     1  |    78      5   2378  |    89     239    6  ||  368*     5     9  |   468*  1468*  1238  |   248*   24-3*   7  || 3678  23678   267  | 46789   4689  23478  |     1       5  348  |+--------------------+----------------------+---------------------+|    2    178     5  |     3    489   1478  |     6     479   14  || 6789   1678  367@# |     2  14689   1478  |   479  13479@#   5  || 679@#     4  367@# |  679@#   169@     5  |   279#      8  123  |+--------------------+----------------------+---------------------+4-link Advanced Colouring: r5c1=3=r5c6=1=r5c5-1-r9c5=1=r9c9=3=r6c9~3~ => r6c12<>33-celled/2-link Kraken AALS (r4c2=3=r5c1=6=r6c123-6-, r4c8-2-r5c478-6-r5c1=6=r6c123-6-, r4c4-8-r1235c4-6-r5c1=6=r6c123-6-, r4c8-9-r7c8=9=r7c5-9-r6c123569-6-): r4c248=289 => r6c4<>6+--------------------------+-----------------------+--------------------+|     56       26     246  |   1458\$     7      9  |     3   1246  128  ||      1        9    2467  |    458\$     3     48  | 24578   2467  248  ||    357       37       8  |    145\$     2      6  |   457    147    9  |+--------------------------+-----------------------+--------------------+|      4    2378*@      1  |     78*     5   2378  |    89    239*   6  ||  368@#\$       5       9  |   468#\$  1468   1238  |   248#    24#   7  || 678@#\$% 2678@#\$% 267@#\$% | 4789-6   4689% 23478% |     1      5  348% |+--------------------------+-----------------------+--------------------+|      2      178       5  |      3    489%  1478  |     6    479%  14  ||   6789     1678     367  |      2  14689   1478  |   479  13479    5  ||    679        4     367  |    679    169      5  |   279      8  123  |+--------------------------+-----------------------+--------------------+3-celled/2-link Kraken AALS (r4c2-7-r4c4-8-, r4c6-7-r78c6=7=r9c4-7-r4c4-8-, r4c2-8-r13569c1-9-r123459c4-8-, r4c8-9-r7c8=9=r7c5-9-r6c123569-8-): r4c268=789 => r6c4<>8+-------------------+----------------------+--------------------+|   56\$   26   246  |  1458\$     7      9  |     3   1246  128  ||    1     9  2467  |   458\$     3     48  | 24578   2467  248  ||  357\$   37     8  |   145\$     2      6  |   457    147    9  |+-------------------+----------------------+--------------------+|    4  2378*    1  |  78@#\$     5   2378* |    89    239*   6  ||  368\$    5     9  |   468\$  1468   1238  |   248     24    7  || 678\$% 2678%  267% | 479-8   4689% 23478% |     1      5  348% |+-------------------+----------------------+--------------------+|    2   178     5  |     3    489%  1478# |     6    479%  14  || 6789  1678   367  |     2  14689   1478# |   479  13479    5  ||  679\$    4   367  |  679#\$   169      5  |   279      8  123  |+-------------------+----------------------+--------------------+3-celled/2-link Kraken AALS (r7c6|r7c5-4-r7c9-1-r7c2=1=r8c2-1-, r8c6-4-r2c6-8-r2c789|r13c8|r3c7-1-, r7c5-9-r12357c8-1-r1c9=1=r79c9-1-, r7c6-1-r7c2=1=r8c2-1-): r7c56|r8c6=491 => r8c8<>1+--------------------+---------------------+---------------------+|   56     26   246  | 1458      7      9  |     3   1246#\$ 128\$ ||    1      9  2467  |  458      3     48# | 24578#  2467#\$ 248# ||  357     37     8  |  145      2      6  |   457#   147#\$   9  |+--------------------+---------------------+---------------------+|    4   2378     1  |   78      5   2378  |    89     239    6  ||  368      5     9  |  468   1468   1238  |   248      24\$   7  ||  678   2678   267  |  479   4689  23478  |     1       5  348  |+--------------------+---------------------+---------------------+|    2   178@%    5  |    3    489*  1478* |     6     479\$ 14@\$ || 6789  1678@%  367  |    2  14689   1478* |   479  3479-1    5  ||  679      4   367  |  679    169      5  |   279       8  123\$ |+--------------------+---------------------+---------------------+Locked Column Line/Box: r79c9 => r1c9<>1XYZ-wing: r12c9, r2c6 => r2c7<>8Locked Column Line/Box: r12c9 => r6c9<>85-valued/2-link Kraken Cell (r9c4-6-r89c5=6=r56c5-6-r5c478-8-, r9c4-7-r4c4-8-r4c7=8=r5c7-8-, r9c4-9-r7c5=9=r7c8-9-r5c78|r4c8|r6c9-8-): r9c4=679 => r5c156<>8+-------------------+---------------------+-------------------+|   56    26   246  | 1458      7      9  |     3  1246   28  ||    1     9  2467  |  458      3     48  |  2457  2467  248  ||  357    37     8  |  145      2      6  |   457   147    9  |+-------------------+---------------------+-------------------+|    4  2378     1  |   78#     5   2378  |    89#  239\$   6  || 36-8     5     9  |  468@ 146-8@ 123-8  | 248@#\$  24@\$   7  ||  678  2678   267  |  479   4689@ 23478  |     1     5   34\$ |+-------------------+---------------------+-------------------+|    2   178     5  |    3    489\$  1478  |     6   479\$  14  || 6789  1678   367  |    2  14689@  1478  |   479  3479    5  ||  679     4   367  |  679*   169@     5  |   279     8  123  |+-------------------+---------------------+-------------------+Column X-Wing Fillet-o-Fish: r68c1|r678c5 => r8c6<>81-link Kraken ALS (r9c1-7-r1356c1-8-, r9c4-7-r4c4-8-, r9c5-1-r278c6-8-): r9c145=71 => r6c6<>8+-------------------+----------------------+------------------+|   56@   26   246  | 1458      7       9  |    3  1246   28  ||    1     9  2467  |  458      3      48\$ | 2457  2467  248  ||  357@   37     8  |  145      2       6  |  457   147    9  |+-------------------+----------------------+------------------+|    4  2378     1  |   78#     5    2378  |   89   239    6  ||   36@    5     9  |  468    146     123  |  248    24    7  ||  678@ 2678   267  |  479   4689  2347-8  |    1     5   34  |+-------------------+----------------------+------------------+|    2   178     5  |    3    489    1478\$ |    6   479   14  || 6789  1678   367  |    2  14689     147\$ |  479  3479    5  ||  679*    4   367  |  679*   169*      5  |  279     8  123  |+-------------------+----------------------+------------------+2-link Kraken ALS (r4c2-3-r5c1-6-r1235c4-4-, r4c8-3-r6c9-4-r5c78=4=r5c45-4-, r4c2-8-r6c12369-4-, r4c7-8-r5c7=8=r5c4-8-r123c4-4-, r4c4-8-r123c4-4-): r4c2478=38 => r6c4<>4+-------------------+---------------------+------------------+|   56    26   246  | 1458@%     7     9  |    3  1246   28  ||    1     9  2467  |  458@%     3    48  | 2457  2467  248  ||  357    37     8  |  145@%     2     6  |  457   147    9  |+-------------------+---------------------+------------------+|    4  2378*    1  |    78*     5  2378  |   89*  239*   6  ||   36@    5     9  | 468@#%   146#  123  | 248#%   24#   7  ||  678\$ 2678\$  267\$ |  79-4   4689  2347\$ |    1     5  34#\$ |+-------------------+---------------------+------------------+|    2   178     5  |     3    489  1478  |    6   479   14  || 6789  1678   367  |     2  14689   147  |  479  3479    5  ||  679     4   367  |   679    169     5  |  279     8  123  |+-------------------+---------------------+------------------+5-element Grouped Nice Loop: ALS:r278c6-1-r9c5=1=r9c9=3=r6c9=4=r6c56-4-ALS:r4569c4~8~ => r4c6<>82-link Kraken ALS (r9c1-7-r89c3-6-r2c34679-2-, r9c4-7-r4c47-9-r45c8|r6c9-2-, r9c7-7-r2c4679-2-, r9c7-2-r5c7=2=r45c8-2-): r9c147=72 => r2c8<>2+-------------------+--------------------+---------------------+|   56    26   246  | 1458      7     9  |     3   1246    28  ||    1     9  2467@ | 458@\$     3   48@\$ | 2457@\$ 467-2  248@\$ ||  357    37     8  |  145      2     6  |   457    147     9  |+-------------------+--------------------+---------------------+|    4  2378     1  |   78#     5   237  |    89#  239#%    6  ||   36     5     9  |  468    146   123  |   248%   24#%    7  ||  678  2678   267  |   79   4689  2347  |     1      5    34# |+-------------------+--------------------+---------------------+|    2   178     5  |    3    489  1478  |     6    479    14  || 6789  1678   367@ |    2  14689   147  |   479   3479     5  ||  679*    4   367@ |  679*   169     5  |   279*     8   123  |+-------------------+--------------------+---------------------+2-celled/2-link Kraken AALS (r8c6-1-r8c2=1=r7c2-1-r7c9-4-, r8c7-9-r4c7=9=r4c8-9-r12357c8-4-): r8c67=149 => r8c8<>4+-------------------+--------------------+-------------------+|   56    26   246  | 1458      7     9  |    3   1246#  28  ||    1     9  2467  |  458      3    48  | 2457    467# 248  ||  357    37     8  |  145      2     6  |  457    147#   9  |+-------------------+--------------------+-------------------+|    4  2378     1  |   78      5   237  |   89#   239#   6  ||   36     5     9  |  468    146   123  |  248     24#   7  ||  678  2678   267  |   79   4689  2347  |    1      5   34  |+-------------------+--------------------+-------------------+|    2   178@    5  |    3    489  1478  |    6    479#  14@ || 6789  1678@  367  |    2  14689   147* |  479* 379-4    5  ||  679     4   367  |  679    169     5  |  279      8  123  |+-------------------+--------------------+-------------------+3-celled/2-link Kraken AALS (r7c2-1-r7c9-4-r6c9=4=r6c56-4-, r7c2-8-r7c6=8=r2c6=4=r123c4-4-, r8c3=3=r8c8-3-r45c78-4-, r9c3-6-r8c123=6=r8c5-6-r123469c4-4-): r7c2|r89c3=186 => r5c4<>4+--------------------+---------------------+------------------+|   56    26    246  | 1458#%     7     9  |    3  1246   28  ||    1     9   2467  |  458#%     3    48# | 2457   467  248  ||  357    37      8  |  145#%     2     6  |  457   147    9  |+--------------------+---------------------+------------------+|    4  2378      1  |    78%     5   237  |   89\$  239\$   6  ||   36     5      9  |  68-4    146   123  |  248\$   24\$   7  ||  678  2678    267  |    79%  4689@ 2347@ |    1     5   34@ |+--------------------+---------------------+------------------+|    2   178*     5  |     3    489  1478# |    6   479   14@ || 6789% 1678% 367*\$% |     2  14689%  147  |  479   379\$   5  ||  679     4    367* |   679%   169     5  |  279     8  123  |+--------------------+---------------------+------------------+Locked Column Box/Box: r5678c5|r678c6 => r2c6<>4Locked Column Line/Box: r78c5 => r6c5<>8Locked Row Box/Box: r45c47 => r4c2<>83-element Grouped Nice Loop: r7c9-1-r9c9=1=r9c5-1-ALS:r78c6~4~ => r7c5<>44-valued/2-link Kraken Cell (r4c8-2-r5c78=2=r5c6=1=r5c5-1-, r4c8-3-r67c9-1-r7c2=1=r8c2-1-, r4c8-9-r7c689-1-r7c2=1=r8c2-1-): r4c8=239 => r8c5<>1+--------------------+--------------------+------------------+|   56     26   246  | 145       7     9  |    3  1246    8  ||    1      9  2467  |  45       3     8  | 2457   467   24  ||  357     37     8  | 145       2     6  |  457   147    9  |+--------------------+--------------------+------------------+|    4    237     1  |  78       5   237  |   89   239*   6  ||   36      5     9  |  68     146@  123@ |  248@   24@   7  ||  678   2678   267  |  79     469  2347  |    1     5   34# |+--------------------+--------------------+------------------+|    2   178#\$    5  |   3      89   147\$ |    6   479\$ 14#\$ || 6789  1678#\$  367  |   2  4689-1   147  |  479   379    5  ||  679      4   367  | 679     169     5  |  279     8  123  |+--------------------+--------------------+------------------+5-valued/2-link Kraken House (r8c1=9=r9c1-9-r9c45|r78c6-6-, r7c2-8-r7c5=8=r8c5=6=r9c45-6-, r8c2=1=r8c6-1-r9c145-6-): r8c12|r7c2=8 => r9c3<>6+---------------------+---------------------+------------------+|    56     26   246  |   145      7     9  |    3  1246    8  ||     1      9  2467  |    45      3     8  | 2457   467   24  ||   357     37     8  |   145      2     6  |  457   147    9  |+---------------------+---------------------+------------------+|     4    237     1  |    78      5   237  |   89   239    6  ||    36      5     9  |    68    146   123  |  248    24    7  ||   678   2678   267  |    79    469  2347  |    1     5   34  |+---------------------+---------------------+------------------+|     2    178*    5  |     3     89#  147@ |    6   479   14  || 6789*@ 1678*\$  367  |     2   4689# 147@\$ |  479   379    5  ||  679@\$     4  37-6  | 679@#\$ 169@#\$    5  |  279     8  123  |+---------------------+---------------------+------------------+4-valued/2-link Kraken Cell (r4c8-2-r34c2-7-, r4c8-3-r8c8=3=r8c3-3-r9c3-7-, r4c8-9-r4c7=9=r89c7-9-r7c689-7-): r4c8=239 => r7c2<>7+-------------------+------------------+------------------+|   56    26   246  | 145     7     9  |    3  1246    8  ||    1     9  2467  |  45     3     8  | 2457   467   24  ||  357    37@    8  | 145     2     6  |  457   147    9  |+-------------------+------------------+------------------+|    4   237@    1  |  78     5   237  |   89\$  239*   6  ||   36     5     9  |  68   146   123  |  248    24    7  ||  678  2678   267  |  79   469  2347  |    1     5   34  |+-------------------+------------------+------------------+|    2  18-7     5  |   3    89   147\$ |    6   479\$  14\$ || 6789  1678   367# |   2  4689   147  |  479\$  379#   5  ||  679     4    37# | 679   169     5  |  279\$    8  123  |+-------------------+------------------+------------------+Overlap 3-element Grouped Nice Loop: ALS:r8c7|r7c8-4-ALS:r7c259-9-ALS:r12357c8~7~ => r8c8<>7Multiple Overlap 4-element Grouped Nice Loop: ALS:r34c2-2-ALS:r48c8-9-ALS:r7c689-1-ALS:r13467c2~3~ => r8c2<>7Bivalued/2-link Kraken Cell (r6c5-4-r6c9-3-r9c9=3=r8c8-3-, r6c5-6-r6c123=6=r5c1=3=r4c2-3-, r6c5-9-r7c5=9=r7c8-9-r8c8-3-): r6c5=469 => r4c8<>3+-------------------+------------------+------------------+|   56    26   246  | 145     7     9  |    3  1246    8  ||    1     9  2467  |  45     3     8  | 2457   467   24  ||  357    37     8  | 145     2     6  |  457   147    9  |+-------------------+------------------+------------------+|    4   237#    1  |  78     5   237  |   89  29-3    6  ||   36#    5     9  |  68   146   123  |  248    24    7  ||  678# 2678#  267# |  79   469* 2347  |    1     5   34@ |+-------------------+------------------+------------------+|    2    18     5  |   3    89\$  147  |    6   479\$  14  || 6789   168   367  |   2  4689   147  |  479   39@\$   5  ||  679     4    37  | 679   169     5  |  279     8  123@ |+-------------------+------------------+------------------+Locked Row Line/Box: r5c78 => r5c5<>44-element Nice Loop: r4c6=3=r4c2-3-r5c1-6-r5c4-8-r4c4~7~r4c6 => r4c6<>74-element Nice Loop: r4c4-8-r4c7-9-r4c8=9=r7c8=7=r7c6~7~ => r6c6<>7,r9c4<>7XYZ-wing: r9c45, r5c5 => r8c5<>6Locked Row Line/Box: r9c45 => r9c1<>6WXYZ-wing: r79c5|r9c4, r7c2 => r7c6<>15-link Advanced Colouring: r4c8=9=r4c7=8=r4c4=7=r6c4=9=r9c4-9-r7c5=9=r7c8-9-r4c8 => r89c5<>9Naked Column Pair: r59c5 => r6c5<>6Locked Row Line/Box: r5c45 => r5c1<>6Locked Column Box/Box: r389c1|r28c3 => r6c13<>7Row X-Wing Fillet-o-Fish: r3c178|r9c17 => r2c7<>7Naked Row Triple: r2c479 => r2c3<>24,r2c8<>4`
Mike Barker

Posts: 458
Joined: 22 January 2006