Sudoku Players' Forum (clone)

by **Mike Barker** » Sun Sep 24, 2006 1:23 pm

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.

by **daj95376** » Sun Sep 24, 2006 9:01 pm

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.

by **Mike Barker** » Mon Sep 25, 2006 4:09 am

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.

by **gurth** » Mon Sep 25, 2006 8:51 am

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.

by **gurth** » Mon Sep 25, 2006 8:58 am

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.

by **Myth Jellies** » Mon Sep 25, 2006 4:03 pm

gurth wrote: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.

gurth wrote: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.

gurth wrote: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.

by **daj95376** » Mon Sep 25, 2006 5:01 pm

Mike Barker wrote: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.

by **Mike Barker** » Thu Sep 28, 2006 1:49 pm

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).

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

by **Myth Jellies** » Sat Sep 30, 2006 3:06 am

Mike Barker's Program wrote: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.

by **Mike Barker** » Sun Oct 01, 2006 12:10 am

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

by **AndrewStuart** » Fri Oct 06, 2006 3:44 pm

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.

by **Mike Barker** » Sun Oct 08, 2006 1:57 pm

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

by **AndrewStuart** » Sun Oct 08, 2006 3:38 pm

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

by **Mike Barker** » Sat Jan 26, 2008 7:07 pm

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!

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

by **champagne** » Sat Jan 26, 2008 10:33 pm

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

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Challenge: New set of 11 'Unsolvables'

Unsolvables #31, #32 and #23

GET and GMET

Re: GET and GMET

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