The New Sudoku Players' Forum

[Mike Barker]

The elimination for r8c1 for r3c1<>9 is direct (they share the same house) so I don't show it on the list of links. For r1c2<>9 the use of the strong link and the direct link get reversed. Since they are almost identical I had manually combined the two. Also I agree that SUM does include multi-implication chains.

Note that SUM probably shouldn't be directly compared to triple implication chains (TICs). They are really not the same thing. It would be kind of like comparing an apple to a tree. An apple is a fruit useful for apple pies. The tree is good for explaining where apples come from and for making more apples (not a great analogy, but hopefully it conveys the point). TICs are a technique, SUM is a model. The purpose of TICs is to solve puzzles; the purpose of SUM is to explain where techniques come from and to help create new techniques. So SUM can explain why TICs work, provide a framework for doing things like combining the properties of Kraken fish and Death Blossom to create Kraken Blossom, and create a different technique like Kraken House.

[daj95376]

Mike Barker, Okay on SUM being a model. However, I'd like to now digress to your solution for U#24.

My modest solver doesn't have anywhere near the number of techniques of your solver. However, I decided to compare my solution against yours and see what I could learn -- besides humility. The listing below appends your eliminations after mine. It's seldom that your eliminations are out of order.

Code: Select all

6...8...5.4...128..8.....6...7..23.....5.8.....17......6.....4...43...2.3...9...6

    b4  -  8     Locked Candidate (1)
    b6  -  5     Locked Candidate (1)
    b6  -  8     Locked Candidate (1)
  c8    -  9     Locked Candidate (2)
Locked Column Line/Box: r46c1 => r78c1<>8
Locked Column Line/Box: r46c8 => r9c8<>5
Locked Column Line/Box: r46c9 => r8c9<>8
Locked Column Box/Box: r137c7|r238c9 => r5c7<>9,r456c9<>9

r3c1    <> 5     Templates (A)
Column Swordfish Fillet-o-Fish: r2379c3|r379c6|r79c7 => r3c1<>5

r3c6    <> 4     Forcing Chain/Net on [r8c5]
r6c6    <> 4     Forcing Chain/Net on [r8c5]
r7c5    <> 5     Forcing Chain/Net on [r8c5]
r9c2    <> 5     Forcing Chain/Net on [r8c5]
A=2 cell ALS xz-rule: r1c34-9-r179c6 => r3c6<>4
A=2 cell ALS xz-rule: r56c5-1-r8c5|r79c6 => r6c6<>4
B=2 cell ALS xy-rule: r179c6-9-r13c4-2-r234568c5 => r7c5<>5
4-element Grouped Nice Loop: r4c2-9-r4c4=9=r6c6-9-ALS:r179c6-5-r8c5=5=r8c12~5~ => r9c2<>5

r6c8    =  5     Forcing Chain/Net on [r6c6]
Bivalued Kraken House (SUM Exclusion) (r5c2|r5c1-9-r4c2-5-, r5c8-9-r6c8-5-): r5c128 => r4c8<>5,r6c21<>5

r2c4    =  6     Forcing Chain/Net on [r2c1]
r4c5    =  6     Forcing Chain/Net on [r2c1]
Multiple 4-element Grouped Nice Loop: ALS:r2c19-5-r23c3=5=r79c3-5-r8c12=5=r8c5-5-ALS:r179c6~7~ => r2c4<>9

r1c2    <> 9     Templates (A)
r3c1    <> 9     Templates (A)
Bivalued Kraken House (SUM Exclusion) (r8c2-9-r456c2=9=r456c1-9-, r8c9-9-r2c9=9=r2c13-9-): r8c129 => r3c1|r1c2<>9

r3c3    <> 2     Forcing Chain/Net on [r2c5]
Overlap 4-element Grouped Nice Loop: r1c3-9-ALS:r179c6-5-r8c5=5=r8c12-5-ALS:r179c3~2~ => r3c3<>2

r9c7    <> 1     Forcing Chain/Net on [r1c7]
Bivalued Kraken House (SUM Exclusion) (r1c2=1=r1c7-1-, r1c6-7-r7c6-5-r7c7|r9c8|r8c9-1-, r1c7-7-r2c9-9-r9c8|r8c9-1-): r1c267 => r9c7<>1

r9c3    <> 5     Forcing Chain/Net on [r2c1]
(none)

r4c9    <> 1     Forcing Chain/Net on [r3c9]
3-element Grouped Nice Loop: ALS:r4c48-4-r56c5=4=r3c5-4-ALS:r238c9~1~ => r4c9<>1


At this point, there's no further overlap in our eliminations. I think it's because I restrict my Forcing Chain/Net routine to resolve only Naked/Hidden Singles. Therefore, I was missing the (2x) Locked Candidates (2) logic necessary to obtain your r4c4<>4 elimination. Still, I'm surprised that there was even this much similarity between our solutions.

[Mike Barker]

Thanks Daj. When I was first programming up grouped nice loops I moved them right after singles in my solver. Not surprisingly these were the only techniques which were used in solutions besides a few unique rectangles. Forcing chains take the solving power up one level and for those that can employ them allow solutions to problems I'll probably never be able to touch. Its not surprising that forcing chains can solve U#24. What was surprising and rather cool was that the forcing chain and Kraken eliminations were so close.

So what's the difference? I could move grouped strong links back up to the top of my solving list, but just having solved a puzzle is only part of the challenge. I actually find application of the different logical techniques to solve a puzzle as something rather beautiful and creating new techniques as pleasurable. It is also useful, in that, saying an X-wing or ALS or even a Kraken Blossom has solved a puzzle says a lot more than it was solved with a forcing chain. Besides there's always a chance that a new / more powerful technique will be discovered. I keep trying out new techniques in my solver to see what they can do which is why I have so many. They are not needed, but you never know where an idea like SHuisman's APE extension or Anne Morelot's fish elimination might lead. Besides if you've ever visited the zoo there are a lot of colorful critters there.

[gurth]

Unsolvable #31 solved by a very fine Ruby :

Note that no candidates are placeable using SSTS. But does this make U#31 a pearl ? Definitely not ! There are superfluous clues.

However, the puzzle is immediately solvable by using the magnificent Ruby in cell c1. Putting -5c1 in this 4-candidate cell leads to a contradiction via singles only, whereupon putting 5c1 leads to solution via nothing more difficult than singles, locked candidates, and one naked trip.




Unsolvable #32 solved by 2 Forcing Nets :

1. SSTS. No cell establishable. Is THIS a pearl? No, it's not. It also has at least one superfluous clue. As these puzzles are not symmetrical anyway, why not spare us these redundant clues? And even if they were...

2. ? -2b9, singles only, ?? 2b9, singles, naked pair, 3 XY Wings, etc (SSTS).

3. ? -8c2, singles only, ?? 8c2, SINGLES TO END.




Unsolvable #23 solved by Forcing Nets :

1. SSTS.

2. ? 8d2, singles and one naked pair, ?? -8d2, 8d1.

3. ? 8c3, SSTS, ?? -8c3, 8c2, SSTS.

4. ? 4d2, singles, ?? -4d2.

5. ? 3d2, many singles only, ?? -3d2, 2d2, SSTS to solution.

[gurth]

Myth,
I see no reason to prefer "one of the candidates must be true" to "if (not A) then B". In fact I much prefer the latter, because it directly makes the vital implication, whereas your preferred version defers that deduction.

But sooner or later the deduction must be made, otherwise the chain is meaningless. And the only DEDUCTION that can be made from a strong inference is this: "if A is false, then B is true."

We cannot logically dispense with the assumption of a false statement.

Every "if" statement consists of two parts: the "if" part and the "then" part. The "if" part is the ASSUMPTION. (Note that the "if" statement does NOT assert that the ASSUMPTION is true.) The "if statement" then proceeds, in the "then" part, to describe the consequences IF THE ASSUMPTION IS TRUE. It does NOT state anything about the truth or falsehood of the ASSUMPTION.

In the case of a STRONG INFERENCE, we examine the consequences of the assumption of a statement A being false, without implying that the assumption itself is true or false.

I hope you now understand what I mean. I am sure there is no disagreement between us about how AIC chains work, it is just a matter of preferred terminology. Whether or not there is any less "bifurcation" in an AIC than in a Forcing Net, is another matter. I would say not.

[Myth Jellies]

But sooner or later the deduction must be made, otherwise the chain is meaningless. And the only DEDUCTION that can be made from a strong inference is this: "if A is false, then B is true."

We cannot logically dispense with the assumption of a false statement

I disagree. The relationship of the strong links to each other, forming an AIC, implies that one of the endpoints of an AIC must be true. You can then make deductions based purely on this knowledge.

In the case of a STRONG INFERENCE, we examine the consequences of the assumption of a statement A being false, without implying that the assumption itself is true or false.

No, I examine the consequences of the existance in the puzzle of fairly obvious pieces where a binary choice exists without ever assuming that one of those choices is either true or false. You are assuming a candidate has a particular truth value and then traveling up that branch to discover what contradictions you can find.

I hope you now understand what I mean. I am sure there is no disagreement between us about how AIC chains work, it is just a matter of preferred terminology. Whether or not there is any less "bifurcation" in an AIC than in a Forcing Net, is another matter. I would say not.

It is far more than just a matter of preferred terminology. In coloring if you assume one color has a particular truth value to make an additional discovery, you have in effect plugged in or erased values in all of your colored cells. Equivalently, in a chain if you make a move that relies on the assumption that one element of the chain has a particular truth value, you are no longer documenting a potential deduction due to the existance of forks in the puzzle, you are documenting the result of taking a particular fork in the puzzle--you are guessing a value and taking that path.

[daj95376]

Thanks Daj. When I was first programming up grouped nice loops I moved them right after singles in my solver. Not surprisingly these were the only techniques which were used in solutions besides a few unique rectangles. Forcing chains take the solving power up one level and for those that can employ them allow solutions to problems I'll probably never be able to touch. Its not surprising that forcing chains can solve U#24. What was surprising and rather cool was that the forcing chain and Kraken eliminations were so close.

So what's the difference? I could move grouped strong links back up to the top of my solving list, but just having solved a puzzle is only part of the challenge. I actually find application of the different logical techniques to solve a puzzle as something rather beautiful and creating new techniques as pleasurable. It is also useful, in that, saying an X-wing or ALS or even a Kraken Blossom has solved a puzzle says a lot more than it was solved with a forcing chain. Besides there's always a chance that a new / more powerful technique will be discovered. I keep trying out new techniques in my solver to see what they can do which is why I have so many. They are not needed, but you never know where an idea like SHuisman's APE extension or Anne Morelot's fish elimination might lead. Besides if you've ever visited the zoo there are a lot of colorful critters there.

I respect your solver and all of the interesting techniques it employs. I was not trying to imply that Forcing Chains/Nets should be used instead of your techniques. I was simply amazed at the coincidence in the solution path from the two different approaches. I hope your SUM model produces some powerful new techniques that everyone can use.

[Mike Barker]

Here is an alternative solution to #23 using SUM derived techniques: Kraken Blossom - Death Blossom with strong links, Kraken Unit - with the contradiction being an incomplete house (eg no 2 in a row and in this case trilocal rows), and Kraken Fish (an improved version over its predecessor). The second Kraken swordfish, although valid, is a pretty extreme elimination, but shows the ability of the approach which can address both the highly difficult and the relatively simple. It also demonstrates some pretty extreme overlap in the links. Only the really hard eliminations are done with the Kraken techniques as they are last in my solving hierarchy. Most, if not all, of the other steps could be done with much simpler Kraken steps. I left in UR's since I was pulling out the kitchen sink on this one (and personally I think they are an excellent solving technique).

1) Naked Single (42)
2) Hidden Single (12)
3) Locked Line/Box (6)
4) Locked Box/Box (3)
5) Strong Nice Loops with 5 GSL/BV Cells (3)
6) UR+2(X,D,B)/1SL (Type 4,...) (2)
7) Nice Loops with 3 Strong Links/BV Cells (2)
8) Nice Loops with 5 Strong Links/BV Cells (2)
9) Grouped Nice Loops with 3 GSL/ALS (2)
10) Grouped Nice Loops with 5 GSL/ALS (2)
11) B=1 cell ALS-xy rule (2)
12) 5-valued/2-link Kraken Blossom (2)
13) Hidden Pair (1)
14) Finned X-wing (1)
15) 5-node XY-chain (1)
16) UR+3(X,C,N,U,E)/2SL (1)
17) Advanced Colouring with 4 Links (1)
18) Advanced Colouring with 5 Links (1)
19) Grouped Nice Loops with 4 GSL/ALS (1)
20) A=2 cell ALS-xz rule (1)
21) B=2 cell ALS-xy rule (1)
22) 4-valued/2-link Kraken Blossom (1)
23) Bivalued/2-link Kraken Unit (1)
24) 3-valued/2-link Kraken Unit (1)
25) Bivalued/1-link Kraken Swordfish (1)
26) Bivalued/2-link Kraken Swordfish (1)
27) 2-link Kraken Swordfish (1)

Code: Select all

Row Finned X-Wing: r6c59|r9c579 => r8c9<>5
B=1 cell ALS xy-rule: r7c4-6-r3c4-5-r137c5 => r9c5<>7
A=2 cell ALS xz-rule: r69c8-2-r4c189 => r9c1<>8
Overlap 4-element Grouped Nice Loop: ALS:r69c1-7-r2c1=7=r2c6-7-r5c6-5-ALS:r6c1258~9~ => r5c1<>9
+-----------------------+------------------+----------------------+
|      5   3479    479  |    8   679    2  |      1   346   3469  |
|    379*     6      2  |    1     4  379* |     39     5      8  |
|      1   3489    489  |   56   569  359  |   3469     7      2  |
+-----------------------+------------------+----------------------+
|    348   2348      5  |    9     1    6  |      7   234     34  |
| 3467-9  23479   4679  |  257     8   57* | 234569     1  34569  |
|   679*b  279*b     1  |    3  257*b   4  |      8   26*b   569  |
+-----------------------+------------------+----------------------+
|      2      5  46789  |   67   679  789  |    346  3468      1  |
|   4678      1   4678  | 2567     3  578  |   2456     9    467  |
|    679*   789      3  |    4  2569    1  |    256   268    567  |
+-----------------------+------------------+----------------------+
Bivalued/1-element Kraken Row Swordfish (r269/c12, fins=r2c6|r6c5|r9c9) (r2c6-7-r5c6-5-, r6c5=5=r6c9-5-, r9c9=5=r56c9-5-): r2c16|r6c125|r9c129=7 => r5c7<>5
+---------------------+------------------+-----------------------+
|    5   3479    479  |    8   679    2  |       1   346   3469  |
|  379*     6      2  |    1     4  379* |      39     5      8  |
|    1   3489    489  |   56   569  359  |    3469     7      2  |
+---------------------+------------------+-----------------------+
|  348   2348      5  |    9     1    6  |       7   234     34  |
| 3467  23479   4679  |  257     8   57b | 23469-5     1  34569d |
|  679*   279*     1  |    3  257*c   4  |       8    26   569cd |
+---------------------+------------------+-----------------------+
|    2      5  46789  |   67   679  789  |     346  3468      1  |
| 4678      1   4678  | 2567     3  578  |    2456     9    467  |
|  679*   789*     3  |    4  2569    1  |     256   268   567*d |
+---------------------+------------------+-----------------------+
Locked Column Line/Box: r56c9 => r9c9<>5
3-element Grouped Nice Loop: ALS:r7c456-8-r7c8=8=r9c8-8-ALS:r9c129~6~ => r9c5<>6,r7c3<>9,r9c5<>9
+----------------------+-------------------+---------------------+
|    5   3479     479  |    8    679    2  |     1   346   3469  |
|  379      6       2  |    1      4  379  |    39     5      8  |
|    1   3489     489  |   56    569  359  |  3469     7      2  |
+----------------------+-------------------+---------------------+
|  348   2348       5  |    9      1    6  |     7   234     34  |
| 3467  23479    4679  |  257      8   57  | 23469     1  34569  |
|  679    279       1  |    3    257    4  |     8    26    569  |
+----------------------+-------------------+---------------------+
|    2      5  4678-9  |   67*   679* 789* |   346  3468*     1  |
| 4678      1    4678  | 2567      3  578  |  2456     9    467  |
| 679*b  789*b      3  |    4  25-69    1  |   256   268*   67*b |
+----------------------+-------------------+---------------------+
3-element Nice Loop: r8c4=2=r8c7=5=r9c7-5-r9c5-2-r8c4 => r8c7=25
5-element Strong Nice Loop: r2c1=7=r2c6-7-r5c6-5-r6c5=5=r6c9=9=r6c12-9-r5c3=9=r13c3~9~r2c1 => r2c1<>9
UR+3N/2SL (3,9): r23c67 => r3c7<>9
3-valued/2-element Kraken Row (r1c5=7=r2c6-7-r2c1-3-, r1c8-6-r6c8-2-r4c89-3-, r1c9-6-r89c9-4-r4c9-3-): r1c589=6 => r4c1<>3
+--------------------+------------------+---------------------+
|    5   3479   479  |    8  679*b   2  |     1   346*  3469* |
|   37b     6     2  |    1     4  379b |    39     5      8  |
|    1   3489   489  |   56   569  359  |   346     7      2  |
+--------------------+------------------+---------------------+
| 48-3   2348     5  |    9     1    6  |     7   234c   34cd |
| 3467  23479  4679  |  257     8   57  | 23469     1  34569  |
|  679    279     1  |    3   257    4  |     8    26c   569  |
+--------------------+------------------+---------------------+
|    2      5  4678  |   67   679  789  |   346  3468      1  |
| 4678      1  4678  | 2567     3  578  |    25     9    467d |
|  679    789     3  |    4    25    1  |   256   268     67d |
+--------------------+------------------+---------------------+
B=2 cell ALS xy-rule: r1489c9-9-r2c17-7-r45689c1 => r5c9<>3
4-valued/2-element Kraken Blossom (r5c7-2-r5c4=2=r8c4=6=r7c45-6-, r5c7-3-r4c89-2-r6c8-6-, r5c7-4-r4c89-2-r6c8-6-, r5c7-6-r5c13=6=r6c1-6-r9c1=6=r9c789-6-, r5c7-9-r2c7-3-r7c7|r89c9-6-): r5c7=23469 => r7c8<>6
+--------------------+-----------------+---------------------+
|    5   3479   479  |    8  679    2  |     1    346  3469  |
|   37      6     2  |    1    4  379  |    39e     5     8  |
|    1   3489   489  |   56  569  359  |   346      7     2  |
+--------------------+-----------------+---------------------+
|   48   2348     5  |    9    1    6  |     7   234cd  34cd |
| 3467e 23479  4679e |  257b   8   57  | 23469*     1  4569  |
|  679e   279     1  |    3  257    4  |     8    26cd  569  |
+--------------------+-----------------+---------------------+
|    2      5  4678  |   67b 679b 789  |   346e 348-6     1  |
| 4678      1  4678  | 2567b   3  578  |    25      9   467e |
|  679e   789     3  |    4   25    1  |   256e   268e   67e |
+--------------------+-----------------+---------------------+
5-valued/2-element Kraken Blossom (r8c1-4-r7c3=4=r7c78-4-r89c9-6-, r8c1-6-r6c1=6=r5c13-6-, r8c1-7-r2c17-9-r1489c9-6-, r8c1-8-r4c1-4-r46c8|r4c9-6-): r8c1=4678 => r5c9<>6
+--------------------+-----------------+--------------------+
|    5   3479   479  |    8  679    2  |     1  346   3469d |
|   37d     6     2  |    1    4  379  |    39d   5      8  |
|    1   3489   489  |   56  569  359  |   346    7      2  |
+--------------------+-----------------+--------------------+
|   48e  2348     5  |    9    1    6  |     7  234e   34de |
| 3467c 23479  4679c |  257    8   57  | 23469    1  459-6  |
|  679c   279     1  |    3  257    4  |     8   26e   569  |
+--------------------+-----------------+--------------------+
|    2      5  4678b |   67  679  789  |   346b 348b     1  |
| 4678*     1  4678  | 2567    3  578  |    25    9   467bd |
|  679    789     3  |    4   25    1  |   256  268    67bd |
+--------------------+-----------------+--------------------+
5-valued/2-element Kraken Blossom (r1c3-4-r7c3=4=r8c13-4-r89c9-6-, r1c3-7-r2c17-9-r1489c9-6-, r1c3-9-r1489c9-6-): r1c3=479 => r6c9<>6
+--------------------+-----------------+--------------------+
|    5   3479   479* |    8  679    2  |     1  346  3469cd |
|   37c     6     2  |    1    4  379  |    39c   5      8  |
|    1   3489   489  |   56  569  359  |   346    7      2  |
+--------------------+-----------------+--------------------+
|   48   2348     5  |    9    1    6  |     7  234    34cd |
| 3467  23479  4679  |  257    8   57  | 23469    1    459  |
|  679    279     1  |    3  257    4  |     8   26   59-6  |
+--------------------+-----------------+--------------------+
|    2      5  4678b |   67  679  789  |   346  348      1  |
| 4678b     1  4678b | 2567    3  578  |    25    9  467bcd |
|  679    789     3  |    4   25    1  |   256  268   67bcd |
+--------------------+-----------------+--------------------+
Bivalued/2-element Kraken Row (r1c5=7=r2c6=9=r2c7-9-, r1c8-6-r6c8=6=r5c7=9=r2c7-9-): r1c589=6 => r1c9<>9
+--------------------+------------------+--------------------+
|    5   3479   479  |    8  679*b   2  |     1  346* 346-9* |
|   37      6     2  |    1     4  379b |   39bc   5      8  |
|    1   3489   489  |   56   569  359  |   346    7      2  |
+--------------------+------------------+--------------------+
|   48   2348     5  |    9     1    6  |     7  234     34  |
| 3467  23479  4679  |  257     8   57  | 23469c   1    459  |
|  679    279     1  |    3   257    4  |     8   26c    59  |
+--------------------+------------------+--------------------+
|    2      5  4678  |   67   679  789  |   346  348      1  |
| 4678      1  4678  | 2567     3  578  |    25    9    467  |
|  679    789     3  |    4    25    1  |   256  268     67  |
+--------------------+------------------+--------------------+
Hidden Column Pair: r56c9 => r5c9=59
Overlap 5-element Strong Nice Loop: r1c23=9=r1c5=7=r2c6-7-r5c6-5-r5c9-9-r5c3=9=r13c3~9~ => r3c2<>9
2-element Kraken Row Swordfish (r158/c93, fins=r1c58|r5c17|r8c14) (r1c5-6-r7c5=6=r78c4-6-r3c4-5-, r5c7|r1c8-6-r6c8-2-r6c5|r5c6-5-, r5c1=3=r2c1-3-r25c6-5-, r8c1-6-r269c1-3-r25c6-5-, r8c4-6-r7c5=6=r13c5-6-r3c4-5-): r1c589|r5c137|r8c1349=6 => r5c4<>5
+---------------------+--------------------+-----------------+
|     5   3479   479  |     8  679*e    2  |    1  346* 346* |
|   37de     6     2  |     1     4   37de |    9    5    8  |
|     1    348   489  |   56be  569e  359  |  346    7    2  |
+---------------------+--------------------+-----------------+
|    48   2348     5  |     9     1     6  |    7  234   34  |
| 3467*d 23479  4679* |  27-5     8  57cde | 2346*   1   59  |
|   679e   279     1  |     3   257c    4  |    8   26c  59  |
+---------------------+--------------------+-----------------+
|     2      5  4678  |    67b 679be  789  |  346  348    1  |
|  4678*     1  4678* | 2567*b    3   578  |   25    9  467* |
|   679e   789     3  |     4    25     1  |  256  268   67  |
+---------------------+--------------------+-----------------+
5-element Grouped Nice Loop: ALS:r7c456|r8c6-5-r5c6=5=r5c9=9=r6c9-9-r6c1=9=r9c1=6=r9c789~6~ => r7c7<>6
+--------------------+-----------------+------------------+
|    5   3479   479  |    8  679    2  |    1   346  346  |
|   37      6     2  |    1    4   37  |    9     5    8  |
|    1    348   489  |   56  569  359  |  346     7    2  |
+--------------------+-----------------+------------------+
|   48   2348     5  |    9    1    6  |    7   234   34  |
| 3467  23479  4679  |   27    8   57* | 2346     1   59* |
|  679*   279     1  |    3  257    4  |    8    26   59* |
+--------------------+-----------------+------------------+
|    2      5  4678  |   67* 679* 789* | 34-6   348    1  |
| 4678      1  4678  | 2567    3  578* |   25     9  467  |
|  679*   789     3  |    4   25    1  | 256*b 268*b 67*b |
+--------------------+-----------------+------------------+
5-element Grouped Nice Loop: r4c1=8=r8c1-8-ALS:r58c6-7-ALS:r23c6|r3c45-6-ALS:r37c7-4-r5c7=4=r5c123~4~r4c1 => r4c1<>4
+-----------------------+-------------------+-----------------+
|     5    3479    479  |    8   679     2  |    1  346  346  |
|    37       6      2  |    1     4   37*c |    9    5    8  |
|     1     348    489  |  56*c 569*c 359*c | 346*d   7    2  |
+-----------------------+-------------------+-----------------+
|   8-4*   2348      5  |    9     1     6  |    7  234   34  |
| 3467*e 23479*e 4679*e |   27     8   57*b | 2346*   1   59  |
|   679     279      1  |    3   257     4  |    8   26   59  |
+-----------------------+-------------------+-----------------+
|     2       5   4678  |   67   679   789  |  34*d 348    1  |
|  4678*      1   4678  | 2567     3  578*b |   25    9  467  |
|   679     789      3  |    4    25     1  |  256  268   67  |
+-----------------------+-------------------+-----------------+
3-element Grouped Nice Loop: ALS:r2689c1-4-ALS:r89c9|r9c78|r8c7-8-ALS:r14569c2~3~ => r3c2<>3
+---------------------+-----------------+-------------------+
|    5   3479*c  479  |    8  679    2  |    1   346   346  |
|   37*      6     2  |    1    4   37  |    9     5     8  |
|    1    48-3   489  |   56  569  359  |  346     7     2  |
+---------------------+-----------------+-------------------+
|    8    234*c    5  |    9    1    6  |    7   234    34  |
| 3467  23479*c 4679  |   27    8   57  | 2346     1    59  |
|  679*   279*c    1  |    3  257    4  |    8    26    59  |
+---------------------+-----------------+-------------------+
|    2       5  4678  |   67  679  789  |   34   348     1  |
|  467*      1  4678  | 2567    3  578  |  25*b    9  467*b |
|  679*   789*c    3  |    4   25    1  | 256*b 268*b  67*b |
+---------------------+-----------------+-------------------+
5-element Strong Nice Loop: r1c2=3=r2c1=7=r2c6-7-r5c6-5-r5c9-9-r5c3=9=r13c3~9~r1c2 => r1c2<>9
Locked Column Line/Box: r13c3 => r5c3<>9
Bivalued/2-element Kraken Column Swordfish (c137/r5b7, fins=r13c3|r37c7) (r1c3=9=r3c3=8=r3c2-8-, r3c3=8=r3c2-8-, r3c7-4-r1c89=4=r1c23-4-r3c2-8-, r7c7=3=r7c8=8=r9c8-8-): r58c1|r13578c3|r357c7=4 => r9c2<>8
+---------------------+-----------------+-----------------+
|    5    347d 479*bd |    8  679    2  |    1  346d 346d |
|   37      6      2  |    1    4   37  |    9    5    8  |
|    1   48bcd 489*bc |   56  569  359  |  346*   7    2  |
+---------------------+-----------------+-----------------+
|    8    234      5  |    9    1    6  |    7  234   34  |
| 3467* 23479    467* |   27    8   57  | 2346*   1   59  |
|  679    279      1  |    3  257    4  |    8   26   59  |
+---------------------+-----------------+-----------------+
|    2      5   4678* |   67  679  789  |  34*e 348e   1  |
|  467*     1   4678* | 2567    3  578  |   25    9  467  |
|  679   79-8      3  |    4   25    1  |  256  268e  67  |
+---------------------+-----------------+-----------------+
Locked Row Line/Box Pair: r7c78 => r7c3<>4
Locked Row Line/Box: r8c13 => r8c9<>4
Locked Column Line/Box: r89c7 => r5c7<>2
Locked Column Line/Box Pair: r89c9 => r1c9<>6
Locked Column Box/Box: r35c7|r16c8 => r9c7<>6
UR+2B/1SL (6,7): r89c19 => r8c1<>7
4-element Advanced Colouring: r8c1=4=r8c3-4-r3c3=4=r3c7=6=r5c7-6-r6c8=6=r6c1~6~r8c1 => r8c1<>6
5-element Advanced Colouring: r2c6=7=r1c5=6=r1c8-6-r6c8=6=r6c1-6-r9c1=6=r9c9=7=r8c9~7~ => r8c6<>7
5-element Nice Loop: r7c3=8=r8c3-8-r8c6-5-r5c6=5=r5c9=9=r5c2-9-r9c2~7~r7c3 => r7c3<>7
Locked Row Box/Box: r8c39|r9c129 => r8c4<>7
5-element Nice Loop: r7c4=7=r5c4=2=r6c5-2-r6c8-6-r1c8=6=r1c5=7=r2c6~7~ => r7c6<>7
B=1 cell ALS xy-mer: r69c2-2-r6c8-6-r1c289-7-r69c2|r5c7 => r5c2<>7,r5c2<>9,r1c3<>4
Locked Row Box/Box: r8c39|r9c19 => r8c4<>6
UR+2B/1SL (3,4): r14c89 => r4c8<>4
5-node XY-chain (r5c4-2-r8c4-5-r3c4-6-r1c5-9-r1c3-7-) => r5c3<>7
3-element Nice Loop: r3c3=4=r3c7=6=r5c7-6-r5c3-4-r3c3 => r5c1<>6

[Myth Jellies]

2-link Kraken Row Swordfish (r2c6-7-r5c6-5-, r6c5=5=r6c9-5, r9c9=5=r56c9-5-): r2c16|r6c125|r9c129 => r5c7<>5

3-valued/2-link Kraken House (SUM Exclusion) (r1c5=7=r2c6-7-r2c1-3-, r1c8-6-r6c8-2-r4c89-3-, r1c9-6-r89c9-4-r4c9-3-): r1c589 => r4c1<>3

4-valued/2-link Kraken Blossom (SUM Exclusion) (r5c7-2-r5c4=2=r8c4=6=r7c45-6-, r5c7-34-r4c89-2-r6c8-6-, r5c7-6-r5c13=6=r6c1-6-r9c1=6=r9c789-6-, r5c7-9-r2c7-3-r7c7|r89c9-6-): r5c7 => r7c8<>6

Mike, I have a request. It would help me to better follow your program's output if you would add just a little more information on your starting seed or branch point. Something like...

2-link Kraken Row Swordfish (r2c6-7-r5c6-5-, r6c5=5=r6c9-5, r9c9=5=r56c9-5-): r2c16|r6c125|r9c129 = 7 => r5c7<>5

3-valued/2-link Kraken House (SUM Exclusion) (r1c5=7=r2c6-7-r2c1-3-, r1c8-6-r6c8-2-r4c89-3-, r1c9-6-r89c9-4-r4c9-3-): r1c589 = 6 => r4c1<>3

4-valued/2-link Kraken Blossom (SUM Exclusion) (r5c7-2-r5c4=2=r8c4=6=r7c45-6-, r5c7-34-r4c89-2-r6c8-6-, r5c7-6-r5c13=6=r6c1-6-r9c1=6=r9c789-6-, r5c7-9-r2c7-3-r7c7|r89c9-6-): r5c7 = 23569 => r7c8<>6

Sometimes it can be difficult to tell what the base start point really is without the extra guidance.

[Mike Barker]

Good idea, I've updated my solver and the above posting.

[AndrewStuart]

The unsolvables have been restored on

http://www.sudoku.org.uk/bifurcation.htm

I wish someone had emailed me about them going missing. It was the stupid content management system - overwrote my work and I didn't notice. Actually a hand crafted page and shouldn't have been in the CMS.

Very sorry about that. I've put credits in for Mike Barker on the solve routes i've 95% followed. I'll update that as I try to follow through everyones attempts.

[Mike Barker]

An ALS can also be used as the Restricted Subset (RS) using the SUM method. In this case the Contradiction Condition is the elimination of two digits from the ALS. One of the simpliest forms is the existing ALS xz-rule where one digit of the RS (the ALS) is directly linked to the Candidate Elimination Cell (CEC) and the second digit is linked via a second ALS. In the xy-rule the links between the RS and the CEC are both ALS. Note that the case of the xz-mutual exclusion rule is really just a special case of the xy-rule where the same ALS is used to link both digits. Now that's serious overlap! The standard ALS rules require that all candidates for one digit of the RS be linked to a single ALS or directly linked to the CEC. Using the SUM method this is not required - each candidate for one of the two digits can be linked to the CEC via a separate linking element or linking chain. Using this approach Unsolvable #33 can be solved using a variety of SUM derived techniques including the Kraken ALS. To show some milder eliminations using SUM derived techniques, I've moved some of the eliminations which do not use ALS except for bivalue cells ealier in the solving hierarchy. Again I've included uniqueness tests. With this all 11 of the latest unsolvables have been solved with SUM derived techniques, overlapping basic and grouped nice loops, or previously existing techniques.

Removed

[AndrewStuart]

Wow, that's a most impressive solution. It will take me a while to work through it, but well done Mike.

Would it be possible for me to email you directly? I have something to discuss I don't want to clutter up the forum with. If so, buzz me on andrew-at-scanraid-dot-com

[Mike Barker]

I find a great way to compare different techniques is to be able to evaluate multiple solutions to the same puzzle. Here's an nrct-chain solution to U#26. It replaces the grouped and simple nice loops of my previous solution with nrct-chains. I've kept xy-chains and hxy-chains (x-cycles and advanced colouring) as these are part of nrct-chains. I hope others will post alternative solutions to the unsolvables, especially if they emphasize human solvability, some new technique, or some clever solution strategy. Who knows maybe you'll end up in Andrew's hall of fame!

1) Hidden Single in a box (51)
2) NRCT chains with 5 Strong Links/BV Cells (5)
3) NRCT chains with 6 Strong Links/BV Cells (5)
4) Naked Single (3)
5) NRCT chains with 7 Strong Links/BV Cells (3)
6) Locked Line/Box - Pointing (2)
7) Locked Box/Box - Claiming (2)
8) Hidden Single (2)
9) NRCT chains with 4 Strong Links/BV Cells (2)
10) Naked Pair (1)
11) Hidden Pair in a box (1)
12) Empty Rectangle (1)
13) X-cycle with 2 Strong Links (1)
14) X-cycle with 3 Strong Links (1)
15) Advanced Colouring with 3 Links (1)
16) Advanced Colouring with 5 Links (1)
17) NRCT chains with 3 Strong Links/BV Cells (1)
18) NRCT chains with 8 Strong Links/BV Cells (1)
19) NRCT chains with 9 Strong Links/BV Cells (1)
20) NRCT chains with 10 Strong Links/BV Cells (1)

Code: Select all

Locked Column Line/Box: r12c1 => r579c1<>7
Locked Column Box/Box: r38c8|r389c9 => r46c8<>6,r5c9<>6
Two Strong Links: r5c9 =1= r5c1 -1- r6c2 =1= r3c2 ~1~  => r3c9<>1
4-element NRCT chain: r2c6 -1- r8c6 =1= r7c5 =7= r7c3 =8= (r7c5)r7c1 ~8~  => r2c1<>8
+--------------------+---------------------+-------------------+
| 13478     2     6  | 3458    1348  1458  |  9    137   1378  |
| 137-8     9     5  |  238       6    18* |  4   1237  12378  |
|  1348   138   348  |    9   12348     7  |  5   1236   2368  |
+--------------------+---------------------+-------------------+
|     9  3678  2378  |    1    2478   468  | 36     37      5  |
|   136     4    37  |   57      79   569  |  2      8   1379  |
|     5  1678   278  |  278    2789     3  | 16    179      4  |
+--------------------+---------------------+-------------------+
|   348*    5  3478* |    6  134789*    2  | 13   1349    139  |
| 23468   368     9  |  348       5   148* |  7  12346   1236  |
|  2346   367     1  |  347    3479    49  |  8      5   2369  |
+--------------------+---------------------+-------------------+
5-element Advanced Colouring: r8c6 =1= r7c5 -1- r7c7 =1= r6c7 -1- r6c2 =1= r5c1 =6= r5c6 =5= r1c6 ~1~ r8c6
   => r1c6<>1
5-element NRCT chain: r3c3 =4= r7c3 -4- r7c8 =4= r8c8 =6= r3c8 =2= (r8c8)r2c8 -2- r2c4 =2= r3c5 ~4~ r3c3 => r3c5<>4
+--------------------+--------------------+-------------------+
| 13478     2     6  | 3458    1348  458  |  9    137   1378  |
|   137     9     5  |  238*      6   18  |  4   1237* 12378  |
|  1348   138   348* |    9  1238-4*   7  |  5   1236*  2368  |
+--------------------+--------------------+-------------------+
|     9  3678  2378  |    1    2478  468  | 36     37      5  |
|   136     4    37  |   57      79  569  |  2      8   1379  |
|     5  1678   278  |  278    2789    3  | 16    179      4  |
+--------------------+--------------------+-------------------+
|   348     5  3478* |    6  134789    2  | 13   1349*   139  |
| 23468   368     9  |  348       5  148  |  7  12346*  1236  |
|  2346   367     1  |  347    3479   49  |  8      5   2369  |
+--------------------+--------------------+-------------------+
Locked Row Line/Box: r1c456 => r1c1<>4
Overlap 5-element NRCT chain: r1c6 =5= r1c4 -5- r5c4 =5= r5c6 =6= r4c6 =4= r4c5 -4- (r1c4)r1c5 ~4~  => r1c6=45
+--------------------+---------------------+-------------------+
|  1378     2     6  | 3458*   1348* 45-8* |  9    137   1378  |
|   137     9     5  |  238       6    18  |  4   1237  12378  |
|  1348   138   348  |    9    1238     7  |  5   1236   2368  |
+--------------------+---------------------+-------------------+
|     9  3678  2378  |    1    2478*  468* | 36     37      5  |
|   136     4    37  |   57*     79   569* |  2      8   1379  |
|     5  1678   278  |  278    2789     3  | 16    179      4  |
+--------------------+---------------------+-------------------+
|   348     5  3478  |    6  134789     2  | 13   1349    139  |
| 23468   368     9  |  348       5   148  |  7  12346   1236  |
|  2346   367     1  |  347    3479    49  |  8      5   2369  |
+--------------------+---------------------+-------------------+
7-element NRCT chain: r5c6 =6= r4c6 =4= r4c5 =2= r4c3 =8= (r4c56)r4c2 -8- (2)r6c3 -7- r7c3 =7= r7c5
   -7- r5c5 ~9~ r5c6 => r5c6<>9
+--------------------+---------------------+-------------------+
|  1378     2     6  | 3458    1348    45  |  9    137   1378  |
|   137     9     5  |  238       6    18  |  4   1237  12378  |
|  1348   138   348  |    9    1238     7  |  5   1236   2368  |
+--------------------+---------------------+-------------------+
|     9  3678* 2378* |    1    2478*  468* | 36     37      5  |
|   136     4    37  |   57      79* 56-9* |  2      8   1379  |
|     5  1678   278* |  278    2789     3  | 16    179      4  |
+--------------------+---------------------+-------------------+
|   348     5  3478* |    6  134789*    2  | 13   1349    139  |
| 23468   368     9  |  348       5   148  |  7  12346   1236  |
|  2346   367     1  |  347    3479    49  |  8      5   2369  |
+--------------------+---------------------+-------------------+
Lasso 10-element NRCT chain: r8c6 =1= r7c5 =7= r7c3 =8= (r7c5)r7c1 =4= (r7c35)r7c8 =9= r7c9 =3= (r7c1358)r7c7
   -3- r4c7 -6- r6c7 =6= r6c2 -6- (8)r8c2 -(367)- r9c2 => r8c89<>1,r2c6<>1,r7c5<>1
+--------------------+--------------------+--------------------+
|  1378     2     6  | 3458    1348   45  |  9     137   1378  |
|   137     9     5  |  238       6  8-1  |  4    1237  12378  |
|  1348   138   348  |    9    1238    7  |  5    1236   2368  |
+--------------------+--------------------+--------------------+
|     9  3678  2378  |    1    2478  468  | 36*     37      5  |
|   136     4    37  |   57      79   56  |  2       8   1379  |
|     5  1678*  278  |  278    2789    3  | 16*    179      4  |
+--------------------+--------------------+--------------------+
|   348*    5  3478* |    6  3478-1*   2  | 13*   1349*   139* |
| 23468   368*    9  |  348       5  148* |  7  2346-1  236-1  |
|  2346   367*    1  |  347     347    9  |  8       5    236  |
+--------------------+--------------------+--------------------+
Three Strong Links: r5c9 =1= r5c1 -1- r6c2 =1= r3c2 -1- r3c5 =1= r1c5 ~1~  => r1c9<>1
6-element NRCT chain: r4c8 -3- r4c7 =3= r7c7 =1= r6c7 -1- r6c2 =1= r3c2 -1- r3c5 =1= r1c5 -1- (3)r1c8 ~7~
    => r26c8<>7
+--------------------+----------------+------------------+
|  1378     2     6  | 345   134* 45  |  9    137*  378  |
|   137     9     5  |  23     6   8  |  4  123-7  1237  |
|  1348   138*  348  |   9   123*  7  |  5   1236  2368  |
+--------------------+----------------+------------------+
|     9  3678  2378  |   1  2478  46  | 36*    37*    5  |
|   136     4    37  |  57    79  56  |  2      8  1379  |
|     5  1678*  278  | 278  2789   3  | 16*  19-7     4  |
+--------------------+----------------+------------------+
|   348     5  3478  |   6  3478   2  | 13*  1349   139  |
| 23468   368     9  | 348     5   1  |  7   2346   236  |
|  2346   367     1  | 347   347   9  |  8      5   236  |
+--------------------+----------------+------------------+
3-element NRCT chain: r7c7 =1= r6c7 -1- r6c8 -9- r7c8 =9= r7c9 ~1~ r7c7 => r7c9<>1
+--------------------+----------------+-----------------+
|  1378     2     6  | 345   134  45  |  9   137   378  |
|   137     9     5  |  23     6   8  |  4   123  1237  |
|  1348   138   348  |   9   123   7  |  5  1236  2368  |
+--------------------+----------------+-----------------+
|     9  3678  2378  |   1  2478  46  | 36    37     5  |
|   136     4    37  |  57    79  56  |  2     8  1379  |
|     5  1678   278  | 278  2789   3  | 16*   19*    4  |
+--------------------+----------------+-----------------+
|   348     5  3478  |   6  3478   2  | 13* 1349* 39-1* |
| 23468   368     9  | 348     5   1  |  7  2346   236  |
|  2346   367     1  | 347   347   9  |  8     5   236  |
+--------------------+----------------+-----------------+
6-element NRCT chain: r4c8 -3- r4c7 =3= r7c7 =1= r7c8 -1- (3)r2c8 -2- r2c4 =2= r6c4 -2- r6c3 =2= r4c3 ~7~ r4c8
   => r4c3<>7
+---------------------+----------------+-----------------+
|  1378     2      6  | 345   134  45  |  9   137   378  |
|   137     9      5  |  23*    6   8  |  4   123* 1237  |
|  1348   138    348  |   9   123   7  |  5  1236  2368  |
+---------------------+----------------+-----------------+
|     9  3678  238-7* |   1  2478  46  | 36*   37*    5  |
|   136     4     37  |  57    79  56  |  2     8  1379  |
|     5  1678    278* | 278* 2789   3  | 16    19     4  |
+---------------------+----------------+-----------------+
|   348     5   3478  |   6  3478   2  | 13* 1349*   39  |
| 23468   368      9  | 348     5   1  |  7  2346   236  |
|  2346   367      1  | 347   347   9  |  8     5   236  |
+---------------------+----------------+-----------------+
7-element NRCT chain: r4c7 =3= r7c7 =1= r7c8 =4= r8c8 =6= r3c8 =2= (r8c8)r2c8 -2- r2c4 =2= r6c4 -2- r6c3
   =2= r4c3 ~3~ r4c7 => r4c3<>3
+--------------------+----------------+-----------------+
|  1378     2     6  | 345   134  45  |  9   137   378  |
|   137     9     5  |  23*    6   8  |  4   123* 1237  |
|  1348   138   348  |   9   123   7  |  5  1236* 2368  |
+--------------------+----------------+-----------------+
|     9  3678  28-3* |   1  2478  46  | 36*   37     5  |
|   136     4    37  |  57    79  56  |  2     8  1379  |
|     5  1678   278* | 278* 2789   3  | 16    19     4  |
+--------------------+----------------+-----------------+
|   348     5  3478  |   6  3478   2  | 13* 1349*   39  |
| 23468   368     9  | 348     5   1  |  7  2346*  236  |
|  2346   367     1  | 347   347   9  |  8     5   236  |
+--------------------+----------------+-----------------+
9-element NRCT chain: r3c3 =4= r7c3 =7= r7c5 =8= r8c4 =4= (r8c1)r8c8 =6= r3c8 =2= (r8c8)r2c8 -2- r2c4 =2= r6c4
   =7= (r9c4)r5c4 -7- r5c3 ~3~ r3c3 => r3c3<>3
+--------------------+----------------+-----------------+
|  1378     2     6  | 345   134  45  |  9   137   378  |
|   137     9     5  |  23*    6   8  |  4   123* 1237  |
|  1348   138  48-3* |   9   123   7  |  5  1236* 2368  |
+--------------------+----------------+-----------------+
|     9  3678    28  |   1  2478  46  | 36    37     5  |
|   136     4    37* |  57*   79  56  |  2     8  1379  |
|     5  1678   278  | 278* 2789   3  | 16    19     4  |
+--------------------+----------------+-----------------+
|   348     5  3478* |   6  3478*  2  | 13  1349    39  |
| 23468   368     9  | 348*    5   1  |  7  2346*  236  |
|  2346   367     1  | 347   347   9  |  8     5   236  |
+--------------------+----------------+-----------------+
Two Strong Links: r5c3 =3= r7c3 -3- r7c7 =3= r4c7 ~3~  => r4c2<>3,r5c9<>3
5-element NRCT chain: r5c3 =3= r7c3 =7= r7c5 =8= (r7c3)r7c1 -8- (3)r8c2 -6- (r6c2)r4c2 =6= r5c1 ~3~ r5c3 => r5c1<>3
+--------------------+----------------+-----------------+
|  1378     2     6  | 345   134  45  |  9   137   378  |
|   137     9     5  |  23     6   8  |  4   123  1237  |
|  1348   138    48  |   9   123   7  |  5  1236  2368  |
+--------------------+----------------+-----------------+
|     9   678*   28  |   1  2478  46  | 36    37     5  |
|  16-3*    4    37* |  57    79  56  |  2     8   179  |
|     5  1678   278  | 278  2789   3  | 16    19     4  |
+--------------------+----------------+-----------------+
|   348*    5  3478* |   6  3478*  2  | 13  1349    39  |
| 23468   368*    9  | 348     5   1  |  7  2346   236  |
|  2346   367     1  | 347   347   9  |  8     5   236  |
+--------------------+----------------+-----------------+
5-element NRCT chain: r1c5 =1= r3c5 -1- r3c2 =1= r6c2 -1- r5c1 -6- r5c6 =6= r4c6 =4= r4c5 ~4~ r1c5 => r1c5<>4
+-------------------+----------------+-----------------+
|  1378     2    6  | 345  13-4* 45  |  9   137   378  |
|   137     9    5  |  23     6   8  |  4   123  1237  |
|  1348   138*  48  |   9   123*  7  |  5  1236  2368  |
+-------------------+----------------+-----------------+
|     9   678   28  |   1  2478* 46* | 36    37     5  |
|    16*    4    3  |  57    79  56* |  2     8   179  |
|     5  1678* 278  | 278  2789   3  | 16    19     4  |
+-------------------+----------------+-----------------+
|   348     5  478  |   6  3478   2  | 13  1349    39  |
| 23468   368    9  | 348     5   1  |  7  2346   236  |
|  2346   367    1  | 347   347   9  |  8     5   236  |
+-------------------+----------------+-----------------+
Hidden Pair in a box: r1c46 => r1c4=45
6-element NRCT chain: r3c2 =1= r6c2 -1- r5c1 -6- r5c6 =6= r4c6 =4= r4c5 =2= r4c3 =8= (r4c5)r4c2 ~8~ r3c2 => r3c2<>8
+-------------------+----------------+-----------------+
|  1378     2    6  |  45    13  45  |  9   137   378  |
|   137     9    5  |  23     6   8  |  4   123  1237  |
|  1348  13-8*  48  |   9   123   7  |  5  1236  2368  |
+-------------------+----------------+-----------------+
|     9   678*  28* |   1  2478* 46* | 36    37     5  |
|    16*    4    3  |  57    79  56* |  2     8   179  |
|     5  1678* 278  | 278  2789   3  | 16    19     4  |
+-------------------+----------------+-----------------+
|   348     5  478  |   6  3478   2  | 13  1349    39  |
| 23468   368    9  | 348     5   1  |  7  2346   236  |
|  2346   367    1  | 347   347   9  |  8     5   236  |
+-------------------+----------------+-----------------+
4-element NRCT chain: r2c1 =7= r2c9 =1= r5c9 -1- r5c1 =1= r6c2 -1- r3c2 ~3~ r2c1 => r2c1<>3
+-------------------+----------------+-----------------+
|  1378     2    6  |  45    13  45  |  9   137   378  |
|  17-3*    9    5  |  23     6   8  |  4   123  1237* |
|  1348    13*  48  |   9   123   7  |  5  1236  2368  |
+-------------------+----------------+-----------------+
|     9   678   28  |   1  2478  46  | 36    37     5  |
|    16*    4    3  |  57    79  56  |  2     8   179* |
|     5  1678* 278  | 278  2789   3  | 16    19     4  |
+-------------------+----------------+-----------------+
|   348     5  478  |   6  3478   2  | 13  1349    39  |
| 23468   368    9  | 348     5   1  |  7  2346   236  |
|  2346   367    1  | 347   347   9  |  8     5   236  |
+-------------------+----------------+-----------------+
6-element NRCT chain: r8c4 =8= r6c4 -8- (r6c5)r4c5 =8= r7c5 =7= r7c3 =4= r3c3 =8= (r67c3)r4c3 -8- (r6c2)r4c2
   =8= r8c2 ~8~ => r8c1<>8
+--------------------+----------------+-----------------+
|   1378     2    6  |  45    13  45  |  9   137   378  |
|     17     9    5  |  23     6   8  |  4   123  1237  |
|   1348    13   48* |   9   123   7  |  5  1236  2368  |
+--------------------+----------------+-----------------+
|      9   678*  28* |   1  2478* 46  | 36    37     5  |
|     16     4    3  |  57    79  56  |  2     8   179  |
|      5  1678  278  | 278* 2789   3  | 16    19     4  |
+--------------------+----------------+-----------------+
|    348     5  478* |   6  3478*  2  | 13  1349    39  |
| 2346-8   368*   9  | 348*    5   1  |  7  2346   236  |
|   2346   367    1  | 347   347   9  |  8     5   236  |
+--------------------+----------------+-----------------+
5-element NRCT chain: r1c4 =4= r1c6 -4- r4c6 =4= r4c5 =2= r4c3 =8= (r4c5)r4c2 -8- r8c2 =8= r8c4 ~4~ r1c4 => r8c4<>4
+------------------+-----------------+-----------------+
| 1378     2    6  |   45*   13  45* |  9   137   378  |
|   17     9    5  |   23     6   8  |  4   123  1237  |
| 1348    13   48  |    9   123   7  |  5  1236  2368  |
+------------------+-----------------+-----------------+
|    9   678*  28* |    1  2478* 46* | 36    37     5  |
|   16     4    3  |   57    79  56  |  2     8   179  |
|    5  1678  278  |  278  2789   3  | 16    19     4  |
+------------------+-----------------+-----------------+
|  348     5  478  |    6  3478   2  | 13  1349    39  |
| 2346   368*   9  | 38-4*    5   1  |  7  2346   236  |
| 2346   367    1  |  347   347   9  |  8     5   236  |
+------------------+-----------------+-----------------+
6-element NRCT chain: r3c8 =6= r8c8 =4= r8c1 =2= r9c1 =6= (r8c1)r5c1 =1= r6c2 -1- r3c2 ~3~ r3c8 => r3c8<>3
+------------------+----------------+------------------+
| 1378     2    6  |  45    13  45  |  9    137   378  |
|   17     9    5  |  23     6   8  |  4    123  1237  |
| 1348    13*  48  |   9   123   7  |  5  126-3* 2368  |
+------------------+----------------+------------------+
|    9   678   28  |   1  2478  46  | 36     37     5  |
|   16*    4    3  |  57    79  56  |  2      8   179  |
|    5  1678* 278  | 278  2789   3  | 16     19     4  |
+------------------+----------------+------------------+
|  348     5  478  |   6  3478   2  | 13   1349    39  |
| 2346*  368    9  |  38     5   1  |  7   2346*  236  |
| 2346*  367    1  | 347   347   9  |  8      5   236  |
+------------------+----------------+------------------+
7-element NRCT chain: r1c5 -3- r2c4 -2- (3)r3c5 -1- r3c2 =1= r6c2 -1- (r6c8)r6c7 =1= r5c9 =7= r4c8 -7- (3)r1c8 ~1~
   => r1c1<>1
+-------------------+----------------+-----------------+
| 378-1     2    6  |  45    13* 45  |  9   137*  378  |
|    17     9    5  |  23*    6   8  |  4   123  1237  |
|  1348    13*  48  |   9   123*  7  |  5   126  2368  |
+-------------------+----------------+-----------------+
|     9   678   28  |   1  2478  46  | 36    37*    5  |
|    16     4    3  |  57    79  56  |  2     8   179* |
|     5  1678* 278  | 278  2789   3  | 16*   19     4  |
+-------------------+----------------+-----------------+
|   348     5  478  |   6  3478   2  | 13  1349    39  |
|  2346   368    9  |  38     5   1  |  7  2346   236  |
|  2346   367    1  | 347   347   9  |  8     5   236  |
+-------------------+----------------+-----------------+
8-element NRCT chain: r3c2 =1= r6c2 -1- (r6c8)r6c7 =1= r5c9 =7= r4c8 -7- (r6c2)r4c2 =7= r6c3 -7- r7c3 =7= r7c5
   =8= r8c4 -8- (7)r6c4 -2- r2c4 =2= r3c5 ~1~ r3c2 => r3c5<>1
+------------------+----------------+-----------------+
|  378     2    6  |  45    13  45  |  9   137   378  |
|   17     9    5  |  23*    6   8  |  4   123  1237  |
| 1348    13*  48  |   9  23-1*  7  |  5   126  2368  |
+------------------+----------------+-----------------+
|    9   678*  28  |   1  2478  46  | 36    37*    5  |
|   16     4    3  |  57    79  56  |  2     8   179* |
|    5  1678* 278* | 278* 2789   3  | 16*   19     4  |
+------------------+----------------+-----------------+
|  348     5  478* |   6  3478*  2  | 13  1349    39  |
| 2346   368    9  |  38*    5   1  |  7  2346   236  |
| 2346   367    1  | 347   347   9  |  8     5   236  |
+------------------+----------------+-----------------+
Naked Column Pair: r14c8 => r278c8<>3
3-element Advanced Colouring: r2c9 =1= r5c9 =7= r4c8 =3= r1c8 ~3~ r2c9 => r2c9<>3
Locked Column Box/Box: r137c1|r37c3 => r6c3<>8

[champagne]

For U26, my solver suggest about 12 AICs.

I picked up the three best to start

.26...9....5.6.4.....9.7...9..1....5.4....28.5....3..4...6.2.....9.5.7....1...85.

Code: Select all

134x7k8 2      6      |34z5a8 1348       145A8  |9    137      1378     
137K8   9      5      |2m38   6          1É8é   |4    1237     12378   
1348    1e38   34l8   |9      12M34x8    7      |5    1236b    1236B8   
-----------------------------------------------------------------------
9       3678   2c378  |1      2C4d78     4D6f8  |3G6g 3Ê7ê     5       
1e36f   4      3ë7Ë   |5A7a   7Ì9ì       5a6F9n |2    8        1E3s7Â9h
5       1E6g78 2C78   |2M78   2789h      3      |1g6G 179H     4       
-----------------------------------------------------------------------
348     5      34L7i8 |6      1p347I8ä9È 2      |1G3g 134o9h   139     
2J3468  368    9      |348    5          1P48   |7    12R34O6B 1236     
2j34y6  367I   1      |347á   3479       4n9N   |8    5        2J36À9È 

[]2r4c5-C/2r6c4_2r2c4/2r2c8_2r38c8/4r8c8|*r38c8_4r7c8/4r7c13_4r89c1/6r89c1|*r89c1_6r4c6/2r4c5-C|*r4c56 r4c3=2
[]9r5c6-n/9r5c5_7r5c5/7r7c5_7r7c3/8r37c3|*r37c3_8r46c3/8r4c2_8r4c356/6r4c6|*r4c356_6r5c6/9r5c6-n r9c6=9
[]1r7c5-p/4r7c135|*r7c135_4r7c8/4r7c13_4r89c1/1r5c1|*r589c1_1r5c9/1r6c7_1r7c7/1r7c5-p r8c6=1

EDIT 1

Although it would have been possible to continue with the first grid, one could expect simplest chains applying the three fixs.
The four AICs below are enough to crack the grid.

Code: Select all

134x7k8b 2       6      |3Ç4z5a 1n34    4a5A |9    137     1378B   
1p37K    9       5      |2G3g   6       8    |4    1237    1237     
1348     1e38    34l8   |9      1N2g34x 7    |5    1236c   1236C8b 
-------------------------------------------------------------------
9        3s6È78d 2      |1      4a78D   4A6a |3H6h 3Å7å    5       
1e36a    4       3Æ7æ   |5A7a   7F9f    5a6A |2    8       1E3s7Â9F
5        1E6h78  7Ä8ä   |2g78m  2G789F  3    |1h6H 179f    4       
-------------------------------------------------------------------
348      5       34L7i8 |6      347I8m  2    |1H3h 1Î34o9F 139f     
2J3468   368     9      |348M   5       1    |7    2R34O6C 236     
2j34y6   367I    1      |347á   347     9    |8    5       2J36À   

[]8r6c4-m/8r6c3_7r6c3/7r7c3_7r7c5/8r7c5-m
[]7r6c4/2r6c4_2r2c4/2r2c8_2r38c8/4r8c8|*r38c8_4r7c8/4r7c13_4r89c1/6r89c1|*r89c1_7r5c4/7r6c4
[]3r7c5/3r7c789_4r7c8/4r7c13_4r89c1/6r89c1|*r89c1_4r4c5/8r4c5_8r4c2/8r6c3_7r6c3/7r7c3_7r7c5/3r7c5
[]6r9c2/7r9c2_7r7c3/7r6c3_8r6c3/8r4c2_8r4c5/4r4c5_6r89c1/6r9c2


Here is the chain found by the solver for 8r6c4 on the first grid.

Code: Select all

[]8r6c4/8r8c4_8r7c5r8c6/7r7c5|*r7c5r8c6_7r7c3/8r37c3|*r37c3_8r46c3/8r4c2_8r4c356/6r4c6|*r4c356_6r89c1/4r89c1|*r89c1_4r7c13/4r7c8_4r8c8/2r38c8|*r38c8_2r2c8/2r2c4_2r6c4/8r6c4