The New Sudoku Players' Forum

Website · by **denis_berthier** » Wed Jan 22, 2025 4:33 pm

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Code: Select all: +-------+-------+-------+ ! . . . ! . 5 . ! . . . ! ! . 5 6 ! 7 . . ! . . 2 ! ! 8 . . ! 2 3 . ! . 6 . ! +-------+-------+-------+ ! . . . ! . . 7 ! 8 . . ! ! . . . ! 3 . . ! . 2 . ! ! . . 1 ! . 2 . ! 3 . 7 ! +-------+-------+-------+ ! 3 4 . ! 9 . . ! . 1 . ! ! . . 2 ! 1 . . ! . 7 . ! ! . . . ! 8 . . ! . 9 . ! +-------+-------+-------+ ....5.....567....28..23..6......78.....3...2...1.2.3.734.9...1...21...7....8...9.

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 12479 12379 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 2369 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 ! +-------------------+-------------------+-------------------+ 186 candidates

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by **Cenoman** » Thu Jan 23, 2025 11:24 am

The puzzle can be solved with three uniqueness steps:
After basics, (including HP 23r14c2, NT467r789c5)

Code: Select all: +-------------------------+----------------------+-----------------------+ | 12479 23 3479 | 6+4 5 1469 | 1479 8 149 | | 149 5 6 | 7 189 1489 | 149 3 2 | | 8 179 479 | 2 3 149 | 14579 6 1459 | +-------------------------+----------------------+-----------------------+ | 2469 23 349 | 456* 19 7 | 8 45* 169 | | 45679 6789 45789 | 3 189 14689 | 169 2 169 | | 469 689 1 | 456* 2 4689 | 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 | +-------------------------+----------------------+-----------------------+

1. UR(45)r46c48 using single external => +4 r1c4; lcls, 2 placements {NT(169)c9, locked(6)r789c7}

Hidden Text: Show

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 45 | +-------------------------+--------------------+-----------------------+ | 2469 23 349 | 56 19 7 | 8 45 169 | | 45679 6789 45789 | 3 189 1489 | 19 2 169 | | 469 689 1 | 56 2 489 | 3 45 7 | +-------------------------+--------------------+-----------------------+ | 3 4 a578* | 9 67* 25* | Y256* 1 b58 | | 59-6 89-6 2 | 1 zBc46 z35 |zZc456 7 3458 | | 1567 167 57* | 8 A467* y235* | Y2456* 9 345 | +-------------------------+--------------------+-----------------------+

2. MUG(2567)r79c3567 (*), having four guardians (internals 8r7c3, 4r9c5, 3r9c6, 4r9c7)
(8)r7c3 - (8=5)r7c9 - (5=46)r8c57
(4)r9c5 - (4=6)r8c5
(3)r9c6 - (3=456)r8c567
(42-6)r79c7 = (6)r8c7
=> -6 r8c12, lcls {HP(16)r9c12, locked(7)r79c3, NP(349)r134c3}

Code: Select all: +-----------------------+--------------------+----------------------+ | b1279 23 39 | 4 5 6 | a179* 8 19* | | 149 5 6 | 7 189 189 | 149 3 2 | | 8 179 49 | 2 3 19 | 14579 6 45 | +-----------------------+--------------------+----------------------+ | B2469 23 B349 | 56 A19* 7 | 8 45 A169* | | c45679 6789 c5-8 | 3 z189* 1489 | 19* 2 169 | | C469 C689 1 | C56 2 489 | 3 C45 7 | +-----------------------+--------------------+----------------------+ | 3 4 578 | 9 67 25 | 256 1 58 | | 59 89 2 | 1 46 35 | 456 7 3458 | | 16 16 57 | 8 47 235 | 245 9 345 | +-----------------------+--------------------+----------------------+

3. DP(19)r45c5, b6p34, r1c79 (*) using internals
(7)r1c7 - r1c1 = (75)r5c13
(61-9)r4c59 = r4c13 - (9=4568)r6c1248
(8)r5c5
=> -8 r5c3; ste

EDIT: added missing guardian @step #2. Thanks eleven !

by **eleven** » Thu Jan 23, 2025 2:26 pm

Cenoman wrote:2. MUG(2567)r79c3567 (*), having three guardians (internals 8r7c3, 4r9c5, 4r9c7)

Think you missed to mention the guardian 3r9c6 - (3=5)r8c6 - (5=46)r8c57.

by **Cenoman** » Thu Jan 23, 2025 4:27 pm

eleven wrote:
Cenoman wrote:2. MUG(2567)r79c3567 (*), having three guardians (internals 8r7c3, 4r9c5, 4r9c7)

Think you missed to mention the guardian 3r9c6 - (3=5)r8c6 - (5=46)r8c57.

Oops, once more !
Thank you for spotting.
I have edited my post, keeping internals; but using externals in b8 (6r8c5 and 5r8c6 instead of 4r9c5, 3r9c6) would make the solution a bit shorter.
(6)r8c5 == (5)r8c6 = (5=46)r8c57 (missed courage to rewrite tags in the pm's)

by **totuan** » Fri Jan 24, 2025 11:22 am

Code: Select all: *---------------------------------------------------------------------* | #1279 23 379 | 4 5 6 |*19+7 8 *19 | | 149 5 6 | 7 189 189 | 149 3 2 | | 8 179 479 | 2 3 19 | 14579 6 45 | |-----------------------+----------------------+----------------------| | 2469 23 349 | 56 *19 7 | 8 45 *19+6 | |ae45679 6789 d4579-8 | 3 *19+8 1489 |*19 2 169 | | 469 689 1 | 56 2 489 | 3 45 7 | |-----------------------+----------------------+----------------------| | 3 4 578 | 9 67 25 | 256 1 58 | | 569 689 2 | 1 46 35 | 456 7 3458 | | b1567 167 c57 | 8 467 235 | 2456 9 345 | *---------------------------------------------------------------------*

This one can finish by one step after Cenoman’s first move.
DP *-marked cells => (6)r4c9=(7)r1c7=(8)r5c5
If r1c1<>7 => loop abcde: (7)r5c1=r9c1-(7=5)r9c3-r5c3=(5)r5c1 => r7c3<>57, r7c3=8
Present as diagram: => r5c3<>8, ste

Code: Select all: (6)r4c9-(169=8)r5c579* || (7)r1c7-(7)r1c1=[(7)r5c1=r9c1-(7=5)r9c3-r5c3=(5)r5c1]* || (8)r5c5*

Thanks for the puzzle!
totuan

Website · by **denis_berthier** » Sun Jan 26, 2025 4:04 am

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With this puzzle, I wanted to show a rare case of several DP6s that are effectively used in the solution.
(There are more DPs than below, but I kept only the useful ones.)
Note that there's a much simpler solution with only DP4s and no longer chains, so that the DP6s are not really necessary.
Also, the addition of DP4s allows to solve the puzzle with chains of maximum length 4 - instead of 10 with whips alone.

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[/code]

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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