The New Sudoku Players' Forum

Website · by **denis_berthier** » Fri Mar 18, 2022 12:17 pm

.
Considering the same list of 246 expanded puzzles as above, I activated the following rules: Subsets, Finned-Fish, Tridagon elimination rule.

- 94 are solved after a tridagon elimination;

- 35 have no tridagon elimination rule (as per the definition here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html:

Code: Select all: 2 6 7 8 13 18 29 36 37 38 39 40 41 48 51 56 60 61 63 72 80 81 102 103 106 118 121 123 126 139 146 147 177 192 217

I didn't check if the rule would appear after whip eliminations.

This doesn't imply they don't have other applications of the "trivalue oddagon contradiction pattern". I don't have yet the means of embedding "tridagon-links" in chains.
If anyone wants to play with those 35 puzzles, I'm particularly interested in tridagon-links appearing between candidates in diagonally opposed blocks or in adjacent blocks but in different rows/columns.

by **mith** » Fri Mar 18, 2022 3:09 pm

denis_berthier wrote:.
Hi mith,
Until recently, problems of redundancy (puzzles with the same Singles-expansion) in the hardest collection hadn't appeared. I don't know if it's because most of the puzzles don't have Singles at the start or because nobody had looked into the problem before or because in such cases only the first found puzzle was kept.

Now that this problem is taking huge proportions, you have introduced a separate database, with contains only the Singles-expansions (most of which will not be minimal puzzles, but that's irrelevant to rating them). This is an interesting step forward.

However, there seems to appear another problem. Puzzles in the collection of Singles-expansions maybe extensions of one another, as you have already noticed. This is another kind of redundancy. It remains interesting to keep all of them in the Singles-expansions collection, but, in order to avoid redundant calculations every time one tries to use it, we also need a means of marking which is a subset of which.

I agree, and I'm open to ideas for how best to do this. There are a few different potential issues:

1. If we only track the "maximal" expanded form for each puzzle, we potentially lose information on where puzzles get "harder". Taking Loki for example, we know that its singles-expansion is 29c, but then there is a 30c form (with two 11.8 minimals) and the 31c form after T&E(singles,2). The 30c version preserves the hardest step (by SE), subject to morph dependency due to the bug discussed, but the 31c does not. Likewise, we had the case of a puzzle with no "basic" trivalue oddagon deduction (what you call the tridagon elimination) which was a subset of one which did solve with a basic deduction (IIRC, the subset puzzle needs a virtual triple from the trivalue oddagon to resolve one of the potential guardians and reduce it to one remaining guardian for the "basic" deduction).
2. It's not entirely clear how "maximal" should be defined here in the first place. Maximal in terms of staying out of T&E(2) makes sense when we're talking about the small subset of puzzles that meet that criteria, but it's easy to imagine puzzles that are even larger being in T&E(2) but still being rated highly enough to be in the database. Uniqueness of a maximal form is also certainly not guaranteed.
3. The alternative is to track these in a tree structure, with each puzzle only pointing to their closest neighbor(s). This could get very complicated to track in the database, though.

This is where solution-minlex may be most useful. If a puzzle is a subset of another puzzle, their solution-minlex forms will be identical other than the extra digit(s). Storing in this form (probably in addition to the puzzle-minlex form, for those who want to avoid the knowledge that r1 is 1-9 in order!) would allow for some quick operations to determine if a puzzle is a subset of another, in lieu of explicitly storing this information.

by **mith** » Fri Mar 18, 2022 4:18 pm

Possibly winding down on the current neighborhood; regardless, I'm going to have to find a stopping point soon so I can tinker with the computer (long story), I'll post a larger update at that point since I will be backing up everything anyway.

Hidden Text: Show

Code: Select all: ........1......234.....5.6...7..2....85.976..92.86.....627.8.9.7.9.56...8..92.... ED=11.6/1.2/1.2 ........1......234.....5.6...728.....895.6...62..97.8..96..2...2.8.7....75.8.96.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6..14.6.7.1...8...2.67..861.7...21..46..77.4.8.... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.47.8.1..86.1.7....18.2.67..2..6...46..7.1... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6...4.6.7.1...8...2.67..861.7...21..46..77.4.8.1.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.47.8.1..86.1.7....18.2.67..2..6..146..7..... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6..14.6.7.....8.1.2.67..861.7...21..46..77.4.8.... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6..14.4.7.1....8..2.67..74.8....12..46..78.61.7... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.47.8.1..86.1.7.....47.1....18.2.67..2..6...4 ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6...4.4.7.1....8..2.67..74.8.1..12..46..78.61.7... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.47.8.1..86.1.7.....47......18.2.67..2..6..14 ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.....174..86..4.8...12.267.....1.42.8..678..16... ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.1...671.8...41..27..8.7.8.1....84.7...26...4.7.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.....671.8...41..27..8.7.8......84.7..126.1.4.7.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.4..8.1..86.1.7....18.2.6..6..7.1...72..6...4 ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.4..8.1..86.1.7....18.2.6..6..7.....72..6..14 ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.....671.8...41..27..8.84.7..122..8.1...6...4.7.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.1...671.8...41..27..8.84.7...22..8.1...6...4.7.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.....671.8...41..27..8.7.8.1....84.7..126...4.7.. ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.4..8.1..86.1.7.....47.1....18.2.6..72..6...4 ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.46..7.4..8.1..86.1.7.....47......18.2.6..72..6..14 ED=11.6/1.2/1.2 ........1.....2.3......4.5...2.6.....174..8..64.8...12.2.7.....1.42.8..678..16... ED=11.6/1.2/1.2 ........1.....2.34..2.1356.........6.7..3.....89.65.......24..3...1.6.454...5.12. ED=11.6/1.2/1.2

Website · by **denis_berthier** » Fri Mar 18, 2022 4:19 pm

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Hi mith,
Supposing that you keep the idea of 2 separate databases, they may have to contain not only different puzzle information, but also different puzzle forms.
For the database of minimals, considering the heritage, it's useful to keep their original form (which often displays the pattern they're supposed to have), with their original SER.

For the database of expanded forms, I agree that the solution-minlex form is the simplest one for all the operations one might want to do with it.
Probably, we would want a few standard scripts for extracting typical parts of the database of expanded forms:
- minimum puzzles in it (wrt to the inclusion relation),
- maximum puzzles in it (wrt to the inclusion relation),
- all the puzzles larger than one and smaller than another (wrt to the inclusion relation)...
The order in which the puzzles are printed could be obtained by applying a depth-first "search" of the inclusion tree, with decreasing SERs within each level, with maybe a leading space added at each level. (The number of clues is totally irrelevant when rating a puzzle and should not be used for ordering them.)

The relation between the two databases would be easy if each puzzle in this second database has a unique identifier: add it as a new field in the database of minimals.

As for the possibly multiple SER values for different morphs of a puzzle, I think it's a false problem, entirely due to the inconsistencies of SE. The simplest solution is to keep the SER value for the form of the puzzle present in the database (including for the 2nd database); no need to re-compute all the old puzzles of the first.
That a puzzle can have morphs with higher or lower SER is a curiosity with no intrinsic meaning; we all have known this for a long time (though your recent examples beat all the previous ones). And in any case, the SER value of a puzzle is only an indication about its complexity. (This is true for any rating.)

Now, that's only my view of it, based on what I do with it. Indeed, I don't know who else uses this database. It would be interesting to have some input from other users.

by **mith** » Fri Mar 18, 2022 4:32 pm

To be clear, the concern in point 1 above is not about the morph-dependency caused by the known SE bug, but rather the (morph-independent) differences between a puzzle and its subset puzzles. Differences here give an indication of the complexity of obtaining the extra digit(s). (And this isn't specific to SE - we may at some point want to determine the complexity according to SudoRules for obtaining these extra digits, for example, or any other rating system. Likewiwse for T&E classification differences, if any turn out to be in T&E(subsets,2) but not T&E(singles,2), etc.)

Anyway, all that can be handled however people find it useful, so long as we can easily link puzzles by the inclusion relation. Since I'll be writing scripts to generate all this information in the first place, it will be easy to modify those for extraction purposes.

by **marek stefanik** » Fri Mar 18, 2022 4:35 pm

denis_berthier wrote:35 have no tridagon elimination rule (as per the definition here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html:
Code: Select all
2 6 7 8 13 18 29 36 37 38 39 40 41 48 51 56 60 61 63 72 80 81 102 103 106 118 121 123 126 139 146 147 177 192 217

Thank you for the list.
In all of them, the guardians are confined in a single band.
For each puzzle I've added its skfr after that step (or btte, if it is solvable with basics), for a few hardest puzzles I've also added a rating after relabeling in one of the TH boxes.

Most puzzles only require an AIC with one extra strong link.

Code: Select all: .-----------------------.-------------------.---------------------. | 235689 34567 235678 | 35679 3568 3569 | 479 2489 1 | | 5689 14567 5678 | 15679 568 2 | 479 3 48 | | 2389 137 2378 | 1379 4 39 | 5 6 28 | :-----------------------+-------------------+---------------------: | 28 #356 28 | 3569 #356 7 | 13469 1459 3456 | |#356 3567 4 | 8 1 #356+9| 2 b57–9 356 | | 1 9 #356+7 |#356 2 4 | 8 a57 356 | :-----------------------+-------------------+---------------------: |#356 8 9 | 24 #356 1 | 346 245 7 | | 4 2 #356 |#356 7 8 | 136 15 9 | | 7 #356 1 | 24 9 #356 | 346 2458 234568 | '-----------------------'-------------------'---------------------' 9r5c6 == 7r6c3 – 7r6c8 = 7r5c8 => –9r5c8

AIC 1+TH; 2, 8, 13, 29, 39, 41, 48, 51, 56, 103, 106, 118, 121, 126, 139, 177, 192: Show

Some require an AHP.

Code: Select all: .------------------.-----------------.--------------------. |b3457 3468 a3479 | 3489 347 346 | 589 1 2 | |b257–1 1268 a279 | 289 27 26 | 3 589 4 | | 234 2348 2349 | 23489 1 5 | 6 7 89 | :------------------+-----------------+--------------------: | 2347 #234 8 | 15 9 #234 | 12457 2345 6 | |#234 5 1 |#234 6 7 | 2489 23489 389 | | 6 9 #234+7| 15 #234 8 | 12457 2345 1357 | :------------------+-----------------+--------------------: |#234+1 7 5 |#234 8 9 | 124 6 13 | | 8 1234 #234 | 6 5 #234 | 12479 2349 1379 | | 9 #234 6 | 7 #234 1 | 2458 23458 358 | '------------------'-----------------'--------------------' 1r7c1 == 7r6c3 – 7r12c3 = 57r12c1 => –1r2c1

AIC AHP+TH; 18, 60, 61, 63, 80, 81, 123, 146, 147: Show

Some have three guardians and require a short kraken or whip.

Code: Select all: .-----------------------.---------------------.-------------------. | 24 24 #789+6 | 3789 A3569 A35678 | 5789 #789 1 | | 16789 #789 16789 | 789 bB169–5 2 | 3 4 #789+5| |#789+1 3 5 | 789 a19 4 |#789 6 2 | :-----------------------+---------------------+-------------------: | 1289 2589 189 | 23 7 135 | 4 289 6 | | 12467 2457 167 | 24 8 9 | 157 3 57 | | 124789 245789 3 | 6 245 15 | 15789 2789 5789 | :-----------------------+---------------------+-------------------: |#789 1 2 | 5 349 378 | 6 #789 34 | | 3 #789 4 | 1 69 678 | 2 5 #789 | | 5 6 #789 | 234789 2349 378 |#789 1 34 | '-----------------------'---------------------'-------------------' TH 789b1379 using internals: | 6r1c3 – 6r1c56 = 6r2c5 | 1r3c1 – 1r3c5 = 1r2c5 | 5r2c9 => -5r2c5

Kraken 2+TH; 7, 38, 40, 72: Show

A few (all subgrids of the last one) have a short nice loop with two bi-value cells.

Code: Select all: .-----------------------.-------------------.----------------------. | 3456–9 378–9 345678 | 56789 #789 5789 | 12 12 #789 | |b69 78–9 1 |#789+6 2 3 |#789 4 5 | |a59 78–9 2 | 1 4 #789+5| 3 #789 6 | :-----------------------+-------------------+----------------------: | 123–9 123789 378 | 23 6 #789 | 4 5 #789 | | 23–9 5 378 | 23 #789 4 | 6 #789 1 | | 4–9 6 478 |#789 5 1 |#789 3 2 | :-----------------------+-------------------+----------------------: | 12356 4 356 | 5789 1789 5789 | 125789 126789 3789 | | 7 13 356 | 4589 189 2 | 1589 1689 3489 | | 8 12 9 | 457 3 6 | 1257 127 47 | '-----------------------'-------------------'----------------------' 6r2c4 == 5r3c6 – (5=9)r3c1 – (9=6)r2c1 – Loop => -9r1456c1, -9r123c2

NL 2+TH; 6, 36, 37, 102: Show

Finally, in one we have to use an almost SDC.

Code: Select all: .----------------------.------------------.---------------------. | 23456–9 25–789 2456–9| 56 3789 #789 |#789 4789 1 | |*4569 *5789 *4569 |*56 #789 1 | 2 3 #4789 | |*39 *789 1 |#789+3 2 4 | 5 #789 6 | :----------------------+------------------+---------------------: | 12459 259 3 | 4789 1789 789 | 6 124789 45789 | | 149 6 49 | 34789 5 2 | 3789 14789 4789 | | 7 259 8 | 349 139 6 | 39 1249 459 | :----------------------+------------------+---------------------: | 289 1 279 |#789 6 5 | 4 #789 3 | | 689 3 679 | 2 4 #789 | 1 5 #789 | | 589 4 579 | 1 #789 3 |#789 6 2 | '----------------------'------------------'---------------------' 3r3c4 == 4r2c9 – (4=356789)(r2c1234, r3c12) – Loop => –789r1c123

NL ASDC+TH: Show

The last one shows how beneficial it can be to use the externals, as the same eliminations are available using the virtual triple in r1.
It can also be used in a few other puzzles, such as the four puzzles with the NL 2+TH.

Marek

Website · by **denis_berthier** » Fri Mar 18, 2022 4:40 pm

mith wrote:To be clear, the concern in point 1 above is not about the morph-dependency caused by the known SE bug, but rather the (morph-independent) differences between a puzzle and its subset puzzles. Differences here give an indication of the complexity of obtaining the extra digit(s). (And this isn't specific to SE - we may at some point want to determine the complexity according to SudoRules for obtaining these extra digits, for example, or any other rating system. Likewiwse for T&E classification differences, if any turn out to be in T&E(subsets,2) but not T&E(singles,2), etc.)

That was clear.

Website · by **denis_berthier** » Fri Mar 18, 2022 5:17 pm

marek stefanik wrote:
denis_berthier wrote:35 have no tridagon elimination rule (as per the definition here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html:
Code: Select all
2 6 7 8 13 18 29 36 37 38 39 40 41 48 51 56 60 61 63 72 80 81 102 103 106 118 121 123 126 139 146 147 177 192 217
Thank you for the list.
For each puzzle I've added its skfr after that step (or btte, if it is solvable with basics), for a few hardest puzzles I've also added a rating after relabeling in one of the TH boxes.

This confirms what I had noticed with the puzzles having a tridagon elimination: after it, they become (relatively) easy.
This may be due to the pattern implying few possibilities for the erst of the grid.
At this point, I think we can't draw any general conclusion about the puzzles not in T&E(2). We would need more varied cases.

marek stefanik wrote:In all of them, the guardians are confined in a single band.

=> band or stack by row/col symmetry.
But I can't see any a priori reason why they couldn't be in diagonally opposite blocks. Waiting for more examples.

by **mith** » Fri Mar 18, 2022 6:03 pm

The pattern can certainly exist with guardians in opposing boxes:

Code: Select all: 67..8....8.4..9....5946......5.47..89.75...6..8.6..5.......48......7..93...9...2. ED=9.0/1.2/1.2

(This is in T&E(1), never mind T&E(2). But I don't see why harder puzzles with this pattern couldn't exist as well.)

Perhaps at some point I can code up a check for this type of pattern among the database puzzles and try doing neighborhood T&E searches.

by **marek stefanik** » Fri Mar 18, 2022 8:08 pm

In this one, neither guardian can be eliminated using T&E(1).

Code: Select all: 32.....81......7.........45.62.83...73.61.2..8..7.2....73.61...6.83.....21.8.7...; skfr 10.1; non-minimal .-------------------.-------------------.---------------------. | 3 2 45679 | 459 4579 4569 | 69 8 1 | | 1459 4589 14569 | 1459 3459 45689 | 7 2369 2369 | |*19 8–9 167–9 |*129 237–9 68–9 | 36–9 4 5 | :-------------------+-------------------+---------------------: |#459+1 6 2 |#459 8 3 | 1459 1579 479 | | 7 3 #459 | 6 1 #459 | 2 59 8 | | 8 #459 1459 | 7 #459 2 | 134569 13569 3469 | :-------------------+-------------------+---------------------: |#459 7 3 |#459+2 6 1 | 8 259 249 | | 6 #459 8 | 3 2459 #459 | 1459 12579 2479 | | 2 1 #459 | 8 #459 7 | 34569 3569 3469 | '-------------------'-------------------'---------------------'

(9=1)r3c1 – 1r4c1 == 2r7c4 – (2=19)r3c14 => –9r3c23567; 9.0 skfr, then relabeling in b5 or b8 reduces it to 7.8

Marek

by **mith** » Fri Mar 18, 2022 9:46 pm

marek stefanik wrote:Some have three guardians and require a short kraken or whip.
Code: Select all
.-----------------------.---------------------.-------------------. | 24 24 #789+6 | 3789 A3569 A35678 | 5789 #789 1 | | 16789 #789 16789 | 789 bB169–5 2 | 3 4 #789+5| |#789+1 3 5 | 789 a19 4 |#789 6 2 | :-----------------------+---------------------+-------------------: | 1289 2589 189 | 23 7 135 | 4 289 6 | | 12467 2457 167 | 24 8 9 | 157 3 57 | | 124789 245789 3 | 6 245 15 | 15789 2789 5789 | :-----------------------+---------------------+-------------------: |#789 1 2 | 5 349 378 | 6 #789 34 | | 3 #789 4 | 1 69 678 | 2 5 #789 | | 5 6 #789 | 234789 2349 378 |#789 1 34 | '-----------------------'---------------------'-------------------' TH 789b1379 using internals: | 6r1c3 – 6r1c56 = 6r2c5 | 1r3c1 – 1r3c5 = 1r2c5 | 5r2c9 => -5r2c5
Kraken 2+TH; 7, 38, 40, 72: Show
Code: Select all
........1.....234...5..36.2....7..36....894....46......125...6.4.31..52.56....1..; btte ........1.....234..35..4.62....7.4.6....89.3...36......125..6..3.41..25.56.....1.; btte ........1.....234...5..36.2....784.....49..36..46......125...6.4.31..52.56....1..; btte ........1.....234..35..4.62....78.3....39.4.6..36......125..6..3.41..25.56.....1.; btte

I love these. If either 1 or 6 were a guardian in box 1, you get the virtual triple in row 2 placing the 5 as a guardian in r2c9; and of course if neither 1 nor 6 is a guardian, 5 has to be by TH. (Note only the first two still have three guardians after basics - the others place a 1 in r3c5.)

by **mith** » Fri Mar 18, 2022 10:38 pm

mith wrote:Ok, here are the results - I've gone from 88 expanded puzzles to 246:

Here are another 118 (for a total of 375), generated from the existing batch by digit swapping within isolated patterns (999_Springs' algorithm):

[edit]There may be a few duplicates in this list, I was missing a check at one point in my code. This is fixed, but I've generated a whole lot more from the minimals so I'm going to finish this process and then post the whole list again.[/edit]

Code: Select all: ..............1.23....45.67.23....898.7...3.696.8..27..39......28.......7.6.9..32;29 ..............1.23....45.67.27....8986....37.9.38..2.6.39......28.......7.6.9..32;29 .............12.34.13..45.2....6..25....789.3...5......491...5.1.5.....932.9...41;29 ........1.....2.34....56....17......2.8......65..27..8.2678.1..17..65...8.52.17..;29 ........1....23.....4..5..........67..6.781.9..7.6948..48...69.16.9..7.87.9....14;30 ..............1.23....45.67.23....898.7...3.696.8..27..39......28.....9.7.6.9..32;30 ........1....23.....4..5..........67..6.7819...7.694.8.48...9.616.9..78.7.9....14;30 ..............1.23....45.67.27....8986....37.9.38..2.6.39......28.....9.7.6.9..32;30 ........1....12.3....45.67.....8529....9......5912.8...48..1..95.2.98..491..4....;30 .............12.34.13..45.2....6..25...2789.3...5......491...5.1.5.....932.9...41;30 ........1.....2.34.35.......56......7.3..58.682...67...8725..6.3.26.8...56..37...;30 ........1.....2.34....56....17..8...2.8......65..27..8.2678.1..17..65...8.52.17..;30 ........1.....2.34....56....17......2.8.....765..27..8.2678.1..17..65...8.52.17..;30 ........1.....2.......3..45.16.23...27.81.6..3.87.61...32.87..61.7.6...868.......;30 ........1.....2..3....45.67.1485.9..5.2.91...89.2.41...29......1.8......45..28.1.;30 ........1.....2..3....45.67.158.49..2.8.91...49.25.1...54.28.1.8.1......92.......;30 ........1.....2.3..45.3..26.....716......834..6.52.....24...61.13.2...545.6...2.3;30 ........1....23....24....35....46....4273.6.8.6.8.27...76..4.8.2.3.78...48..6....;30 ........1....23....24....35....46....4273.6.8.6.8.27...86..4...2.3.87...47..6..8.;30 ........1..2..1.3..3..45.26....5261....46.2.32....3.45..7...16...8...35.42.5.....;30 ........1.12.34....561.7..........68......9.5.65.9.21..98...5.652....18.6.18...92;31 ........1.12.34....561.7..........68......9.5.65.9.21..98...1.65.1....8262.8..59.;31 ........1....12.3....45678.....9526....6......5612.9...49..1..65.2.69..461..4....;31 ........1.....2....34...256....37.....78.4.9..4392..78.794.3...3.827....42..89...;31 ........1.....2..3....45.67.1485.9..5.2.91...89.2.41...29.1....1.8......45..28.1.;31 ........1.....2..3....45.67.158.49..2.8.91...49.25.1...54.28.1.8.1......92..1....;31 ........1.....2.3.....45..6.1457.8..5.28.17..78..241...51..8...42..57..88.7......;31 ........1.....2.3.....45..6.15..7..82.4.58..787........2857.1..1.7.24...54.8.17..;31 ........1.....2.3.....45..6.15..7..82.4.58..787........4857.1..1.7.24...52.8.17..;31 ........1.....2.3.....45..6.15..7..84.2.58..787........4857.1..52.8.17..7.1.24...;31 ........1.....2.3..45.3..26.....716......834.46.52.....24...61.13.2...545.6...2.3;31 ........1.....2.34....56....17..8...2.8.....756..27..8.2587.1..18..65...7.62.18..;31 ........1.....2.34....56....17..8...2.8.....756..27..8.2687.1..18..65...7.52.18..;31 ........1.....2.34....56....17..8...2.8.....765..27..8.2587.1..18..65...7.62.18..;31 ........1.....2.34....56....17..8...2.8.....765..27..8.2678.1..17..65...8.52.17..;31 ........1.....2.34....56....17..8...2.8.....765..27..8.2687.1..18..65...7.52.18..;31 ........1.....2.34.35.......56......7.3.258.682...67...8725..6.3.26.8...56..37...;31 ........1....12.3....45.672....8529....9......5912.8...48..1..95.2.98..491..4....;31 ........1....12.3....45678.....2596....6......561.92...19.6...45.2.94..664...1...;31 ........1....12.3....45678.....6925....5......5612.9...94..1..55.1.4....62..95..4;31 ........1....23.....4..5..........67..7.684.9.46.7918..78....144.9...8.661.8..79.;31 ........1....23.....4..5..........67..7.6849..46.791.8.78....144.9...68.61.8..7.9;31 ........1....23....24....35....46....4273...8.6.8.27...762.4.8.2.3.78...48.36....;31 ........1....23....24....35....62.....273.8.6..68.47...6834....24..78.6.3.72.6...;31 ........1....23.45234........2.67....6.2.8.7..7834.....26......7.3..68..84..326.7;31 ........1....23.45234........2.67.8..6.2.8....7834.....26......74..326.88.3..67..;31 ........1..2..1.3..3..45.26....5261....46.2.32....3.45..7...16...8...35.42.5.6...;31 ........1..2..3.4..4.56..23....362.4...25.31.2..4.1.65..7...13...8...45.62.3.....;31 ........1..2.13.4..4.56..23...3.52.4...62.31.2...4..56..7...13...8...46.52...6...;31 .......12......3.4..1.2356.....14.23.1.3.54.6.4.26.15.1.......57...3....8.9.56...;31 .......12......3.4..1.2356.....14.53.1.36.42..4.2.51.61.......57...3....8.9.56...;31 ........1....23..4....56.78.12......3.9..1...56..32.1..23.19...1.63.59..95.26.1..;32 ........1....23..4....56.78.12......3.9..1...56..32.1..53.19...12.36.9..9.62.51..;32 ........1.12.34....561.7..........85..8...6.9.65.8.12..89....5652....91.6.19..8.2;32 ........1.12.34....561.7..........85..8...6.9.65.8.12..89....165.1...9.262.9..85.;32 ........1....12.3....456782....9526....6......5612.9...49..1..65.2.69..461..4....;32 ........1....23..4....56.78.12......3.9.1....65..32.1..63.91...1.526.9..92.3.51..;32 .......12......3.4..1.2356.....14.53.1.36.42..4.2.51.61.......57.8.56...95..3....;32 ........1....23..4....56.78.12......3.9.1....65..32.1..23.91...16.2.59..9.536.1..;32 .......12......3.4..1.2356.....14.23.1.3.54.6.4.26.15.1.......57.8.56...95..3....;32 ........1.....2....34...256....37.....78.4.9..4392..78.794.38..3.827....42..89...;32 ........1.....2.3.....45..6.1457.8..5.28.17..78..241...51..7..842..58..78.7......;32 ........1.....2.3.....45..6.1457.8..5.28.17..78..241...51..8..742..57..88.7......;32 ........1.....2.3.....45..6.15..7..82.4.58..787........2857.1..1.7.248..54.8.17..;32 ........1.....2.3.....45..6.15..7..82.4.58..787........4857.1..1.7.248..52.8.17..;32 ........1.....2.3.....45..6.15..7..84.2.58..787........4857.1..52.8.17..7.1.248..;32 ........1.....2.34....56..7.16..8..92.5.69..898........5968.1..1.8.259..62.9.1...;32 ........1.....234.235.......56.278..32.6.8...7.835.....732.5.6858...6.7.6.2......;32 ........1....12.3....456782....2596....6......561.92...19.6...45.2.94..664...1...;32 ........1....12.3....456782....6925....5......5612.9...94..1..55.1.4....62..95..4;32 ........1....23....24....35....62.....67.48..4.283.7.6.6734....24..87.6.3.82.6...;32 ........1....23....24....35....62....6.7.48..24.83.7.6.7634....38.2.6...4.2.78.6.;32 ........1....23....24....35....62....6.7.48..24.83.7.6.7634....38.2.6...4.2.87.6.;32 ........1....23....24....35....62....6.7.48..24.83.7.6.8634....37.2.6...4.2.78.6.;32 ........1....23....24....35....62....6.7.48..24.83.7.6.8634....37.2.6...4.2.87.6.;32 ........1....23....24...35.....46....4273..68.6.8.2.7..762.48..2.3.78...48.36....;32 ........1....23....24...35.....46....4273..68.6.8.2.7..762.48..2.3.87...48.36....;32 ........1....23....24...35.....46....4273..68.6.8.2.7..863.4...2.3.87...47.26.8..;32 ........1....23..4....56.78.12......3.9..1...56..32.1..23.19...15.26.9..9.63.51..;32 ........1....23..4....56.78.12......3.9..1...56..32.1..53.19...1.62.59..92.36.1..;32 ........1....23.4..25...3.6....57....5283...437.4.2.8..872.54..2...84...54.37....;32 ........1..2..3.4..4.56..23....362.4...25.31.2..4.1.65..7...13...8...45.62.3.5...;32 ........1..2.13.4..4.56..23...3.52.4...62.31.2...4..56..7...13...8...46.52..36...;32 .......12.....13.4..1.2356.....14.23.1.3.54.6.4.26.15.1.......57...3....8.9.56...;32 .......12.....13.4..1.2356.....14.53.1.36.42..4.2.51.61.......57...3....8.9.56...;32 .......12.....13.4..1.3256.....14.23.1.26.45..4.3.51.61.......57...23...8.9.56...;33 .......12.....13.4..1.3256.....14.53.1.2.54.6.4.36.12.1.......57...23...8.9.56...;33 ........1.....2.3..45...267....48.....83.5.9..5492..83.895.43..4.328....52..39...;33 .......12.....13.4..1.2356.....14.53.1.36.42..4.2.51.61.......57.8.56...95..3....;33 .......12.....13.4..1.2356.....14.23.1.3.54.6.4.26.15.1.......57.8.56...95..3....;33 ........1.....234.235....6..57.286..32.7.6...8.635.....832.5.7656...7.8.7.2......;33 ........1....23.4..25...36.....57....5283..74.7.4.2.8..872.54..2.3.84...54.37....;33 ........1....23.4..25...36.....57....5283..74.7.4.2.8..872.54..2.3.48...54.37....;33 ........1....23.4..25...36.....57....5283..74.7.4.2.8..473.5...2.3.48...58.27.4..;33 .......12.....13.4..1.2356.....14.23.1.3.54.6.4.26.15.1.......57...32...8.9.56...;33 .......12.....13.4..1.2356.....14.53.1.36.42..4.2.51.61.......57...32...8.9.56...;33 ........1....23....24...356...2.7.....7.84.9..4239..78.79.428..2.873....43.8.9...;33 ........1....23.4..25...3.6....57....5283...437.4.2.85.872.54..2...84...54.37....;33 ........1....2345.234.......23.678..6.82.4...74.38.....82.7..6.3.7......46..32.78;33 ........1....2345.234.......23.678..74.38....8.62.4....82.7..6.3.7......46..32.78;33 ........1....2345.234.......26......7.3..6.8.84..32.67.7834....36.2.87..4.2.67...;33 ........1....2345.234.......26..7.8.37.......4.8.32.76.43.76...68.24....7.23.86..;33 ........1....2345.234.......26.7..8.37.......4.8.32.76.43.67...68.2.4...7.238.6..;33 .......12.....13.4..1.3256.....14.23.1.3.54.6.4.26.15.1.......57...23...8.9.56...;33 .......12.....13.4..1.3256.....14.53.1.36.42..4.2.51.61.......57...23...8.9.56...;33 .......12.....13.4..1.3256.....14.23.1.26.45..4.3.51.61.......57.8.56...95..23...;34 .......12.....13.4..1.3256.....14.53.1.2.54.6.4.36.12.1.......57.8.56...95..23...;34 .......12.....13.4..1.2356.....14.53.1.36.42..4.2.51.61.......57.8.56...95..32...;34 .......12.....13.4..1.2356.....14.23.1.3.54.6.4.26.15.1.......57.8.56...95..32...;34 ........1....2345.234....6..26.7..8.37.......4.8.32.76.43.67...68.2.4...7.238.6..;34 ........1....2345.234....6..26..7.8.37.......4.8.32.76.43.76...68.24....7.23.86..;34 ........1....2345.234....6..23.786..7.62.4...84.36.....62.8..7.3.8......47..32.86;34 ........1....2345.234....6..23.786..6.72.4...84.36.....62.8..7.3.8......47..32.86;34 ........1....2345.234....6..27......6.3..7.8.84..32.76.6834....37.2.86..4.2.76...;34 ........1....23.4..25...367...2.8.....8.45.9..5239..84.89.524..2.483....53.4.9...;34 .......12.....13.4..1.3256.....14.23.1.3.54.6.4.26.15.1.......57.8.56...95..23...;34 .......12.....13.4..1.3256.....14.53.1.36.42..4.2.51.61.......57.8.56...95..23...;34 ........1....23.4..25...367...2.8....5239..84.8..45.9..98.524..24.83....5.39.4...;34

Going to check minimals next, I would guess most are not in the database yet. Given the way they were produced, there won't be anything novel in terms of trivalue oddagon patterns in the expanded forms, but it's always possible there will be some minimals that expand only to a subset of one of these.

Website · by **denis_berthier** » Sat Mar 19, 2022 5:53 am

marek stefanik wrote:
denis_berthier wrote:35 have no tridagon elimination rule (as per the definition here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html:
Code: Select all
2 6 7 8 13 18 29 36 37 38 39 40 41 48 51 56 60 61 63 72 80 81 102 103 106 118 121 123 126 139 146 147 177 192 217
Thank you for the list.
In all of them, the guardians are confined in a single band.

I've coded "tridagons links" between pairs of candidates (no triplets and no rule using these links are coded).
It seems I find fewer than you: only 18. [Edit: ah no, wait; I hadn't tried the full list; I now find 33)
Also, the numbers of candidates and the paths suggest there remains much redundancy in the list.

#2

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 235689 34567 235678 ! 35679 3568 3569 ! 479 2489 1 ! ! 5689 14567 5678 ! 15679 568 2 ! 479 3 48 ! ! 2389 137 2378 ! 1379 4 39 ! 5 6 28 ! +----------------------+----------------------+----------------------+ ! 23568 356 23568 ! 3569 356 7 ! 13469 1459 3456 ! ! 356 3567 4 ! 8 1 3569 ! 2 579 356 ! ! 1 9 3567 ! 356 2 4 ! 8 57 356 ! +----------------------+----------------------+----------------------+ ! 356 8 9 ! 23456 356 1 ! 346 245 7 ! ! 4 2 356 ! 356 7 8 ! 136 15 9 ! ! 7 356 1 ! 23456 9 356 ! 346 2458 234568 ! +----------------------+----------------------+----------------------+ 196 candidates.

hidden-pairs-in-a-column: c4{n2 n4}{r7 r9} ==> r9c4≠6, r9c4≠5, r9c4≠3, r7c4≠6, r7c4≠5, r7c4≠3
hidden-pairs-in-a-row: r4{n2 n8}{c1 c3} ==> r4c3≠6, r4c3≠5, r4c3≠3, r4c1≠6, r4c1≠5, r4c1≠3
extended tridagon for digits 3, 5 and 6 in blocks:
b4, with cells: r6c3 (link cell), r5c1, r4c2
b5, with cells: r6c4, r5c6 (link cell), r4c5
b7, with cells: r8c3, r7c1, r9c2
b8, with cells: r8c4, r7c5, r9c6
==> tridagon-link(n7r6c3, n9r5c6)

#6

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 34569 3789 345678 ! 56789 6789 5789 ! 12789 12789 789 ! ! 69 789 1 ! 6789 2 3 ! 789 4 5 ! ! 59 789 2 ! 1 4 5789 ! 3 789 6 ! +----------------------+----------------------+----------------------+ ! 1239 123789 378 ! 236789 6789 789 ! 4 5 789 ! ! 239 5 378 ! 23789 789 4 ! 6 789 1 ! ! 49 6 478 ! 789 5 1 ! 789 3 2 ! +----------------------+----------------------+----------------------+ ! 12356 4 356 ! 5789 1789 5789 ! 125789 126789 3789 ! ! 7 13 356 ! 4589 189 2 ! 1589 1689 3489 ! ! 8 12 9 ! 457 3 6 ! 1257 127 47 ! +----------------------+----------------------+----------------------+ 196 candidates.

hidden-pairs-in-a-column: c4{n2 n3}{r4 r5} ==> r5c4≠9, r5c4≠8, r5c4≠7, r4c4≠9, r4c4≠8, r4c4≠7, r4c4≠6
hidden-single-in-a-block ==> r4c5=6
hidden-pairs-in-a-row: r1{n1 n2}{c7 c8} ==> r1c8≠9, r1c8≠8, r1c8≠7, r1c7≠9, r1c7≠8, r1c7≠7
extended tridagon for digits 7, 8 and 9 in blocks:
b2, with cells: r3c6 (link cell), r2c4 (link cell), r1c5
b3, with cells: r3c8, r2c7, r1c9
b5, with cells: r4c6, r6c4, r5c5
b6, with cells: r4c9, r6c7, r5c8
==> tridagon-link(n5r3c6, n6r2c4)

#8

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 236789 234789 6789 ! 4789 4569 45678 ! 789 5789 1 ! ! 16789 789 16789 ! 789 569 2 ! 3 4 5789 ! ! 789 4789 5 ! 4789 1 3 ! 6 789 2 ! +----------------------+----------------------+----------------------+ ! 1289 2589 189 ! 124 7 145 ! 289 3 6 ! ! 12367 2357 167 ! 123 8 9 ! 4 157 57 ! ! 123789 235789 4 ! 6 235 15 ! 2789 15789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 349 478 ! 789 6 34789 ! ! 4 789 3 ! 1789 69 1678 ! 5 2 789 ! ! 5 6 789 ! 234789 2349 478 ! 1 789 34789 ! +----------------------+----------------------+----------------------+ 200 candidates.

hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-row: r1{n2 n3}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c2≠4, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
hidden-single-in-a-block ==> r3c2=4
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c7, r3c8
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c7, r9c8
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#13

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 23568 2379 234568 ! 23567 2389 236 ! 479 4789 1 ! ! 1568 179 14568 ! 1567 189 16 ! 479 2 3 ! ! 1238 12379 1238 ! 1237 12389 4 ! 5 789 6 ! +----------------------+----------------------+----------------------+ ! 12368 123 12368 ! 4 5 123 ! 12379 1379 279 ! ! 123 4 7 ! 8 123 9 ! 6 135 25 ! ! 9 5 123 ! 123 6 7 ! 8 134 24 ! +----------------------+----------------------+----------------------+ ! 1235 6 9 ! 123 4 8 ! 1237 1357 257 ! ! 4 123 1235 ! 9 7 1236 ! 123 1356 8 ! ! 7 8 123 ! 1236 123 5 ! 12349 13469 249 ! +----------------------+----------------------+----------------------+ 192 candidates.

hidden-pairs-in-a-row: r4{n6 n8}{c1 c3} ==> r4c3≠3, r4c3≠2, r4c3≠1, r4c1≠3, r4c1≠2, r4c1≠1
extended tridagon for digits 1, 2 and 3 in blocks:
b7, with cells: r7c1 (link cell), r8c2, r9c3
b8, with cells: r7c4, r8c6 (link cell), r9c5
b4, with cells: r5c1, r4c2, r6c3
b5, with cells: r5c5, r4c6, r6c4
==> tridagon-link(n5r7c1, n6r8c6)

#18

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 34568 3479 346 ! 3489 34567 34567 ! 289 2789 1 ! ! 1568 179 16 ! 189 1567 2 ! 89 3 4 ! ! 1348 13479 2 ! 13489 1347 1347 ! 5 6 789 ! +-------------------+-------------------+-------------------+ ! 1346 134 5 ! 7 8 134 ! 12349 1249 2369 ! ! 134 2 7 ! 6 1345 9 ! 1348 1458 358 ! ! 9 8 1346 ! 134 2 1345 ! 134 1457 3567 ! +-------------------+-------------------+-------------------+ ! 134 6 8 ! 5 9 134 ! 7 124 23 ! ! 2 5 134 ! 134 13467 13467 ! 13489 1489 389 ! ! 7 134 9 ! 2 134 8 ! 6 145 35 ! +-------------------+-------------------+-------------------+ 189 candidates.

hidden-pairs-in-a-row: r8{n6 n7}{c5 c6} ==> r8c6≠4, r8c6≠3, r8c6≠1, r8c5≠4, r8c5≠3, r8c5≠1
extended tridagon for digits 1, 3 and 4 in blocks:
b5, with cells: r5c5 (link cell), r6c4, r4c6
b4, with cells: r5c1, r6c3 (link cell), r4c2
b8, with cells: r9c5, r8c4, r7c6
b7, with cells: r9c2, r8c3, r7c1
==> tridagon-link(n5r5c5, n6r6c3)

#29

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 23569 34567 23567 ! 35679 3568 3569 ! 479 489 1 ! ! 569 14567 567 ! 15679 568 2 ! 479 3 48 ! ! 389 137 378 ! 1379 4 39 ! 5 6 2 ! +-------------------+-------------------+-------------------+ ! 23568 356 23568 ! 3569 356 7 ! 13469 1459 3456 ! ! 356 3567 4 ! 8 1 3569 ! 2 579 356 ! ! 1 9 3567 ! 356 2 4 ! 8 57 356 ! +-------------------+-------------------+-------------------+ ! 356 8 9 ! 23456 356 1 ! 346 245 7 ! ! 4 2 356 ! 356 7 8 ! 136 15 9 ! ! 7 356 1 ! 23456 9 356 ! 346 2458 34568 ! +-------------------+-------------------+-------------------+ 186 candidates.

hidden-pairs-in-a-column: c4{n2 n4}{r7 r9} ==> r9c4≠6, r9c4≠5, r9c4≠3, r7c4≠6, r7c4≠5, r7c4≠3
hidden-pairs-in-a-row: r4{n2 n8}{c1 c3} ==> r4c3≠6, r4c3≠5, r4c3≠3, r4c1≠6, r4c1≠5, r4c1≠3
extended tridagon for digits 3, 5 and 6 in blocks:
b4, with cells: r6c3 (link cell), r5c1, r4c2
b5, with cells: r6c4, r5c6 (link cell), r4c5
b7, with cells: r8c3, r7c1, r9c2
b8, with cells: r8c4, r7c5, r9c6
==> tridagon-link(n7r6c3, n9r5c6)

#36

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 34569 3789 345678 ! 56789 789 5789 ! 12789 12789 789 ! ! 69 789 1 ! 6789 2 3 ! 789 4 5 ! ! 59 789 2 ! 1 4 5789 ! 3 789 6 ! +----------------------+----------------------+----------------------+ ! 1239 123789 378 ! 23789 6 789 ! 4 5 789 ! ! 239 5 378 ! 23789 789 4 ! 6 789 1 ! ! 49 6 478 ! 789 5 1 ! 789 3 2 ! +----------------------+----------------------+----------------------+ ! 12356 4 356 ! 5789 1789 5789 ! 125789 126789 3789 ! ! 7 13 356 ! 4589 189 2 ! 1589 1689 3489 ! ! 8 12 9 ! 457 3 6 ! 1257 127 47 ! +----------------------+----------------------+----------------------+ 190 candidates.

hidden-pairs-in-a-column: c4{n2 n3}{r4 r5} ==> r5c4≠9, r5c4≠8, r5c4≠7, r4c4≠9, r4c4≠8, r4c4≠7
hidden-pairs-in-a-row: r1{n1 n2}{c7 c8} ==> r1c8≠9, r1c8≠8, r1c8≠7, r1c7≠9, r1c7≠8, r1c7≠7
extended tridagon for digits 7, 8 and 9 in blocks:
b2, with cells: r3c6 (link cell), r2c4 (link cell), r1c5
b3, with cells: r3c8, r2c7, r1c9
b5, with cells: r4c6, r6c4, r5c5
b6, with cells: r4c9, r6c7, r5c8
==> tridagon-link(n5r3c6, n6r2c4)

#37

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 34569 3789 345678 ! 56789 6789 5789 ! 12789 12789 789 ! ! 69 789 1 ! 6789 2 3 ! 789 4 5 ! ! 59 789 2 ! 1 4 5789 ! 3 789 6 ! +----------------------+----------------------+----------------------+ ! 1239 13789 378 ! 236789 6789 789 ! 4 5 789 ! ! 239 5 378 ! 23789 789 4 ! 6 789 1 ! ! 49 6 478 ! 789 5 1 ! 789 3 2 ! +----------------------+----------------------+----------------------+ ! 1356 4 356 ! 5789 1789 5789 ! 25789 26789 3789 ! ! 7 13 356 ! 4589 189 2 ! 589 689 3489 ! ! 8 2 9 ! 457 3 6 ! 157 17 47 ! +----------------------+----------------------+----------------------+ 186 candidates.

hidden-pairs-in-a-column: c4{n2 n3}{r4 r5} ==> r5c4≠9, r5c4≠8, r5c4≠7, r4c4≠9, r4c4≠8, r4c4≠7, r4c4≠6
hidden-single-in-a-block ==> r4c5=6
hidden-pairs-in-a-row: r1{n1 n2}{c7 c8} ==> r1c8≠9, r1c8≠8, r1c8≠7, r1c7≠9, r1c7≠8, r1c7≠7
extended tridagon for digits 7, 8 and 9 in blocks:
b2, with cells: r3c6 (link cell), r2c4 (link cell), r1c5
b3, with cells: r3c8, r2c7, r1c9
b5, with cells: r4c6, r6c4, r5c5
b6, with cells: r4c9, r6c7, r5c8
==> tridagon-link(n5r3c6, n6r2c4)

#39

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 236789 234789 6789 ! 4789 4569 45678 ! 789 5789 1 ! ! 16789 789 16789 ! 789 569 2 ! 3 4 5789 ! ! 789 4789 5 ! 4789 1 3 ! 6 789 2 ! +----------------------+----------------------+----------------------+ ! 1289 2589 189 ! 24 7 145 ! 289 3 6 ! ! 12367 2357 167 ! 23 8 9 ! 4 157 57 ! ! 123789 235789 4 ! 6 235 15 ! 2789 15789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 349 478 ! 789 6 34789 ! ! 4 789 3 ! 1 69 678 ! 5 2 789 ! ! 5 6 789 ! 234789 2349 478 ! 1 789 34789 ! +----------------------+----------------------+----------------------+ 193 candidates.

hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-row: r1{n2 n3}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c2≠4, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
hidden-single-in-a-block ==> r3c2=4
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c7, r3c8
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c7, r9c8
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#40

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 236789 234789 6789 ! 789 4568 45679 ! 789 5789 1 ! ! 16789 789 16789 ! 789 1568 2 ! 3 4 5789 ! ! 1789 4789 5 ! 789 148 3 ! 6 789 2 ! +----------------------+----------------------+----------------------+ ! 12369 2359 169 ! 23 7 8 ! 4 159 59 ! ! 1278 2578 178 ! 4 9 15 ! 278 3 6 ! ! 123789 235789 4 ! 6 235 15 ! 2789 15789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 348 479 ! 789 6 34789 ! ! 4 789 3 ! 1 68 679 ! 5 2 789 ! ! 5 6 789 ! 23789 2348 479 ! 1 789 34789 ! +----------------------+----------------------+----------------------+ 192 candidates.

naked-pairs-in-a-column: c6{r5 r6}{n1 n5} ==> r1c6≠5
whip[1]: c6n5{r6 .} ==> r6c5≠5
hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-column: c4{n2 n3}{r4 r9} ==> r9c4≠9, r9c4≠8, r9c4≠7
whip[1]: b8n7{r9c6 .} ==> r1c6≠7
whip[1]: b8n8{r9c5 .} ==> r1c5≠8, r2c5≠8, r3c5≠8
whip[1]: b8n9{r9c6 .} ==> r1c6≠9
hidden-pairs-in-a-row: r1{n2 n3}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c2≠4, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
hidden-single-in-a-block ==> r3c2=4
naked-single ==> r3c5=1
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c7, r3c8
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c7, r9c8
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#41

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 246789 24789 6789 ! 3789 3569 35678 ! 5789 789 1 ! ! 16789 789 16789 ! 789 569 2 ! 3 4 5789 ! ! 789 3 5 ! 789 1 4 ! 789 6 2 ! +----------------------+----------------------+----------------------+ ! 1289 2589 189 ! 123 7 135 ! 4 289 6 ! ! 12467 2457 167 ! 124 8 9 ! 157 3 57 ! ! 124789 245789 3 ! 6 245 15 ! 15789 2789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 349 378 ! 6 789 34789 ! ! 3 789 4 ! 1789 69 1678 ! 2 5 789 ! ! 5 6 789 ! 234789 2349 378 ! 789 1 34789 ! +----------------------+----------------------+----------------------+ 194 candidates.

hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-row: r1{n2 n4}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c8, r3c7
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c8, r9c7
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#48

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 246789 24789 6789 ! 789 3568 35679 ! 5789 789 1 ! ! 16789 789 16789 ! 789 568 2 ! 3 4 5789 ! ! 789 3 5 ! 789 1 4 ! 789 6 2 ! +----------------------+----------------------+----------------------+ ! 12469 2459 169 ! 124 7 8 ! 159 3 59 ! ! 1278 2578 178 ! 3 9 15 ! 4 278 6 ! ! 124789 245789 3 ! 6 245 15 ! 15789 2789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 348 379 ! 6 789 34789 ! ! 3 789 4 ! 1789 68 1679 ! 2 5 789 ! ! 5 6 789 ! 24789 2348 379 ! 789 1 34789 ! +----------------------+----------------------+----------------------+ 188 candidates.

naked-pairs-in-a-column: c6{r5 r6}{n1 n5} ==> r8c6≠1, r1c6≠5
hidden-single-in-a-block ==> r8c4=1
whip[1]: c6n5{r6 .} ==> r6c5≠5
hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-column: c4{n2 n4}{r4 r9} ==> r9c4≠9, r9c4≠8, r9c4≠7
whip[1]: b8n7{r9c6 .} ==> r1c6≠7
whip[1]: b8n8{r9c5 .} ==> r1c5≠8, r2c5≠8
whip[1]: b8n9{r9c6 .} ==> r1c6≠9
hidden-pairs-in-a-row: r1{n2 n4}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c8, r3c7
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c8, r9c7
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#51

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 23679 2358 234679 ! 2389 237 2367 ! 458 4589 1 ! ! 1679 158 14679 ! 189 17 167 ! 458 2 3 ! ! 1239 1238 1239 ! 12389 4 5 ! 6 89 7 ! +----------------------+----------------------+----------------------+ ! 12379 123 12379 ! 6 123 4 ! 12358 1358 258 ! ! 123 4 5 ! 123 8 9 ! 7 136 26 ! ! 8 6 123 ! 7 5 123 ! 9 134 24 ! +----------------------+----------------------+----------------------+ ! 1236 7 8 ! 4 9 123 ! 1235 1356 256 ! ! 4 123 1236 ! 5 1237 8 ! 123 1367 9 ! ! 5 9 123 ! 123 6 1237 ! 12348 13478 248 ! +----------------------+----------------------+----------------------+ 184 candidates.

hidden-pairs-in-a-row: r4{n7 n9}{c1 c3} ==> r4c3≠3, r4c3≠2, r4c3≠1, r4c1≠3, r4c1≠2, r4c1≠1
extended tridagon for digits 1, 2 and 3 in blocks:
b7, with cells: r7c1 (link cell), r8c2, r9c3
b8, with cells: r7c6, r8c5 (link cell), r9c4
b4, with cells: r5c1, r4c2, r6c3
b5, with cells: r5c4, r4c5, r6c6
==> tridagon-link(n6r7c1, n7r8c5)

#56

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 34568 5679 34568 ! 34567 5689 456 ! 2789 279 1 ! ! 1568 15679 1568 ! 1567 15689 2 ! 789 3 4 ! ! 1348 179 2 ! 1347 189 14 ! 5 79 6 ! +-------------------+-------------------+-------------------+ ! 14568 156 14568 ! 2 3 156 ! 1679 15679 579 ! ! 156 2 7 ! 8 156 9 ! 136 4 35 ! ! 9 3 156 ! 156 4 7 ! 126 8 25 ! +-------------------+-------------------+-------------------+ ! 1356 4 9 ! 156 2 8 ! 1367 1567 357 ! ! 2 156 1356 ! 9 7 1456 ! 1346 156 8 ! ! 7 8 156 ! 1456 156 3 ! 12469 12569 259 ! +-------------------+-------------------+-------------------+ 185 candidates.

hidden-pairs-in-a-row: r4{n4 n8}{c1 c3} ==> r4c3≠6, r4c3≠5, r4c3≠1, r4c1≠6, r4c1≠5, r4c1≠1
extended tridagon for digits 1, 5 and 6 in blocks:
b7, with cells: r7c1 (link cell), r8c2, r9c3
b8, with cells: r7c4, r8c6 (link cell), r9c5
b4, with cells: r5c1, r4c2, r6c3
b5, with cells: r5c5, r4c6, r6c4
==> tridagon-link(n3r7c1, n4r8c6)

#60

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3457 3468 3479 ! 3489 347 346 ! 589 1 2 ! ! 1257 1268 279 ! 289 27 26 ! 3 589 4 ! ! 234 2348 2349 ! 23489 1 5 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 2347 234 8 ! 12345 9 234 ! 12457 2345 6 ! ! 234 5 1 ! 234 6 7 ! 2489 23489 389 ! ! 6 9 2347 ! 12345 234 8 ! 12457 2345 1357 ! +-------------------+-------------------+-------------------+ ! 1234 7 5 ! 234 8 9 ! 124 6 13 ! ! 8 1234 234 ! 6 5 234 ! 12479 2349 1379 ! ! 9 234 6 ! 7 234 1 ! 2458 23458 358 ! +-------------------+-------------------+-------------------+ 184 candidates.

hidden-pairs-in-a-column: c4{n1 n5}{r4 r6} ==> r6c4≠4, r6c4≠3, r6c4≠2, r4c4≠4, r4c4≠3, r4c4≠2
extended tridagon for digits 2, 3 and 4 in blocks:
b7, with cells: r7c1 (link cell), r8c3, r9c2
b8, with cells: r7c4, r8c6, r9c5
b4, with cells: r5c1, r6c3 (link cell), r4c2
b5, with cells: r5c4, r6c5, r4c6
==> tridagon-link(n1r7c1, n7r6c3)

#61

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3457 3468 3479 ! 3489 346 347 ! 589 1 2 ! ! 1257 1268 279 ! 289 26 27 ! 3 589 4 ! ! 234 2348 2349 ! 23489 1 5 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 2347 234 8 ! 12345 9 234 ! 12457 2345 6 ! ! 234 5 1 ! 234 7 6 ! 2489 23489 389 ! ! 6 9 2347 ! 12345 234 8 ! 12457 2345 1357 ! +-------------------+-------------------+-------------------+ ! 1234 7 5 ! 234 8 9 ! 124 6 13 ! ! 8 1234 234 ! 6 5 234 ! 12479 2349 1379 ! ! 9 234 6 ! 7 234 1 ! 2458 23458 358 ! +-------------------+-------------------+-------------------+ 184 candidates.

hidden-pairs-in-a-column: c4{n1 n5}{r4 r6} ==> r6c4≠4, r6c4≠3, r6c4≠2, r4c4≠4, r4c4≠3, r4c4≠2
extended tridagon for digits 2, 3 and 4 in blocks:
b7, with cells: r7c1 (link cell), r8c3, r9c2
b8, with cells: r7c4, r8c6, r9c5
b4, with cells: r5c1, r6c3 (link cell), r4c2
b5, with cells: r5c4, r6c5, r4c6
==> tridagon-link(n1r7c1, n7r6c3)

#63

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 26789 2459 4689 ! 389 379 369 ! 459 45789 1 ! ! 6789 459 4689 ! 189 179 169 ! 459 2 3 ! ! 13789 139 1389 ! 2 4 5 ! 6 789 789 ! +----------------------+----------------------+----------------------+ ! 123689 1239 13689 ! 7 139 4 ! 1239 139 5 ! ! 139 1349 5 ! 139 8 2 ! 7 1349 6 ! ! 1239 7 1349 ! 5 6 139 ! 8 1349 249 ! +----------------------+----------------------+----------------------+ ! 139 8 7 ! 4 12359 139 ! 12359 6 29 ! ! 4 6 139 ! 139 12359 7 ! 12359 13589 289 ! ! 5 139 2 ! 6 139 8 ! 1349 13479 479 ! +----------------------+----------------------+----------------------+ 188 candidates.

hidden-pairs-in-a-column: c5{n2 n5}{r7 r8} ==> r8c5≠9, r8c5≠3, r8c5≠1, r7c5≠9, r7c5≠3, r7c5≠1
hidden-pairs-in-a-row: r4{n6 n8}{c1 c3} ==> r4c3≠9, r4c3≠3, r4c3≠1, r4c1≠9, r4c1≠3, r4c1≠2, r4c1≠1
extended tridagon for digits 1, 3 and 9 in blocks:
b4, with cells: r4c2 (link cell), r6c3 (link cell), r5c1
b5, with cells: r4c5, r6c6, r5c4
b7, with cells: r9c2, r8c3, r7c1
b8, with cells: r9c5, r8c4, r7c6
==> tridagon-link(n2r4c2, n4r6c3)

#72

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 246789 24789 6789 ! 789 3568 35679 ! 5789 789 1 ! ! 16789 789 16789 ! 789 1568 2 ! 3 4 5789 ! ! 1789 3 5 ! 789 18 4 ! 789 6 2 ! +----------------------+----------------------+----------------------+ ! 12469 2459 169 ! 24 7 8 ! 159 3 59 ! ! 1278 2578 178 ! 3 9 15 ! 4 278 6 ! ! 124789 245789 3 ! 6 245 15 ! 15789 2789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 348 379 ! 6 789 34789 ! ! 3 789 4 ! 1 68 679 ! 2 5 789 ! ! 5 6 789 ! 24789 2348 379 ! 789 1 34789 ! +----------------------+----------------------+----------------------+ 186 candidates.

naked-pairs-in-a-column: c6{r5 r6}{n1 n5} ==> r1c6≠5
whip[1]: c6n5{r6 .} ==> r6c5≠5
hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-column: c4{n2 n4}{r4 r9} ==> r9c4≠9, r9c4≠8, r9c4≠7
whip[1]: b8n7{r9c6 .} ==> r1c6≠7
whip[1]: b8n8{r9c5 .} ==> r1c5≠8, r2c5≠8, r3c5≠8
naked-single ==> r3c5=1
whip[1]: b8n9{r9c6 .} ==> r1c6≠9
hidden-pairs-in-a-row: r1{n2 n4}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c8, r3c7
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c8, r9c7
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#80

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3489 347 3467 ! 34678 3459 34678 ! 259 2589 1 ! ! 189 17 167 ! 1678 159 2 ! 59 3 4 ! ! 13489 2 5 ! 1348 1349 1348 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 134 6 1347 ! 5 8 134 ! 12349 1249 2379 ! ! 2 8 134 ! 1346 7 9 ! 1345 1456 356 ! ! 5 1347 9 ! 2 134 1346 ! 134 1468 3678 ! +-------------------+-------------------+-------------------+ ! 134 9 8 ! 134 6 5 ! 7 124 23 ! ! 6 134 2 ! 13478 134 13478 ! 13459 1459 359 ! ! 7 5 134 ! 9 2 134 ! 8 146 36 ! +-------------------+-------------------+-------------------+ 181 candidates.

hidden-pairs-in-a-row: r8{n7 n8}{c4 c6} ==> r8c6≠4, r8c6≠3, r8c6≠1, r8c4≠4, r8c4≠3, r8c4≠1
extended tridagon for digits 1, 3 and 4 in blocks:
b5, with cells: r5c4 (link cell), r6c5, r4c6
b4, with cells: r5c3, r6c2 (link cell), r4c1
b8, with cells: r7c4, r8c5, r9c6
b7, with cells: r7c1, r8c2, r9c3
==> tridagon-link(n6r5c4, n7r6c2)

#81

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3489 347 3467 ! 34678 3459 34678 ! 259 2589 1 ! ! 189 17 167 ! 1678 159 2 ! 59 3 4 ! ! 13489 2 5 ! 1348 1349 1348 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 134 6 1347 ! 5 8 134 ! 12349 1249 2379 ! ! 2 8 134 ! 1346 7 9 ! 1345 1456 356 ! ! 5 1347 9 ! 2 134 1346 ! 134 1468 3678 ! +-------------------+-------------------+-------------------+ ! 134 9 8 ! 134 2 5 ! 7 146 36 ! ! 6 134 2 ! 13478 134 13478 ! 13459 1459 359 ! ! 7 5 134 ! 9 6 134 ! 8 124 23 ! +-------------------+-------------------+-------------------+ 181 candidates.

hidden-pairs-in-a-row: r8{n7 n8}{c4 c6} ==> r8c6≠4, r8c6≠3, r8c6≠1, r8c4≠4, r8c4≠3, r8c4≠1
extended tridagon for digits 1, 3 and 4 in blocks:
b5, with cells: r5c4 (link cell), r6c5, r4c6
b4, with cells: r5c3, r6c2 (link cell), r4c1
b8, with cells: r7c4, r8c5, r9c6
b7, with cells: r7c1, r8c2, r9c3
==> tridagon-link(n6r5c4, n7r6c2)

Website · by **denis_berthier** » Sat Mar 19, 2022 6:48 am

...
#102

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 34569 3789 345678 ! 56789 789 5789 ! 12789 12789 789 ! ! 69 789 1 ! 6789 2 3 ! 789 4 5 ! ! 59 789 2 ! 1 4 5789 ! 3 789 6 ! +----------------------+----------------------+----------------------+ ! 1239 13789 378 ! 23789 6 789 ! 4 5 789 ! ! 239 5 378 ! 23789 789 4 ! 6 789 1 ! ! 49 6 478 ! 789 5 1 ! 789 3 2 ! +----------------------+----------------------+----------------------+ ! 1356 4 356 ! 5789 1789 5789 ! 25789 26789 3789 ! ! 7 13 356 ! 4589 189 2 ! 589 689 3489 ! ! 8 2 9 ! 457 3 6 ! 157 17 47 ! +----------------------+----------------------+----------------------+ 180 candidates.

hidden-pairs-in-a-column: c4{n2 n3}{r4 r5} ==> r5c4≠9, r5c4≠8, r5c4≠7, r4c4≠9, r4c4≠8, r4c4≠7
hidden-pairs-in-a-row: r1{n1 n2}{c7 c8} ==> r1c8≠9, r1c8≠8, r1c8≠7, r1c7≠9, r1c7≠8, r1c7≠7
extended tridagon for digits 7, 8 and 9 in blocks:
b2, with cells: r3c6 (link cell), r2c4 (link cell), r1c5
b3, with cells: r3c8, r2c7, r1c9
b5, with cells: r4c6, r6c4, r5c5
b6, with cells: r4c9, r6c7, r5c8
==> tridagon-link(n5r3c6, n6r2c4)

#103

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 246789 24789 6789 ! 3789 3569 35678 ! 5789 789 1 ! ! 16789 789 16789 ! 789 569 2 ! 3 4 5789 ! ! 789 3 5 ! 789 1 4 ! 789 6 2 ! +----------------------+----------------------+----------------------+ ! 1289 2589 189 ! 23 7 135 ! 4 289 6 ! ! 12467 2457 167 ! 24 8 9 ! 157 3 57 ! ! 124789 245789 3 ! 6 245 15 ! 15789 2789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 349 378 ! 6 789 34789 ! ! 3 789 4 ! 1 69 678 ! 2 5 789 ! ! 5 6 789 ! 234789 2349 378 ! 789 1 34789 ! +----------------------+----------------------+----------------------+ 187 candidates.

hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-row: r1{n2 n4}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c8, r3c7
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c8, r9c7
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#106

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 4589 457 456 ! 34579 4568 3456 ! 289 289 1 ! ! 189 17 16 ! 179 168 2 ! 3 4 5 ! ! 14589 2 3 ! 1459 1458 145 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 12345 145 8 ! 145 9 1456 ! 1245 12356 7 ! ! 145 6 7 ! 2 3 145 ! 14589 1589 489 ! ! 12345 9 145 ! 8 1456 7 ! 1245 12356 2346 ! +-------------------+-------------------+-------------------+ ! 145 8 9 ! 6 2 1345 ! 7 135 34 ! ! 6 3 145 ! 145 7 8 ! 12459 1259 249 ! ! 7 145 2 ! 1345 145 9 ! 1458 13568 3468 ! +-------------------+-------------------+-------------------+ 180 candidates.

hidden-pairs-in-a-column: c1{n2 n3}{r4 r6} ==> r6c1≠5, r6c1≠4, r6c1≠1, r4c1≠5, r4c1≠4, r4c1≠1
extended tridagon for digits 1, 4 and 5 in blocks:
b8, with cells: r7c6 (link cell), r9c5, r8c4
b7, with cells: r7c1, r9c2, r8c3
b5, with cells: r5c6, r6c5 (link cell), r4c4
b4, with cells: r5c1, r6c3, r4c2
==> tridagon-link(n3r7c6, n6r6c5)

#118

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 246789 24789 6789 ! 789 3568 35679 ! 5789 789 1 ! ! 16789 789 16789 ! 789 568 2 ! 3 4 5789 ! ! 789 3 5 ! 789 1 4 ! 789 6 2 ! +----------------------+----------------------+----------------------+ ! 12469 2459 169 ! 24 7 8 ! 159 3 59 ! ! 1278 2578 178 ! 3 9 15 ! 4 278 6 ! ! 124789 245789 3 ! 6 245 15 ! 15789 2789 5789 ! +----------------------+----------------------+----------------------+ ! 789 1 2 ! 5 348 379 ! 6 789 34789 ! ! 3 789 4 ! 1 68 679 ! 2 5 789 ! ! 5 6 789 ! 24789 2348 379 ! 789 1 34789 ! +----------------------+----------------------+----------------------+ 182 candidates.

naked-pairs-in-a-column: c6{r5 r6}{n1 n5} ==> r1c6≠5
whip[1]: c6n5{r6 .} ==> r6c5≠5
hidden-pairs-in-a-column: c9{n3 n4}{r7 r9} ==> r9c9≠9, r9c9≠8, r9c9≠7, r7c9≠9, r7c9≠8, r7c9≠7
hidden-pairs-in-a-column: c4{n2 n4}{r4 r9} ==> r9c4≠9, r9c4≠8, r9c4≠7
whip[1]: b8n7{r9c6 .} ==> r1c6≠7
whip[1]: b8n8{r9c5 .} ==> r1c5≠8, r2c5≠8
whip[1]: b8n9{r9c6 .} ==> r1c6≠9
hidden-pairs-in-a-row: r1{n2 n4}{c1 c2} ==> r1c2≠9, r1c2≠8, r1c2≠7, r1c1≠9, r1c1≠8, r1c1≠7, r1c1≠6
extended tridagon for digits 7, 8 and 9 in blocks:
b3, with cells: r2c9 (link cell), r1c8, r3c7
b1, with cells: r2c2, r1c3 (link cell), r3c1
b9, with cells: r8c9, r7c8, r9c7
b7, with cells: r8c2, r7c1, r9c3
==> tridagon-link(n5r2c9, n6r1c3)

#121

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 489 469 479 ! 36789 5689 359 ! 259 25789 1 ! ! 189 169 179 ! 6789 5689 2 ! 59 3 4 ! ! 2 3 5 ! 14789 1489 149 ! 6 789 789 ! +----------------------+----------------------+----------------------+ ! 149 2 6 ! 1349 1459 7 ! 8 1459 359 ! ! 3 149 8 ! 149 12459 1459 ! 7 12459 6 ! ! 5 7 149 ! 134689 124689 1349 ! 12349 1249 239 ! +----------------------+----------------------+----------------------+ ! 149 5 2 ! 149 7 8 ! 1349 6 39 ! ! 6 8 149 ! 2 3 149 ! 1459 14579 579 ! ! 7 149 3 ! 5 149 6 ! 1249 12489 289 ! +----------------------+----------------------+----------------------+ 183 candidates.

hidden-pairs-in-a-row: r6{n6 n8}{c4 c5} ==> r6c5≠9, r6c5≠4, r6c5≠2, r6c5≠1, r6c4≠9, r6c4≠4, r6c4≠3, r6c4≠1
hidden-single-in-a-block ==> r5c5=2
extended tridagon for digits 1, 4 and 9 in blocks:
b5, with cells: r6c6 (link cell), r4c5 (link cell), r5c4
b4, with cells: r6c3, r4c1, r5c2
b8, with cells: r8c6, r9c5, r7c4
b7, with cells: r8c3, r9c2, r7c1
==> tridagon-link(n3r6c6, n5r4c5)

#123

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 26789 2459 4689 ! 389 379 369 ! 459 45789 1 ! ! 6789 459 4689 ! 189 179 169 ! 459 2 3 ! ! 13789 139 1389 ! 2 4 5 ! 6 789 789 ! +----------------------+----------------------+----------------------+ ! 123689 1239 13689 ! 7 139 4 ! 12359 1359 259 ! ! 139 1349 5 ! 139 8 2 ! 7 1349 6 ! ! 1239 7 1349 ! 5 6 139 ! 8 1349 249 ! +----------------------+----------------------+----------------------+ ! 139 8 7 ! 4 5 139 ! 1239 6 29 ! ! 4 6 139 ! 139 2 7 ! 1359 13589 589 ! ! 5 139 2 ! 6 139 8 ! 1349 13479 479 ! +----------------------+----------------------+----------------------+ 181 candidates

.
hidden-pairs-in-a-row: r4{n6 n8}{c1 c3} ==> r4c3≠9, r4c3≠3, r4c3≠1, r4c1≠9, r4c1≠3, r4c1≠2, r4c1≠1
extended tridagon for digits 1, 3 and 9 in blocks:
b4, with cells: r4c2 (link cell), r6c3 (link cell), r5c1
b5, with cells: r4c5, r6c6, r5c4
b7, with cells: r9c2, r8c3, r7c1
b8, with cells: r9c5, r8c4, r7c6
==> tridagon-link(n2r4c2, n4r6c3)

#126

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3489 347 3467 ! 34678 3459 34678 ! 259 2589 1 ! ! 189 17 167 ! 1678 159 2 ! 59 3 4 ! ! 13489 2 5 ! 1348 1349 1348 ! 6 89 7 ! +-------------------+-------------------+-------------------+ ! 134 6 1347 ! 5 8 134 ! 12349 12479 239 ! ! 2 8 134 ! 1346 7 9 ! 1345 1456 356 ! ! 5 1347 9 ! 2 134 1346 ! 134 1467 8 ! +-------------------+-------------------+-------------------+ ! 134 9 8 ! 134 2 5 ! 7 146 36 ! ! 6 134 2 ! 13478 134 13478 ! 13459 1459 359 ! ! 7 5 134 ! 9 6 134 ! 8 124 23 ! +-------------------+-------------------+-------------------+ 177 candidates.

hidden-pairs-in-a-row: r8{n7 n8}{c4 c6} ==> r8c6≠4, r8c6≠3, r8c6≠1, r8c4≠4, r8c4≠3, r8c4≠1
extended tridagon for digits 1, 3 and 4 in blocks:
b5, with cells: r5c4 (link cell), r6c5, r4c6
b4, with cells: r5c3, r6c2 (link cell), r4c1
b8, with cells: r7c4, r8c5, r9c6
b7, with cells: r7c1, r8c2, r9c3
==> tridagon-link(n6r5c4, n7r6c2)

#139

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 489 469 479 ! 36789 5689 359 ! 259 25789 1 ! ! 189 169 179 ! 6789 5689 2 ! 59 3 4 ! ! 2 3 5 ! 14789 1489 149 ! 6 789 789 ! +----------------------+----------------------+----------------------+ ! 149 2 6 ! 1349 1459 7 ! 8 1459 359 ! ! 3 149 8 ! 149 2 1459 ! 7 1459 6 ! ! 5 7 149 ! 134689 14689 1349 ! 1349 149 2 ! +----------------------+----------------------+----------------------+ ! 149 5 12349 ! 149 7 8 ! 12349 6 39 ! ! 6 8 149 ! 2 3 149 ! 1459 14579 579 ! ! 7 149 12349 ! 5 149 6 ! 12349 12489 389 ! +----------------------+----------------------+----------------------+ 183 candidates.

hidden-pairs-in-a-column: c3{n2 n3}{r7 r9} ==> r9c3≠9, r9c3≠4, r9c3≠1, r7c3≠9, r7c3≠4, r7c3≠1
hidden-pairs-in-a-row: r6{n6 n8}{c4 c5} ==> r6c5≠9, r6c5≠4, r6c5≠1, r6c4≠9, r6c4≠4, r6c4≠3, r6c4≠1
extended tridagon for digits 1, 4 and 9 in blocks:
b5, with cells: r6c6 (link cell), r4c5 (link cell), r5c4
b4, with cells: r6c3, r4c1, r5c2
b8, with cells: r8c6, r9c5, r7c4
b7, with cells: r8c3, r9c2, r7c1
==> tridagon-link(n3r6c6, n5r4c5)

#146

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3457 3468 3479 ! 3489 347 346 ! 589 1 2 ! ! 1257 1268 279 ! 289 27 26 ! 3 58 4 ! ! 234 2348 2349 ! 23489 1 5 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 2347 234 8 ! 12345 9 234 ! 12457 2345 6 ! ! 234 5 1 ! 234 6 7 ! 2489 2348 389 ! ! 6 9 2347 ! 12345 234 8 ! 12457 2345 1357 ! +-------------------+-------------------+-------------------+ ! 1234 7 5 ! 234 8 9 ! 124 6 13 ! ! 8 1234 234 ! 6 5 234 ! 1247 9 137 ! ! 9 234 6 ! 7 234 1 ! 2458 23458 358 ! +-------------------+-------------------+-------------------+ 176 candidates.

hidden-pairs-in-a-column: c4{n1 n5}{r4 r6} ==> r6c4≠4, r6c4≠3, r6c4≠2, r4c4≠4, r4c4≠3, r4c4≠2
extended tridagon for digits 2, 3 and 4 in blocks:
b7, with cells: r7c1 (link cell), r8c3, r9c2
b8, with cells: r7c4, r8c6, r9c5
b4, with cells: r5c1, r6c3 (link cell), r4c2
b5, with cells: r5c4, r6c5, r4c6
==> tridagon-link(n1r7c1, n7r6c3)

#147

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3457 3468 3479 ! 3489 346 347 ! 589 1 2 ! ! 1257 1268 279 ! 289 26 27 ! 3 58 4 ! ! 234 2348 2349 ! 23489 1 5 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 2347 234 8 ! 12345 9 234 ! 12457 2345 6 ! ! 234 5 1 ! 234 7 6 ! 2489 2348 389 ! ! 6 9 2347 ! 12345 234 8 ! 12457 2345 1357 ! +-------------------+-------------------+-------------------+ ! 1234 7 5 ! 234 8 9 ! 124 6 13 ! ! 8 1234 234 ! 6 5 234 ! 1247 9 137 ! ! 9 234 6 ! 7 234 1 ! 2458 23458 358 ! +-------------------+-------------------+-------------------+ 176 candidates.

hidden-pairs-in-a-column: c4{n1 n5}{r4 r6} ==> r6c4≠4, r6c4≠3, r6c4≠2, r4c4≠4, r4c4≠3, r4c4≠2
extended tridagon for digits 2, 3 and 4 in blocks:
b7, with cells: r7c1 (link cell), r8c3, r9c2
b8, with cells: r7c4, r8c6, r9c5
b4, with cells: r5c1, r6c3 (link cell), r4c2
b5, with cells: r5c4, r6c5, r4c6
==> tridagon-link(n1r7c1, n7r6c3)

#177

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 4589 457 456 ! 34579 4568 3456 ! 29 289 1 ! ! 189 17 16 ! 179 168 2 ! 3 4 5 ! ! 14589 2 3 ! 1459 1458 145 ! 6 7 89 ! +-------------------+-------------------+-------------------+ ! 12345 145 8 ! 145 9 1456 ! 1245 12356 7 ! ! 145 6 7 ! 2 3 145 ! 1459 1589 489 ! ! 12345 9 145 ! 8 1456 7 ! 1245 12356 2346 ! +-------------------+-------------------+-------------------+ ! 145 8 9 ! 6 2 1345 ! 7 135 34 ! ! 6 3 145 ! 145 7 8 ! 12459 1259 249 ! ! 7 145 2 ! 1345 145 9 ! 8 1356 346 ! +-------------------+-------------------+-------------------+ 172 candidates.

hidden-pairs-in-a-column: c1{n2 n3}{r4 r6} ==> r6c1≠5, r6c1≠4, r6c1≠1, r4c1≠5, r4c1≠4, r4c1≠1
extended tridagon for digits 1, 4 and 5 in blocks:
b8, with cells: r7c6 (link cell), r9c5, r8c4
b7, with cells: r7c1, r9c2, r8c3
b5, with cells: r5c6, r6c5 (link cell), r4c4
b4, with cells: r5c1, r6c3, r4c2
==> tridagon-link(n3r7c6, n6r6c5)

#192

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 489 469 479 ! 36789 5689 359 ! 259 25789 1 ! ! 189 169 179 ! 6789 5689 2 ! 59 3 4 ! ! 2 3 5 ! 14789 1489 149 ! 6 789 789 ! +----------------------+----------------------+----------------------+ ! 149 2 6 ! 1349 1459 7 ! 8 1459 359 ! ! 3 149 8 ! 149 2 1459 ! 7 1459 6 ! ! 5 7 149 ! 134689 14689 1349 ! 12349 1249 239 ! +----------------------+----------------------+----------------------+ ! 149 5 2 ! 149 7 8 ! 1349 6 39 ! ! 6 8 149 ! 2 3 149 ! 1459 14579 579 ! ! 7 149 3 ! 5 149 6 ! 1249 12489 289 ! +----------------------+----------------------+----------------------+ 176 candidates.

hidden-pairs-in-a-row: r6{n6 n8}{c4 c5} ==> r6c5≠9, r6c5≠4, r6c5≠1, r6c4≠9, r6c4≠4, r6c4≠3, r6c4≠1
extended tridagon for digits 1, 4 and 9 in blocks:
b5, with cells: r6c6 (link cell), r4c5 (link cell), r5c4
b4, with cells: r6c3, r4c1, r5c2
b8, with cells: r8c6, r9c5, r7c4
b7, with cells: r8c3, r9c2, r7c1
==> tridagon-link(n3r6c6, n5r4c5)

#217

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 234569 25789 24569 ! 356789 3789 789 ! 789 4789 1 ! ! 4569 5789 4569 ! 56789 789 1 ! 2 3 4789 ! ! 39 789 1 ! 3789 2 4 ! 5 789 6 ! +----------------------+----------------------+----------------------+ ! 12459 259 3 ! 4789 1789 789 ! 6 124789 45789 ! ! 149 6 49 ! 34789 5 2 ! 3789 14789 4789 ! ! 7 259 8 ! 349 139 6 ! 39 1249 459 ! +----------------------+----------------------+----------------------+ ! 289 1 279 ! 789 6 5 ! 4 789 3 ! ! 689 3 679 ! 2 4 789 ! 1 5 789 ! ! 589 4 579 ! 1 789 3 ! 789 6 2 ! +----------------------+----------------------+----------------------+ 179 candidates.

hidden-pairs-in-a-column: c4{n5 n6}{r1 r2} ==> r2c4≠9, r2c4≠8, r2c4≠7, r1c4≠9, r1c4≠8, r1c4≠7, r1c4≠3
extended tridagon for digits 7, 8 and 9 in blocks:
b2, with cells: r3c4 (link cell), r2c5, r1c6
b3, with cells: r3c8, r2c9 (link cell), r1c7
b8, with cells: r7c4, r9c5, r8c6
b9, with cells: r7c8, r9c7, r8c9
==> tridagon-link(n3r3c4, n4r2c9)

The only two that don't have a tridagon elimination or a tridagon link between two candidates are #7 and #38

by **marek stefanik** » Sat Mar 19, 2022 7:06 am

Edit: no longer relevant, missed your edit

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