The New Sudoku Players' Forum

Website · by **denis_berthier** » Wed Jan 15, 2025 4:54 am

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Code: Select all: +-------+-------+-------+ ! . . . ! 4 5 6 ! 7 . 9 ! ! . . . ! . 8 9 ! . . . ! ! 6 9 . ! 3 2 . ! . . . ! +-------+-------+-------+ ! . . . ! . . . ! . . 7 ! ! . 3 6 ! . 7 . ! . 9 1 ! ! 7 . 9 ! . . . ! . 2 . ! +-------+-------+-------+ ! 3 6 . ! . . . ! . . . ! ! . 7 2 ! . . 3 ! . 6 . ! ! 9 . 1 ! . . 2 ! . . 3 ! +-------+-------+-------+ ...4567.9....89...69.32............7.36.7..917.9....2.36........72..3.6.9.1..2..3

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 128 128 38 ! 4 5 6 ! 7 138 9 ! ! 145 145 3457 ! 17 8 9 ! 123456 1345 2456 ! ! 6 9 4578 ! 3 2 17 ! 1458 1458 458 ! +----------------------+----------------------+----------------------+ ! 12458 12458 458 ! 125689 13469 1458 ! 34568 3458 7 ! ! 2458 3 6 ! 258 7 458 ! 458 9 1 ! ! 7 1458 9 ! 1568 1346 1458 ! 34568 2 4568 ! +----------------------+----------------------+----------------------+ ! 3 6 458 ! 15789 149 14578 ! 124589 14578 2458 ! ! 458 7 2 ! 1589 149 3 ! 14589 6 458 ! ! 9 458 1 ! 5678 46 2 ! 458 4578 3 ! +----------------------+----------------------+----------------------+ 194 candidates.

by **Cenoman** » Wed Jan 15, 2025 8:38 pm

As a start, a solution in four steps, among which tridagon and UR.

Code: Select all: +-------------------------+-------------------------+--------------------------+ | 128 128 38 | 4 5 6 | 7 138 9 | | 145 145 3457 | z17 8 9 | 26 1345 26 | | 6 9 4578 | 3 2 7-1 |Zb1458 1458 458 | +-------------------------+-------------------------+--------------------------+ | B12458 B12458 458* | 69 369 C1458 | b36 458* 7 | | 458* 3 6 | 2 7 458 | b458* 9 1 | | 7 A1458* 9 | 1568 1346 1458 | b34568 2 a4568* | +-------------------------+-------------------------+--------------------------+ | 3 6 458* | 15789 149 14578 | Y124589 Xw14578* 2458 | | 458* 7 2 | 1589 149 3 | Y14589 6 458* | | 9 458* 1 | y5678 46 2 | b458* x4578 3 | +-------------------------+-------------------------+--------------------------+

1. Tridagon (458)b4679 (*) having four guardians (1r6c2, 6r6c9, 17r7c8)
(6)r6c9 - (6=34581)r34569c7
(1)r6c2 - r4c12 = (1)r4c6
(7)r7c8 - r9c8 = r9c4 - (7=1)r2c4
(1)r7c8 - r78c7 = (1)r3c7
=> -1 r3c6; lcls, 7 placements

Code: Select all: +--------------------+-----------------------+-------------------------+ | 12* 12* 3 | 4 5 6 | 7 8 9 | | 45 45 7 | 1 8 9 | 26 3 26 | | 6 9 8 | 3 2 7 | 145 145 45 | +--------------------+-----------------------+-------------------------+ | 12* 12+8* 45 | 69 369 18 | 36 45 7 | | 458 3 6 | 2 7 458 | 458 9 1 | | 7 1458 9 | 568 1346 1458 | 34568 2 4568 | +--------------------+-----------------------+-------------------------+ | 3 6 45 | 5789 149 1458 | 124589 1457 2458 | | 458 7 2 | 589 149 3 | 14589 6 458 | | 9 458 1 | 5678 46 2 | 458 457 3 | +--------------------+-----------------------+-------------------------+

2. UR(12)r14c12, using single internal => +8 r4c2; lcls, 7 placements

Code: Select all: +-----------------+---------------------+-----------------------+ | 1 2 3 | 4 5 6 | 7 8 9 | | 45 45 7 | 1 8 9 | 26 3 26 | | 6 9 8 | 3 2 7 | 145 145 45 | +-----------------+---------------------+-----------------------+ | 2 8 45* | 69 369 1 | 36 45* 7 | | 45 3 6 | 2 7 458^ | 458^ 9 1 | | 7 1 9 | 56-8 346 458 | 34568 2 68 | +-----------------+---------------------+-----------------------+ | 3 6 45* | 5789 149 45-8 | 124589 1457 28 | | 8 7 2 | 59 149 3 | 1459 6 45 | | 9 45* 1 | 5678^ 46 2 | 458^ 7-45 3 | +-----------------+---------------------+-----------------------+

3. Remote Pair (45)r4c38, b7p38 => -45 r9c8
4. Skyscraper (8): r5c6 = r5c7 - r9c7 = r9c4 => -8 r6c4, r7c6; lclste

Added. Solution w/o uniqueness.
After step #1, the rating is lowered to skfr 6.6, but the end is'nt that easy.
This puzzle is a nice example of a solution simplified by uniqueness.
After step #1:

Code: Select all: +--------------------+-----------------------+-------------------------+ | 12 12 3 | 4 5 6 | 7 8 9 | | 45 45 7 | 1 8 9 | 26 3 26 | | 6 9 8 | 3 2 7 | 145 145 d45 | +--------------------+-----------------------+-------------------------+ | 12 128* C45 | 69 369 18* | 36 Ba45 7 | | b458 3 6 | 2 7 F458 |Aa45-8* 9 1 | | 7 1458 9 | G56-8 1346 F1458 | 34568* 2 G68*-45 | +--------------------+-----------------------+-------------------------+ | 3 6 D45 | 5789 149 E1458 | 124589 1457 2458 | | c458 7 2 | 589 149 3 | 14589 6 d458 | | 9 458* 1 | 5678* 46 2 | 458* 457 3 | +--------------------+-----------------------+-------------------------+

2. (8): r9c4 =' [r4c6 = r4c2 - r9c2 '= r9c7 - r56c7 = r6c7] => -8 r6c4 (*)
3. ALS W-Wing (45=8)b6b24 - r5c1 = r8c1 - (8=45)r38c9 => -45 r6c9
4. (5)r5c7 = r4c8 - r4c3 = r7c3 - r7c6 = r56c6 - (5=68)r6c49 => -8 r5c7; lcls

Code: Select all: +-------------------+----------------------+-------------------------+ | 12 12 3 | 4 5 6 | 7 8 9 | | 45 45 7 | 1 8 9 | 26 3 26 | | 6 9 8 | 3 2 7 | 145 145 45 | +-------------------+----------------------+-------------------------+ | 12 128 45#^ | 69 369 18 | 36 45#^ 7 | | 458 3 6 | 2 7 458 | 45# 9 1 | | 7 145 9 | 56 1346 145 | 368 2 68 | +-------------------+----------------------+-------------------------+ | 3 6 45*#^| 78-59 149* 145* | 1289-45 17-45 28-45 | | 458*# 7 2 | 589 149* 3 | 19-458 6 45-8*# | | 9 45-8^ 1 | 5678 46 2 | 458# 7-45 3 | +-------------------+----------------------+-------------------------+

5. MSLS; 7 cells (*): r7c356, r8c1459; 7 links: 45r78, 8r8, 19b8 => -59 r7c4, -45 r7c789, -458 r8c7; HP(45)r38c9 => -8 r8c9
6. 7-link bivalue oddagon (45)r48,c37,b679 (#), having two guardians: (8): r8c1 = r9c7 => -8 r9c2; lcls, 9 placements
7. Remote Pair (45)r4c38, b7p38 (^) => -45 r9c8; ste

Website · by **denis_berthier** » Thu Jan 16, 2025 5:37 am

Cenoman wrote:As a start, a solution in four steps, among which tridagon and UR.
Code: Select all
+-------------------------+-------------------------+--------------------------+ | 128 128 38 | 4 5 6 | 7 138 9 | | 145 145 3457 | z17 8 9 | 26 1345 26 | | 6 9 4578 | 3 2 7-1 |Zb1458 1458 458 | +-------------------------+-------------------------+--------------------------+ | B12458 B12458 458* | 69 369 C1458 | b36 458* 7 | | 458* 3 6 | 2 7 458 | b458* 9 1 | | 7 A1458* 9 | 1568 1346 1458 | b34568 2 a4568* | +-------------------------+-------------------------+--------------------------+ | 3 6 458* | 15789 149 14578 | Y124589 Xw14578* 2458 | | 458* 7 2 | 1589 149 3 | Y14589 6 458* | | 9 458* 1 | y5678 46 2 | b458* x4578 3 | +-------------------------+-------------------------+--------------------------+

The question is, how do you reach this starting point, where many candidates not eliminated by W1 or Basics are already missing?

by **eleven** » Thu Jan 16, 2025 10:45 pm

denis_berthier wrote:
Cenoman wrote:The question is, how do you reach this starting point, where many candidates not eliminated by W1 or Basics are already missing?

The hidden pair 26r2c79 (naked quad 13458) in box 3 and the hidden triple 369 (naked quint 12458) in row 4 are basics.

Website · by **denis_berthier** » Fri Jan 17, 2025 12:40 pm

eleven wrote:
denis_berthier wrote:
Cenoman wrote:The question is, how do you reach this starting point, where many candidates not eliminated by W1 or Basics are already missing?

The hidden pair 26r2c79 (naked quad 13458) in box 3 and the hidden triple 369 (naked quint 12458) in row 4 are basics.

OK. I had missed them because I grant Deadly Patterns higher priorities. In this puzzle, it's better to look for them later.
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Website · by **denis_berthier** » Fri Jan 17, 2025 1:01 pm

Cenoman wrote:This puzzle is a nice example of a solution simplified by uniqueness.

That's why I chose it among those with a tridagon.
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Website · by **denis_berthier** » Fri Jan 24, 2025 6:39 am

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This puzzle is #4921 in mith's T&E(3) collection of minimals.
The two reasons I chose it are:
- using deadly patterns in addition to tridagon rules allows to drastically reduce the max length of chains, from 10 to 4;
- it has a deadly pattern on 6 cells which effectively leads to an ORk-whip elimination (but not to a reduction of the max-length of chains beyond what can be achieved using only DP4s). Note that the resolution path using only DP4s is much shorter. But DP6s are very rarely used in ORk-whips and I wanted to show an example of them.

The following path (manually reduced to show only the useful deadly patterns and useful steps) shows how early detection of the patterns and ultra-persistency of the ORk-relations allows to use the patterns even after they have degenerated (no need to re-add candidates that have been eliminated).

biv-chain[2]: c2n2{r1 r4} - c2n3{r4 r1} ==> r1c2≠1, r1c2≠7, r1c2≠9
biv-chain[2]: c2n2{r4 r1} - c2n3{r1 r4} ==> r4c2≠6, r4c2≠9

Code: Select all: DP4-2-1s-OR5-relation for digits: 46 in cells (marked #): (r6c6 r6c4 r1c6 r1c4) with 5 guardians (in cells marked @) : n8r6c6 n9r6c6 n5r6c4 n1r1c6 n9r1c6 +----------------------+----------------------+----------------------+ ! 12479 23 3479 ! 46# 5 1469#@ ! 1479 8 149 ! ! 149 5 6 ! 7 1489 1489 ! 149 3 2 ! ! 8 179 479 ! 2 3 149 ! 14579 6 1459 ! +----------------------+----------------------+----------------------+ ! 2469 23 349 ! 456 1469 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 14689 14689 ! 169 2 169 ! ! 469 689 1 ! 456#@ 2 4689#@ ! 3 45 7 ! +----------------------+----------------------+----------------------+ ! 3 4 578 ! 9 67 256 ! 256 1 568 ! ! 569 689 2 ! 1 46 3456 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 23456 ! 2456 9 3456 ! +----------------------+----------------------+----------------------+ DP4-2-1s-OR4-relation for digits: 69 in cells (marked #): (r8c2 r8c1 r6c2 r6c1) with 4 guardians (in cells marked @) : n8r8c2 n5r8c1 n8r6c2 n4r6c1 +-------------------+-------------------+-------------------+ ! 12479 23 3479 ! 46 5 1469 ! 1479 8 149 ! ! 149 5 6 ! 7 1489 1489 ! 149 3 2 ! ! 8 179 479 ! 2 3 149 ! 14579 6 1459 ! +-------------------+-------------------+-------------------+ ! 2469 23 349 ! 456 1469 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 14689 14689 ! 169 2 169 ! ! 469#@ 689#@ 1 ! 456 2 4689 ! 3 45 7 ! +-------------------+-------------------+-------------------+ ! 3 4 578 ! 9 67 256 ! 256 1 568 ! ! 569#@ 689#@ 2 ! 1 46 3456 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 23456 ! 2456 9 3456 ! +-------------------+-------------------+-------------------+ DP4-2-1-OR2-relation for digits: 45 in cells (marked #): (r6c8 r6c4 r4c8 r4c4) with 2 guardians (in cells marked @) : n6r6c4 n6r4c4 +-------------------+-------------------+-------------------+ ! 12479 23 3479 ! 46 5 1469 ! 1479 8 149 ! ! 149 5 6 ! 7 1489 1489 ! 149 3 2 ! ! 8 179 479 ! 2 3 149 ! 14579 6 1459 ! +-------------------+-------------------+-------------------+ ! 2469 23 349 ! 456#@ 1469 7 ! 8 45# 169 ! ! 45679 6789 45789 ! 3 14689 14689 ! 169 2 169 ! ! 469 689 1 ! 456#@ 2 4689 ! 3 45# 7 ! +-------------------+-------------------+-------------------+ ! 3 4 578 ! 9 67 256 ! 256 1 568 ! ! 569 689 2 ! 1 46 3456 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 23456 ! 2456 9 3456 ! +-------------------+-------------------+-------------------+

DP4-2-1-OR2-whip[1]: OR2{{n6r4c4 n6r6c4 | .}} ==> r6c6≠6, r1c4≠6, r4c5≠6, r5c5≠6, r5c6≠6
singles ==> r1c4=4, r1c6=6

Code: Select all: +-------------------+-------------------+-------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 149 5 6 ! 7 189 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +-------------------+-------------------+-------------------+ ! 2469 23 349 ! 56 149 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 1489 1489 ! 169 2 169 ! ! 469 689 1 ! 56 2 489 ! 3 45 7 ! +-------------------+-------------------+-------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345 ! 2456 9 3456 ! +-------------------+-------------------+-------------------+

At least one candidate of a previous DP4-2-1s-OR5-relation between candidates n8r6c6 n9r6c6 n5r6c4 n1r1c6 n9r1c6 has just been eliminated.
There remains a DP4-2-1s-OR3-relation between candidates: n8r6c6 n9r6c6 n5r6c4

DP4-2-1s-OR3-whip[2]: OR3{{n9r6c6 n8r6c6 | n5r6c4}} - r6c8{n5 .} ==> r6c6≠4

Code: Select all: DP4-2-1-OR5-relation for digits: 25 in cells (marked #): (r9c7 r9c6 r7c7 r7c6) with 5 guardians (in cells marked @) : n4r9c7 n6r9c7 n3r9c6 n4r9c6 n6r7c7 +----------------------+----------------------+----------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 149 5 6 ! 7 189 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +----------------------+----------------------+----------------------+ ! 2469 23 349 ! 56 149 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 1489 1489 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +----------------------+----------------------+----------------------+ ! 3 4 578 ! 9 67 25# ! 256#@ 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345#@ ! 2456#@ 9 3456 ! +----------------------+----------------------+----------------------+ DP4-2-1-OR5-relation for digits: 19 in cells (marked #): (r5c9 r5c5 r4c9 r4c5) with 5 guardians (in cells marked @) : n6r5c9 n4r5c5 n8r5c5 n6r4c9 n4r4c5 +----------------------+----------------------+----------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 149 5 6 ! 7 189 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +----------------------+----------------------+----------------------+ ! 2469 23 349 ! 56 149#@ 7 ! 8 45 169#@ ! ! 45679 6789 45789 ! 3 1489#@ 1489 ! 169 2 169#@ ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +----------------------+----------------------+----------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345 ! 2456 9 3456 ! +----------------------+----------------------+----------------------+ DP6-2-4s-OR6-relation for digits: 19 in cells (marked #): (r1c9 r1c7 r5c5 r5c7 r4c5 r4c9) with 6 guardians (in cells marked @) : n7r1c7 n4r5c5 n8r5c5 n6r5c7 n4r4c5 n6r4c9 +----------------------+----------------------+----------------------+ ! 1279 23 379 ! 4 5 6 ! 179#@ 8 19# ! ! 149 5 6 ! 7 189 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +----------------------+----------------------+----------------------+ ! 2469 23 349 ! 56 149#@ 7 ! 8 45 169#@ ! ! 45679 6789 45789 ! 3 1489#@ 1489 ! 169#@ 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +----------------------+----------------------+----------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345 ! 2456 9 3456 ! +----------------------+----------------------+----------------------+ DP6-3-2s-OR6-relation for digits: 329 in cells (marked #): (r4c2 r4c3 r4c1 r1c2 r1c3 r1c1) with 6 guardians (in cells marked @) : n4r4c3 n4r4c1 n6r4c1 n7r1c3 n1r1c1 n7r1c1 +----------------------+----------------------+----------------------+ ! 1279#@ 23# 379#@ ! 4 5 6 ! 179 8 19 ! ! 149 5 6 ! 7 189 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +----------------------+----------------------+----------------------+ ! 2469#@ 23# 349#@ ! 56 149 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 1489 1489 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +----------------------+----------------------+----------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345 ! 2456 9 3456 ! +----------------------+----------------------+----------------------+ DP6-3-1-OR3-relation for digits: 546 in cells (marked #): (r6c8 r6c4 r6c1 r4c8 r4c4 r4c1) with 3 guardians (in cells marked @) : n9r6c1 n2r4c1 n9r4c1 +----------------------+----------------------+----------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 149 5 6 ! 7 189 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +----------------------+----------------------+----------------------+ ! 2469#@ 23 349 ! 56# 149 7 ! 8 45# 169 ! ! 45679 6789 45789 ! 3 1489 1489 ! 169 2 169 ! ! 469#@ 689 1 ! 56# 2 89 ! 3 45# 7 ! +----------------------+----------------------+----------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345 ! 2456 9 3456 ! +----------------------+----------------------+----------------------+

biv-chain[3]: r3c6{n1 n9} - r6c6{n9 n8} - b2n8{r2c6 r2c5} ==> r2c5≠1
whip[1]: c5n1{r5 .} ==> r5c6≠1
z-chain[3]: b5n1{r5c5 r4c5} - c5n9{r4 r2} - c5n8{r2 .} ==> r5c5≠4

Code: Select all: +-------------------+-------------------+-------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 149 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +-------------------+-------------------+-------------------+ ! 2469 23 349 ! 56 149 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 189 489 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +-------------------+-------------------+-------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 2345 ! 2456 9 3456 ! +-------------------+-------------------+-------------------+

At least one candidate of a previous DP4-2-1-OR5-relation between candidates n6r5c9 n4r5c5 n8r5c5 n6r4c9 n4r4c5 has just been eliminated.
There remains a DP4-2-1-OR4-relation between candidates: n6r5c9 n8r5c5 n6r4c9 n4r4c5

At least one candidate of a previous DP6-2-4s-OR6-relation between candidates n7r1c7 n4r5c5 n8r5c5 n6r5c7 n4r4c5 n6r4c9 has just been eliminated.
There remains a DP6-2-4s-OR5-relation between candidates: n7r1c7 n8r5c5 n6r5c7 n4r4c5 n6r4c9

z-chain[3]: b5n4{r5c6 r4c5} - r4n1{c5 c9} - b6n9{r4c9 .} ==> r5c6≠9
z-chain[3]: b2n9{r2c6 r3c6} - r6n9{c6 c2} - r8n9{c2 .} ==> r2c1≠9
z-chain[3]: c6n2{r9 r7} - c6n5{r7 r8} - c6n3{r8 .} ==> r9c6≠4

Code: Select all: +-------------------+-------------------+-------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 14 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +-------------------+-------------------+-------------------+ ! 2469 23 349 ! 56 149 7 ! 8 45 169 ! ! 45679 6789 45789 ! 3 189 48 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +-------------------+-------------------+-------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 345 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 235 ! 2456 9 3456 ! +-------------------+-------------------+-------------------+

At least one candidate of a previous DP4-2-1-OR5-relation between candidates n4r9c7 n6r9c7 n3r9c6 n4r9c6 n6r7c7 has just been eliminated.
There remains a DP4-2-1-OR4-relation between candidates: n4r9c7 n6r9c7 n3r9c6 n6r7c7

t-whip[3]: r8c5{n4 n6} - r7c5{n6 n7} - r9c5{n7 .} ==> r4c5≠4, r8c6≠4
hidden-single-in-a-column ==> r5c6=4

Code: Select all: +-------------------+-------------------+-------------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 14 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 14579 6 1459 ! +-------------------+-------------------+-------------------+ ! 2469 23 349 ! 56 19 7 ! 8 45 169 ! ! 5679 6789 5789 ! 3 189 4 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +-------------------+-------------------+-------------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 35 ! 456 7 34568 ! ! 1567 167 57 ! 8 467 235 ! 2456 9 3456 ! +-------------------+-------------------+-------------------+

At least one candidate of a previous DP4-2-1-OR4-relation between candidates n6r5c9 n8r5c5 n6r4c9 n4r4c5 has just been eliminated.
There remains a DP4-2-1-OR3-relation between candidates: n6r5c9 n8r5c5 n6r4c9

At least one candidate of a previous DP6-2-4s-OR5-relation between candidates n7r1c7 n8r5c5 n6r5c7 n4r4c5 n6r4c9 has just been eliminated.
There remains a DP6-2-4s-OR4-relation between candidates: n7r1c7 n8r5c5 n6r5c7 n6r4c9

t-whip[3]: r3c6{n1 n9} - r2n9{c6 c7} - r1c9{n9 .} ==> r3c9≠1, r3c7≠1
whip[3]: r1c9{n9 n1} - c7n1{r2 r5} - c7n9{r5 .} ==> r3c9≠9

DP4-2-1-OR3-whip[3]: OR3{{n6r4c9 n6r5c9 | n8r5c5}} - b4n8{r5c2 r6c2} - r8n8{c2 .} ==> r8c9≠6
DP6-3-1-OR3-whip[3]: OR3{{n9r4c1 n2r4c1 | n9r6c1}} - r6c6{n9 n8} - r6c2{n8 .} ==> r4c1≠6

Code: Select all: +----------------+----------------+----------------+ ! 1279 23 379 ! 4 5 6 ! 179 8 19 ! ! 14 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 179 479 ! 2 3 19 ! 4579 6 45 ! +----------------+----------------+----------------+ ! 249 23 349 ! 56 19 7 ! 8 45 169 ! ! 5679 6789 5789 ! 3 189 4 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +----------------+----------------+----------------+ ! 3 4 578 ! 9 67 25 ! 256 1 568 ! ! 569 689 2 ! 1 46 35 ! 456 7 3458 ! ! 1567 167 57 ! 8 467 235 ! 2456 9 3456 ! +----------------+----------------+----------------+

At least one candidate of a previous DP6-3-2s-OR6-relation between candidates n4r4c3 n4r4c1 n6r4c1 n7r1c3 n1r1c1 n7r1c1 has just been eliminated.
There remains a DP6-3-2s-OR5-relation between candidates: n4r4c3 n4r4c1 n7r1c3 n1r1c1 n7r1c1

DP4-2-1-OR4-whip[3]: c7n2{r9 r7} - OR4{{n6r7c7 n4r9c7 n6r9c7 | n3r9c6}} - r9n2{c6 .} ==> r9c7≠5

t-whip[4]: c9n8{r7 r8} - c9n3{r8 r9} - c9n4{r9 r3} - c9n5{r3 .} ==> r7c9≠6
z-chain[4]: r8c5{n6 n4} - r8c7{n4 n5} - r7c9{n5 n8} - b7n8{r7c3 .} ==> r8c2≠6
z-chain[3]: r8c2{n9 n8} - r6n8{c2 c6} - r6n9{c6 .} ==> r5c2≠9
z-chain[4]: r8c2{n9 n8} - r6n8{c2 c6} - r6n9{c6 c1} - r8n9{c1 .} ==> r3c2≠9
biv-chain[4]: r3c2{n7 n1} - r2c1{n1 n4} - c3n4{r3 r4} - c3n3{r4 r1} ==> r1c3≠7

Code: Select all: +----------------+----------------+----------------+ ! 1279 23 39 ! 4 5 6 ! 179 8 19 ! ! 14 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 17 479 ! 2 3 19 ! 4579 6 45 ! +----------------+----------------+----------------+ ! 249 23 349 ! 56 19 7 ! 8 45 169 ! ! 5679 678 5789 ! 3 189 4 ! 169 2 169 ! ! 469 689 1 ! 56 2 89 ! 3 45 7 ! +----------------+----------------+----------------+ ! 3 4 578 ! 9 67 25 ! 256 1 58 ! ! 569 89 2 ! 1 46 35 ! 456 7 3458 ! ! 1567 167 57 ! 8 467 235 ! 246 9 3456 ! +----------------+----------------+----------------+

At least one candidate of a previous DP6-3-2s-OR5-relation between candidates n4r4c3 n4r4c1 n7r1c3 n1r1c1 n7r1c1 has just been eliminated.
There remains a DP6-3-2s-OR4-relation between candidates: n4r4c3 n4r4c1 n1r1c1 n7r1c1

DP6-3-2s-OR4-whip[3]: b1n9{r3c3 r1c1} - OR4{{n7r1c1 n1r1c1 n4r4c3 | n4r4c1}} - c1n2{r4 .} ==> r4c3≠9
DP6-3-2s-OR4-whip[3]: r2c1{n4 n1} - OR4{{n1r1c1 n4r4c1 n4r4c3 | n7r1c1}} - r3c2{n7 .} ==> r6c1≠4

singles ==> r6c8=4, r4c8=5, r4c4=6, r6c4=5

Code: Select all: +----------------+----------------+----------------+ ! 1279 23 39 ! 4 5 6 ! 179 8 19 ! ! 14 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 17 479 ! 2 3 19 ! 4579 6 45 ! +----------------+----------------+----------------+ ! 249 23 34 ! 6 19 7 ! 8 5 19 ! ! 5679 678 5789 ! 3 189 4 ! 169 2 169 ! ! 69 689 1 ! 5 2 89 ! 3 4 7 ! +----------------+----------------+----------------+ ! 3 4 578 ! 9 67 25 ! 256 1 58 ! ! 569 89 2 ! 1 46 35 ! 456 7 3458 ! ! 1567 167 57 ! 8 467 235 ! 246 9 3456 ! +----------------+----------------+----------------+

At least one candidate of a previous DP6-2-4s-OR4-relation between candidates n7r1c7 n8r5c5 n6r5c7 n6r4c9 has just been eliminated.
There remains a DP6-2-4s-OR3-relation between candidates: n7r1c7 n8r5c5 n6r5c7

At least one candidate of a previous DP4-2-1s-OR4-relation between candidates n8r8c2 n5r8c1 n8r6c2 n4r6c1 has just been eliminated.
There remains a DP4-2-1s-OR3-relation between candidates: n8r8c2 n5r8c1 n8r6c2

whip[1]: b6n6{r5c9 .} ==> r5c1≠6, r5c2≠6
biv-chain[2]: r4c9{n9 n1} - r4c5{n1 n9} ==> r4c1≠9
biv-chain[2]: r4c9{n9 n1} - r1c9{n1 n9} ==> r5c9≠9
biv-chain[2]: r1c9{n1 n9} - r4c9{n9 n1} ==> r5c9≠1
naked-single ==> r5c9=6

Code: Select all: +----------------+----------------+----------------+ ! 1279 23 39 ! 4 5 6 ! 179 8 19 ! ! 14 5 6 ! 7 89 189 ! 149 3 2 ! ! 8 17 479 ! 2 3 19 ! 4579 6 45 ! +----------------+----------------+----------------+ ! 24 23 34 ! 6 19 7 ! 8 5 19 ! ! 579 78 5789 ! 3 189 4 ! 19 2 6 ! ! 69 689 1 ! 5 2 89 ! 3 4 7 ! +----------------+----------------+----------------+ ! 3 4 578 ! 9 67 25 ! 256 1 58 ! ! 569 89 2 ! 1 46 35 ! 456 7 3458 ! ! 1567 167 57 ! 8 467 235 ! 246 9 345 ! +----------------+----------------+----------------+

At least one candidate of a previous DP6-2-4s-OR3-relation between candidates n7r1c7 n8r5c5 n6r5c7 has just been eliminated.
There remains a DP6-2-4s-OR2-relation between candidates: n7r1c7 n8r5c5

DP6-2-4s-OR2-whip[2]: OR2{{n7r1c7 | n8r5c5}} - r5n1{c5 .} ==> r1c7≠1
DP4-2-1s-OR3-whip[3]: c2n6{r9 r6} - OR3{{n8r6c2 n5r8c1 | n8r8c2}} - r8n9{c2 .} ==> r8c1≠6

whip[1]: b7n6{r9c2 .} ==> r9c5≠6, r9c7≠6
biv-chain[2]: b7n6{r9c2 r9c1} - b7n1{r9c1 r9c2} ==> r9c2≠7
biv-chain[2]: b7n1{r9c1 r9c2} - b7n6{r9c2 r9c1} ==> r9c1≠5, r9c1≠7
whip[1]: b7n7{r9c3 .} ==> r3c3≠7, r5c3≠7
biv-chain[3]: r2c1{n1 n4} - r3c3{n4 n9} - r3c6{n9 n1} ==> r3c2≠1, r2c6≠1
stte

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