The New Sudoku Players' Forum

Website · by **denis_berthier** » Thu Sep 29, 2022 2:08 pm

#3323

Much easier than my previous one.

Code: Select all: +-------+-------+-------+ ! . . . ! . . . ! 7 . 9 ! ! . 5 7 ! 1 . 9 ! . . 6 ! ! . 6 9 ! 7 3 . ! 1 5 . ! +-------+-------+-------+ ! . . . ! 6 9 . ! . 1 . ! ! 6 . . ! 5 . 3 ! . 7 . ! ! . . . ! . 7 1 ! 6 . . ! +-------+-------+-------+ ! 3 . 4 ! . . . ! 5 . . ! ! 5 . 6 ! 3 . 7 ! . . . ! ! . 8 2 ! . . . ! . . . ! +-------+-------+-------+ ......7.9.571.9..6.6973.15....69..1.6..5.3.7.....716..3.4...5..5.63.7....82......;592;122738 SER = 10.9

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 1248 1234 138 ! 248 24568 24568 ! 7 248 9 ! ! 248 5 7 ! 1 248 9 ! 2348 2348 6 ! ! 248 6 9 ! 7 3 248 ! 1 5 248 ! +-------------------+-------------------+-------------------+ ! 2478 2347 358 ! 6 9 248 ! 2348 1 23458 ! ! 6 1249 18 ! 5 248 3 ! 2489 7 248 ! ! 2489 2349 358 ! 248 7 1 ! 6 23489 23458 ! +-------------------+-------------------+-------------------+ ! 3 179 4 ! 289 1268 268 ! 5 2689 1278 ! ! 5 19 6 ! 3 1248 7 ! 2489 2489 1248 ! ! 179 8 2 ! 49 1456 456 ! 349 3469 1347 ! +-------------------+-------------------+-------------------+

by **yzfwsf** » Thu Sep 29, 2022 11:40 pm

TH was used 4 times。

Hidden Text: Show

by **DEFISE** » Fri Sep 30, 2022 10:13 am

Too hard without general ORk-whips, ORk-forcing-whips, krakens and UR.
This time I think that contrad partial whips are not enough.

After basics :

Code: Select all: |-----------------------------------------------------------| | 1248 1234 138 | 248 56 56 | 7 248 9 | | 248 5 7 | 1 248 9 | 2348 2348 6 | | 248 6 9 | 7 3 248 | 1 5 248 | |-----------------------------------------------------------| | 2478 2347 358 | 6 9 248 | 2348 1 23458 | | 6 1249 18 | 5 248 3 | 2489 7 248 | | 2489 2349 358 | 248 7 1 | 6 23489 23458 | |-----------------------------------------------------------| | 3 179 4 | 289 1268 268 | 5 2689 1278 | | 5 19 6 | 3 1248 7 | 2489 2489 1248 | | 179 8 2 | 49 1456 456 | 349 3469 1347 | |-----------------------------------------------------------|

Tridagon (2,4,8) in b2p159, b3p249, b5p357, b6p168
with 4 guardians: 3r2c7, 3r4c7, 3r6c8, 9r6c8

whip[2]: r5n9{c7 c2}- r8n9{c2 .} => -9r9c7
whip[4]: c1n9{r6 r9}- c4n9{r9 r7}- c4n2{r7 r1}- c2n2{r1 .} => -2r6c1
whip[4]: c1n9{r6 r9}- c4n9{r9 r7}- c4n8{r7 r1}- c3n8{r1 .} => -8r6c1
Contrad. partial whip[3]* from 9r9c8: r9c4{n9 n4}- r9c7{n4 n3}- r2n3{c7 c8}- => -9r9c8
Contrad. partial whip[4]* from 9r6c2: c1n9{r6 r9}- r9c4{n9 n4}- r9c7{n4 n3}- r2n3{c7 c8}- => -9r6c2
Contrad. partial whip[7]* from 1r9c1: r9n7{c1 c9}- r7n7{c9 c2}- b7n9{r7c2 r8c2}- b9n9{r8c7 r7c8}- c8n6{r7 r9}- r9n3{c8 c7}- r2n3{c7 c8}- => -1r9c1
Single(s): 1r1c1, 1r5c3
Then the puzzle is solvable in T&E(1). Not bad for a T&E(3).

* a contrad. partial whip has the property that each guardian is seen by a right-linking candidate.

Website · by **denis_berthier** » Fri Sep 30, 2022 1:25 pm

DEFISE wrote:with 4 guardians: 3r2c7, 3r4c7, 3r6c8, 9r6c8
Contrad. partial whip[3]* from 9r9c8: r9c4{n9 n4}- r9c7{n4 n3}- r2n3{c7 c8}- => -9r9c8

It's a different presentation of the OR4-contrad-whip[4]: r9c4{n9 n4} - r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} => r9c8 ≠ 9

In my definition of an ORk-contrad-whip, the ORk-relation plays the role the last CSP-Variable plays in a whip.

by **DEFISE** » Fri Sep 30, 2022 10:20 pm

I had given up because a semi-automatic solution seemed to me too long in the case of this puzzle, unlike the previous puzzles.
But after integrating the ORk-contrad-whips in my "Simplest-first" software, I automatically obtained this path in W10:

Hidden Text: Show

Website · by **denis_berthier** » Sat Oct 01, 2022 6:02 am

.
I chose this puzzle because it requires the full power of ORk-whips for a solution with relatively short chains (length ≤ 7) and ORk-chains (length ≤ 6).

Like all of you, SudoRules reaches a point with an anti-tridagon with 4 guardians, after only a few easy steps.

Code: Select all: hidden-pairs-in-a-row: r1{n5 n6}{c5 c6} ==> r1c6≠8, r1c6≠4, r1c6≠2, r1c5≠8, r1c5≠4, r1c5≠2 finned-x-wing-in-columns: n9{c1 c8}{r6 r9} ==> r9c7≠9 z-chain[4]: c1n9{r6 r9} - c4n9{r9 r7} - c4n2{r7 r1} - c2n2{r1 .} ==> r6c1≠2 z-chain[4]: c1n9{r6 r9} - c4n9{r9 r7} - c4n8{r7 r1} - c3n8{r1 .} ==> r6c1≠8 +-------------------+-------------------+-------------------+ ! 1248 1234 138 ! 248 56 56 ! 7 248 9 ! ! 248 5 7 ! 1 248 9 ! 2348 2348 6 ! ! 248 6 9 ! 7 3 248 ! 1 5 248 ! +-------------------+-------------------+-------------------+ ! 2478 2347 358 ! 6 9 248 ! 2348 1 23458 ! ! 6 1249 18 ! 5 248 3 ! 2489 7 248 ! ! 49 2349 358 ! 248 7 1 ! 6 23489 23458 ! +-------------------+-------------------+-------------------+ ! 3 179 4 ! 289 1268 268 ! 5 2689 1278 ! ! 5 19 6 ! 3 1248 7 ! 2489 2489 1248 ! ! 179 8 2 ! 49 1456 456 ! 34 3469 1347 ! +-------------------+-------------------+-------------------+ OR4-anti-tridagon[12] for digits 2, 4 and 8 in blocks: b2, with cells: r1c4, r2c5, r3c6 b3, with cells: r1c8, r2c7, r3c9 b5, with cells: r6c4, r5c5, r4c6 b6, with cells: r6c8, r5c9, r4c7 with 4 guardians: n3r2c7 n3r4c7 n3r6c8 n9r6c8

This is where OR4-whips come into play:
Trid-OR4-whip[4]: r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r6c8 n3r4c7 n3r2c7 | n9r6c8}} - b9n9{r7c8 .} ==> r8c7≠4
Trid-OR4-whip[4]: r9c4{n9 n4} - r9c7{n4 n3} - OR4{{n3r4c7 n9r6c8 n3r2c7 | n3r6c8}} - r2n3{c8 .} ==> r9c8≠9
Trid-OR4-whip[5]: r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r6c8 n3r4c7 n3r2c7 | n9r6c8}} - c1n9{r6 r9} - r9n7{c1 .} ==> r9c9≠4
Trid-OR4-whip[5]: r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r6c8 n3r4c7 n3r2c7 | n9r6c8}} - c1n9{r6 r9} - r9c4{n9 .} ==> r9c8≠4
Trid-OR4-whip[5]: r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r6c8 n3r4c7 n3r2c7 | n9r6c8}} - c1n9{r6 r9} - r9c4{n9 .} ==> r9c6≠4
naked-pairs-in-a-column: c6{r1 r9}{n5 n6} ==> r7c6≠6
Trid-OR4-whip[5]: r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r6c8 n3r4c7 n3r2c7 | n9r6c8}} - c1n9{r6 r9} - r9c4{n9 .} ==> r9c5≠4
biv-chain[3]: b8n4{r8c5 r9c4} - r9n9{c4 c1} - r8c2{n9 n1} ==> r8c5≠1
naked-triplets-in-a-column: c5{r2 r5 r8}{n2 n4 n8} ==> r7c5≠8, r7c5≠2
t-whip[4]: r8c2{n1 n9} - b9n9{r8c8 r7c8} - r7n6{c8 c5} - c5n1{r7 .} ==> r9c1≠1
singles ==> r1c1=1, r5c3=1
biv-chain[3]: r4n7{c2 c1} - r9c1{n7 n9} - r6c1{n9 n4} ==> r4c2≠4
z-chain[4]: r7n7{c9 c2} - r7n1{c2 c5} - r9n1{c5 c9} - c9n7{r9 .} ==> r7c9≠8, r7c9≠2
Trid-OR4-whip[5]: c1n9{r6 r9} - r9c4{n9 n4} - r9c7{n4 n3} - OR4{{n3r4c7 n9r6c8 n3r2c7 | n3r6c8}} - r2n3{c8 .} ==> r6c2≠9
Trid-OR4-whip[6]: r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r6c8 n3r4c7 n3r2c7 | n9r6c8}} - r6c1{n9 n4} - c2n4{r6 r1} - c8n4{r1 .} ==> r8c9≠4

The end has nothing noticeable; it is a standard solution in W7:

Code: Select all: finned-x-wing-in-columns: n4{c6 c9}{r3 r4} ==> r4c7≠4 whip[7]: b8n4{r8c5 r9c4} - r9n9{c4 c1} - c1n7{r9 r4} - r4n4{c1 c9} - c9n5{r4 r6} - c9n3{r6 r9} - r9n7{c9 .} ==> r5c5≠4 biv-chain[2]: r9n4{c7 c4} - c5n4{r8 r2} ==> r2c7≠4 biv-chain[4]: c7n4{r5 r9} - r9c4{n4 n9} - c1n9{r9 r6} - b6n9{r6c8 r5c7} ==> r5c7≠2, r5c7≠8 t-whip[7]: r8n2{c9 c5} - r8n4{c5 c8} - r9n4{c7 c4} - b5n4{r6c4 r4c6} - c6n2{r4 r3} - r1c4{n2 n8} - r1c8{n8 .} ==> r7c8≠2 whip[1]: b9n2{r8c9 .} ==> r8c5≠2 whip[6]: r8c5{n4 n8} - b9n8{r8c7 r7c8} - r7c6{n8 n2} - b2n2{r3c6 r1c4} - r1c8{n2 n4} - r8n4{c8 .} ==> r2c5≠4 singles ==> r8c5=4, r9c4=9, r9c1=7, r4c2=7, r7c9=7, r6c1=9, r5c7=9, r9c7=4 whip[1]: r8n8{c9 .} ==> r7c8≠8 whip[5]: r2c5{n2 n8} - r5n8{c5 c9} - r5n4{c9 c2} - r4c1{n4 n8} - r3n8{c1 .} ==> r2c1≠2 z-chain[3]: r2n2{c8 c5} - r5n2{c5 c2} - b1n2{r1c2 .} ==> r3c9≠2 biv-chain[3]: c6n4{r4 r3} - r3c9{n4 n8} - r5n8{c9 c5} ==> r4c6≠8 biv-chain[3]: r2c5{n2 n8} - b5n8{r5c5 r6c4} - c4n4{r6 r1} ==> r1c4≠2 biv-chain[3]: r1n2{c8 c2} - r3n2{c1 c6} - b2n4{r3c6 r1c4} ==> r1c8≠4 finned-swordfish-in-columns: n4{c1 c6 c8}{r2 r3 r4} ==> r4c9≠4 biv-chain[3]: c8n4{r6 r2} - r3c9{n4 n8} - r1c8{n8 n2} ==> r6c8≠2 biv-chain[3]: b1n2{r3c1 r1c2} - r1c8{n2 n8} - r3c9{n8 n4} ==> r3c1≠4 finned-x-wing-in-columns: n4{c8 c1}{r2 r6} ==> r6c2≠4 biv-chain[2]: r3n4{c9 c6} - b5n4{r4c6 r6c4} ==> r6c9≠4 biv-chain[3]: r6n4{c8 c4} - r1n4{c4 c2} - c2n3{r1 r6} ==> r6c8≠3 z-chain[5]: c5n2{r5 r2} - r3n2{c6 c1} - r4n2{c1 c6} - r4n4{c6 c1} - r5c2{n4 .} ==> r5c9≠2 naked-pairs-in-a-block: b6{r5c9 r6c8}{n4 n8} ==> r6c9≠8, r4c9≠8, r4c7≠8 whip[1]: r4n8{c3 .} ==> r6c3≠8 naked-pairs-in-a-column: c9{r3 r5}{n4 n8} ==> r8c9≠8 hidden-pairs-in-a-row: r6{n4 n8}{c4 c8} ==> r6c4≠2 singles ==> r7c4=2, r7c6=8 x-wing-in-columns: n2{c1 c6}{r3 r4} ==> r4c9≠2, r4c7≠2 stte

Note for someone who asked me: do I intend to publish the ORk-whips? Yes, like all my other resolution rules. But don't expect it in the forthcoming week. I'l still testing them.

by **DEFISE** » Sat Oct 01, 2022 8:16 am

Fine !
Have you tried with ORk-forcing-whips only ?

Website · by **denis_berthier** » Sat Oct 01, 2022 8:53 am

DEFISE wrote:Have you tried with ORk-forcing-whips only ?

Yes; it works also, but it requires longer chains: W8 + OR4FW8

Code: Select all: hidden-pairs-in-a-row: r1{n5 n6}{c5 c6} ==> r1c6≠8, r1c6≠4, r1c6≠2, r1c5≠8, r1c5≠4, r1c5≠2 finned-x-wing-in-columns: n9{c1 c8}{r6 r9} ==> r9c7≠9 z-chain[4]: c1n9{r6 r9} - c4n9{r9 r7} - c4n2{r7 r1} - c2n2{r1 .} ==> r6c1≠2 z-chain[4]: c1n9{r6 r9} - c4n9{r9 r7} - c4n8{r7 r1} - c3n8{r1 .} ==> r6c1≠8 whip[8]: r6c1{n9 n4} - b1n4{r1c1 r1c2} - c4n4{r1 r9} - r9c7{n4 n3} - r2n3{c7 c8} - c8n4{r2 r8} - r8n9{c8 c7} - r5n9{c7 .} ==> r6c2≠9 +-------------------+-------------------+-------------------+ ! 1248 1234 138 ! 248 56 56 ! 7 248 9 ! ! 248 5 7 ! 1 248 9 ! 2348 2348 6 ! ! 248 6 9 ! 7 3 248 ! 1 5 248 ! +-------------------+-------------------+-------------------+ ! 2478 2347 358 ! 6 9 248 ! 2348 1 23458 ! ! 6 1249 18 ! 5 248 3 ! 2489 7 248 ! ! 49 234 358 ! 248 7 1 ! 6 23489 23458 ! +-------------------+-------------------+-------------------+ ! 3 179 4 ! 289 1268 268 ! 5 2689 1278 ! ! 5 19 6 ! 3 1248 7 ! 2489 2489 1248 ! ! 179 8 2 ! 49 1456 456 ! 34 3469 1347 ! +-------------------+-------------------+-------------------+ OR4-anti-tridagon[12] for digits 2, 4 and 8 in blocks: b2, with cells: r1c4, r2c5, r3c6 b3, with cells: r1c8, r2c7, r3c9 b5, with cells: r6c4, r5c5, r4c6 b6, with cells: r6c8, r5c9, r4c7 with 4 guardians: n3r2c7 n3r4c7 n3r6c8 n9r6c8

Code: Select all: Trid-OR4-forcing-whip-elim[6]: || n3r2c7 - partial-whip[1]: r9c7{n3 n4} - || n3r4c7 - partial-whip[1]: r9c7{n3 n4} - || n9r6c8 - partial-whip[1]: c7n9{r5 r8} - || n3r6c8 - partial-whip[2]: r2n3{c8 c7} - r9c7{n3 n4} - ==> r8c7≠4 Trid-OR4-forcing-whip-elim[6]: || n3r2c7 - partial-whip[1]: r9c7{n3 n4} - || n3r4c7 - partial-whip[1]: r9c7{n3 n4} - || n3r6c8 - partial-whip[1]: c9n3{r6 r9} - || n9r6c8 - partial-whip[2]: c1n9{r6 r9} - r9n7{c1 c9} - ==> r9c9≠4 Trid-OR4-forcing-whip-elim[7]: || n3r2c7 - partial-whip[1]: r9c7{n3 n4} - || n3r4c7 - partial-whip[1]: r9c7{n3 n4} - || n3r6c8 - partial-whip[2]: r2n3{c8 c7} - r9c7{n3 n4} - || n9r6c8 - partial-whip[2]: c1n9{r6 r9} - r9c4{n9 n4} - ==> r9c8≠4 Trid-OR4-forcing-whip-elim[7]: || n3r2c7 - partial-whip[1]: r9c7{n3 n4} - || n3r4c7 - partial-whip[1]: r9c7{n3 n4} - || n3r6c8 - partial-whip[2]: r2n3{c8 c7} - r9c7{n3 n4} - || n9r6c8 - partial-whip[2]: c1n9{r6 r9} - r9c4{n9 n4} - ==> r9c6≠4 naked-pairs-in-a-column: c6{r1 r9}{n5 n6} ==> r7c6≠6 Trid-OR4-forcing-whip-elim[7]: || n3r2c7 - partial-whip[1]: r9c7{n3 n4} - || n3r4c7 - partial-whip[1]: r9c7{n3 n4} - || n3r6c8 - partial-whip[2]: r2n3{c8 c7} - r9c7{n3 n4} - || n9r6c8 - partial-whip[2]: c1n9{r6 r9} - r9c4{n9 n4} - ==> r9c5≠4 Trid-OR4-forcing-whip-elim[8]: || n9r6c8 - || n3r2c7 - partial-whip[2]: r9c7{n3 n4} - r9c4{n4 n9} - || n3r4c7 - partial-whip[2]: r9c7{n3 n4} - r9c4{n4 n9} - || n3r6c8 - partial-whip[3]: r2n3{c8 c7} - r9c7{n3 n4} - r9c4{n4 n9} - ==> r9c8≠9

Code: Select all: biv-chain[3]: r8c2{n1 n9} - r9n9{c1 c4} - b8n4{r9c4 r8c5} ==> r8c5≠1 naked-triplets-in-a-column: c5{r2 r5 r8}{n2 n4 n8} ==> r7c5≠8, r7c5≠2 t-whip[4]: r8c2{n1 n9} - b9n9{r8c8 r7c8} - r7n6{c8 c5} - c5n1{r7 .} ==> r9c1≠1 singles ==> r1c1=1, r5c3=1 biv-chain[3]: r4n7{c2 c1} - r9c1{n7 n9} - r6c1{n9 n4} ==> r4c2≠4 z-chain[4]: r7n7{c9 c2} - r7n1{c2 c5} - r9n1{c5 c9} - c9n7{r9 .} ==> r7c9≠8, r7c9≠2 whip[7]: r9n4{c7 c4} - r9n9{c4 c1} - r6c1{n9 n4} - b5n4{r6c4 r5c5} - r2n4{c5 c8} - r2n3{c8 c7} - r9c7{n3 .} ==> r4c7≠4 whip[5]: r9n4{c7 c4} - r9n9{c4 c1} - r6c1{n9 n4} - r4n4{c1 c6} - r3n4{c6 .} ==> r8c9≠4 whip[7]: b8n4{r8c5 r9c4} - r9n9{c4 c1} - c1n7{r9 r4} - r4n4{c1 c9} - c9n5{r4 r6} - c9n3{r6 r9} - r9n7{c9 .} ==> r5c5≠4 biv-chain[2]: r9n4{c7 c4} - c5n4{r8 r2} ==> r2c7≠4 biv-chain[4]: c7n4{r5 r9} - r9c4{n4 n9} - c1n9{r9 r6} - b6n9{r6c8 r5c7} ==> r5c7≠2, r5c7≠8 t-whip[7]: r8n2{c9 c5} - r8n4{c5 c8} - r9n4{c7 c4} - b5n4{r6c4 r4c6} - c6n2{r4 r3} - r1c4{n2 n8} - r1c8{n8 .} ==> r7c8≠2 whip[1]: b9n2{r8c9 .} ==> r8c5≠2 whip[6]: r8c5{n4 n8} - b9n8{r8c7 r7c8} - r7c6{n8 n2} - b2n2{r3c6 r1c4} - r1c8{n2 n4} - r8n4{c8 .} ==> r2c5≠4 singles ==> r8c5=4, r9c4=9, r9c1=7, r4c2=7, r7c9=7, r6c1=9, r5c7=9, r9c7=4 whip[1]: r8n8{c9 .} ==> r7c8≠8 whip[5]: r2c5{n2 n8} - r5n8{c5 c9} - r5n4{c9 c2} - r4c1{n4 n8} - r3n8{c1 .} ==> r2c1≠2 z-chain[3]: r2n2{c8 c5} - r5n2{c5 c2} - b1n2{r1c2 .} ==> r3c9≠2 biv-chain[3]: c6n4{r4 r3} - r3c9{n4 n8} - r5n8{c9 c5} ==> r4c6≠8 biv-chain[3]: r2c5{n2 n8} - b5n8{r5c5 r6c4} - c4n4{r6 r1} ==> r1c4≠2 biv-chain[3]: r1n2{c8 c2} - r3n2{c1 c6} - b2n4{r3c6 r1c4} ==> r1c8≠4 finned-swordfish-in-columns: n4{c1 c6 c8}{r2 r3 r4} ==> r4c9≠4 biv-chain[3]: c8n4{r6 r2} - r3c9{n4 n8} - r1c8{n8 n2} ==> r6c8≠2 biv-chain[3]: b1n2{r3c1 r1c2} - r1c8{n2 n8} - r3c9{n8 n4} ==> r3c1≠4 finned-x-wing-in-columns: n4{c8 c1}{r2 r6} ==> r6c2≠4 biv-chain[2]: r3n4{c9 c6} - b5n4{r4c6 r6c4} ==> r6c9≠4 biv-chain[3]: r6n4{c8 c4} - r1n4{c4 c2} - c2n3{r1 r6} ==> r6c8≠3 z-chain[5]: c5n2{r5 r2} - r3n2{c6 c1} - r4n2{c1 c6} - r4n4{c6 c1} - r5c2{n4 .} ==> r5c9≠2 naked-pairs-in-a-block: b6{r5c9 r6c8}{n4 n8} ==> r6c9≠8, r4c9≠8, r4c7≠8 whip[1]: r4n8{c3 .} ==> r6c3≠8 naked-pairs-in-a-column: c9{r3 r5}{n4 n8} ==> r8c9≠8 hidden-pairs-in-a-row: r6{n4 n8}{c4 c8} ==> r6c4≠2 singles ==> r7c4=2, r7c6=8 x-wing-in-columns: n2{c1 c6}{r3 r4} ==> r4c9≠2, r4c7≠2 stte

Website · by **denis_berthier** » Sat Oct 01, 2022 9:00 am

.
It also works using only ORk-contrad-whips (which are a particular case of ORk-whips), but again, it requires longer lengths: 8

Same start as above until the OR4-anti-tridagon[12] for digits 2, 4 and 8 in blocks. Then:

Trid-OR4-ctr-whip[8]: c1n9{r9 r6} - r5n9{c2 c7} - r8n9{c7 c2} - r7n9{c2 c4} - r9c4{n9 n4} - r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n9r6c8 n3r6c8 n3r4c7 n3r2c7 | .}} ==> r9c8≠9
Trid-OR4-ctr-whip[8]: r9c7{n4 n3} - r2n3{c7 c8} - r9c8{n3 n6} - r9c6{n6 n5} - b8n4{r9c6 r8c5} - r9c4{n4 n9} - c1n9{r9 r6} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r9c9≠4
Trid-OR4-ctr-whip[8]: c9n7{r7 r9} - c9n1{r9 r8} - r8c2{n1 n9} - b9n9{r8c8 r7c8} - c8n6{r7 r9} - r9n3{c8 c7} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r7c9≠2
Trid-OR4-ctr-whip[8]: c9n7{r7 r9} - c9n1{r9 r8} - r8c2{n1 n9} - b9n9{r8c8 r7c8} - c8n6{r7 r9} - r9n3{c8 c7} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r7c9≠8
Trid-OR4-ctr-whip[8]: r9n7{c1 c9} - r7n7{c9 c2} - b7n9{r7c2 r8c2} - b9n9{r8c8 r7c8} - c8n6{r7 r9} - r9n3{c8 c7} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r9c1≠1
singles ==> r1c1=1, r5c3=1
biv-chain[3]: r4n7{c2 c1} - r9c1{n7 n9} - r6c1{n9 n4} ==> r4c2≠4
Trid-OR4-ctr-whip[8]: r9n1{c5 c9} - r7n1{c9 c2} - r8c2{n1 n9} - b9n9{r8c7 r7c8} - c8n6{r7 r9} - r9n3{c8 c7} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r8c5≠1
naked-triplets-in-a-column: c5{r2 r5 r8}{n2 n4 n8} ==> r9c5≠4, r7c5≠8, r7c5≠2
Trid-OR4-ctr-whip[7]: c7n9{r8 r5} - r6n9{c8 c1} - r9n9{c1 c4} - r9n4{c4 c6} - r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r8c7≠4
Trid-OR4-ctr-whip[10]: r9n5{c6 c5} - c5n1{r9 r7} - r7c9{n1 n7} - r7c2{n7 n9} - r5n9{c2 c7} - r8n9{c7 c8} - r8n4{c8 c9} - r9c7{n4 n3} - r2n3{c7 c8} - OR4{{n3r2c7 n3r4c7 n3r6c8 n9r6c8 | .}} ==> r9c6≠4

Code: Select all: naked-pairs-in-a-column: c6{r1 r9}{n5 n6} ==> r7c6≠6 whip[7]: r9n4{c8 c4} - r9n9{c4 c1} - r6c1{n9 n4} - b1n4{r2c1 r1c2} - c8n4{r1 r2} - r2n3{c8 c7} - r9c7{n3 .} ==> r8c9≠4 finned-x-wing-in-columns: n4{c6 c9}{r3 r4} ==> r4c7≠4 whip[7]: b8n4{r8c5 r9c4} - r9n9{c4 c1} - c1n7{r9 r4} - r4n4{c1 c9} - c9n5{r4 r6} - c9n3{r6 r9} - r9n7{c9 .} ==> r5c5≠4 t-whip[5]: r5n9{c7 c2} - c1n9{r6 r9} - r9c4{n9 n4} - c5n4{r8 r2} - c7n4{r2 .} ==> r5c7≠2, r5c7≠8 t-whip[7]: r8n2{c9 c5} - r8n4{c5 c8} - r9n4{c7 c4} - b5n4{r6c4 r4c6} - c6n2{r4 r3} - r1c4{n2 n8} - r1c8{n8 .} ==> r7c8≠2 whip[1]: b9n2{r8c9 .} ==> r8c5≠2 whip[6]: r8c5{n4 n8} - b9n8{r8c7 r7c8} - r7c6{n8 n2} - b2n2{r3c6 r1c4} - r1c8{n2 n4} - r8n4{c8 .} ==> r2c5≠4 singles ==> r8c5=4, r9c4=9, r9c1=7, r4c2=7, r7c9=7, r6c1=9, r5c7=9, r7c8≠8 whip[5]: r2c5{n2 n8} - r5n8{c5 c9} - r5n4{c9 c2} - r4c1{n4 n8} - r3n8{c1 .} ==> r2c1≠2 z-chain[3]: r2n2{c8 c5} - r5n2{c5 c2} - b1n2{r1c2 .} ==> r3c9≠2 biv-chain[3]: c6n4{r4 r3} - r3c9{n4 n8} - r5n8{c9 c5} ==> r4c6≠8 biv-chain[3]: r2c5{n2 n8} - b5n8{r5c5 r6c4} - c4n4{r6 r1} ==> r1c4≠2 biv-chain[3]: r1n2{c8 c2} - r3n2{c1 c6} - b2n4{r3c6 r1c4} ==> r1c8≠4 finned-swordfish-in-rows: n4{r1 r5 r6}{c4 c2 c9} ==> r4c9≠4 biv-chain[3]: b1n2{r3c1 r1c2} - r1c8{n2 n8} - r3c9{n8 n4} ==> r3c1≠4 biv-chain[2]: r3n4{c9 c6} - b5n4{r4c6 r6c4} ==> r6c9≠4 finned-swordfish-in-columns: n4{c7 c8 c1}{r2 r9 r6} ==> r6c2≠4 biv-chain[3]: r6n4{c8 c4} - r1c4{n4 n8} - r1c8{n8 n2} ==> r6c8≠2 biv-chain[3]: r6n4{c8 c4} - r1n4{c4 c2} - c2n3{r1 r6} ==> r6c8≠3 z-chain[5]: c5n2{r5 r2} - r3n2{c6 c1} - r4n2{c1 c6} - r4n4{c6 c1} - r5c2{n4 .} ==> r5c9≠2 naked-pairs-in-a-block: b6{r5c9 r6c8}{n4 n8} ==> r6c9≠8, r4c9≠8, r4c7≠8 whip[1]: r4n8{c3 .} ==> r6c3≠8 naked-pairs-in-a-column: c9{r3 r5}{n4 n8} ==> r8c9≠8 hidden-pairs-in-a-row: r6{n4 n8}{c4 c8} ==> r6c4≠2 singles ==> r7c4=2, r7c6=8 x-wing-in-columns: n2{c1 c6}{r3 r4} ==> r4c9≠2, r4c7≠2 stte

by **Cenoman** » Sun Oct 02, 2022 10:27 pm

A solution using the TH not more than once:

Code: Select all: +----------------------+---------------------+-------------------------+ | 1248 1234 138 | 248 56* 56* | 7 248 9 | | 248 5 7 | 1 248 9 | 2348 2348 6 | | 248 6 9 | 7 3 248 | 1 5 248 | +----------------------+---------------------+-------------------------+ | 2478 2347 358 | 6 9 248 | 2348 1 23458 | | 6 1249 18 | 5 248 3 | 2489 7 248 | | 2489 2349 358 | 248 7 1 | 6 23489 23458 | +----------------------+---------------------+-------------------------+ | 3 179 4 | 289 1268 268 | 5 2689 1278 | | 5 19 6 | 3 1248 7 | 2489 2489 1248 | | 179 8 2 | 49 1456* 456* | 349 349+6 1347 | +----------------------+---------------------+-------------------------+

1. UR(56)r19c56 using single external => +6 r9c8

Code: Select all: +----------------------+---------------------+-------------------------+ | 1248 1234 138 | 248* 56 56 | 7 248* 9 | | 248 5 7 | 1 248* 9 | d2348* 2348 6 | | 248 6 9 | 7 3 248* | 1 5 248* | +----------------------+---------------------+-------------------------+ | 2478 2347 358 | 6 9 248* | d2348* 1 23458 | | 6 g1249 18 | 5 248* 3 | f2489 7 248* | | 2489 2349 358 | 248* 7 1 | 6 e23489* 23458 | +----------------------+---------------------+-------------------------+ | 3 179 4 | 289 1268 268 | 5 289 1278 | | 5 19 6 | 3 1248 7 | 2489 2489 1248 | |ha79-1 8 2 | 49 145 45 | c349 6 b37-14 | +----------------------+---------------------+-------------------------+

2. TH(248)b2356 having four guardians: 3r2c7, 3r4c7, 3r6c8, 9r6c8. Note that 3r2c7 and 3r6c8 have the same parity (both conjugates of 3r2c8); only one chain is needed.
(7)r9c1 = (7-3)r9c9 = r9c7 - (3)r24c7 == (9)r6c8 - r5c7 = r5c2 - r6c1 = (9)r9c1 Loop => -1 r9c1, -14 r9c9; lcls, 7 placements

Code: Select all: +----------------------+--------------------+-------------------------+ | 1 A234 A38 |BA248 5 6 | 7 248 9 | | A248 5 7 | 1 248 9 | 2348 2348 6 | | A248 6 9 | 7 3 248 | 1 5 248 | +----------------------+--------------------+-------------------------+ | 278-4 z237-4 358 | 6 9 248 | 238-4 1 23458 | | 6 yd249 1 | 5 248 3 | e2489 7 248 | |xE49-28 y2349 358 |BA248 7 1 | 6 23489 23458 | +----------------------+--------------------+-------------------------+ | 3 z17 4 | 289 6 28 | 5 289 17 | | 5 zc19 6 | 3 248 7 | 289-4 2489 128-4 | |Db79 8 2 |Ca49 1 5 | a34-9 6 37 | +----------------------+--------------------+-------------------------+

3. (3=49)r9c47 - r9c1 = r8c2 - r5c2 = (9)r5c7 => -9r9c7
4. [RP(28):r6c4 = r1c4 - r1c23 = r23c1] = (4)r16c4 - (4=9)r9c4 - r9c1 = (9)r6c1 => -28 r6c1

5. (4=9)r6c1 - r56c2 = (917)r478c2 => -4 r4c2

6. Kraken row (4)r2c1578 (untagged)
(4)r2c1
(4)r2c5 - r3c6 = (4)r4c6
(4)r2c7 - (4=37)r9c79 - r9c1 = (7)r4c1
(4-3)r2c8 = r6c8 - (3=294)b4p578
=>-4r4c1

7. Kraken row (4)r2c1578 (untagged)
(4)r2c1 - (4=9)r6c1 - r9c1 = (9-4)r9c4 = (4)r9c7
(4)r2c5 - r3c6 = (4)r4c6
(4)r2c7
(4-3)r2c8 = r2c7 - (3=4)r9c7
=>-4r4c7

8. Kraken column (4)r1268c8 (untagged)
(4)r1c8 - r1c2 = r23c1 - (4=9)r6c1 - r9c1 = (9-4)r9c4 = (4)r9c7
(4-3)r2c8 = r2c7 - (3=4)r9c7
(4)r6c8 - (4=9)r6c1 - r9c1 = (9-4)r9c4 = (4)r9c7
(4)r8c8
=>-4r8c79

Code: Select all: +---------------------+--------------------+-------------------------+ | 1 234 38 | 248 5 6 | 7 248 9 | | 248 5 7 | 1 248 9 | 2348 2348 6 | | 248 6 9 | 7 3 248 | 1 5 248 | +---------------------+--------------------+-------------------------+ | 278 237 358 | 6 9 f248 | 238 1 e23458 | | 6 249 1 | 5 28-4 3 | 2489 7 248 | | 49 2349 358 | 248 7 1 | 6 23489 e23458 | +---------------------+--------------------+-------------------------+ | 3 17 4 | 289 6 28 | 5 289 17 | | 5 19 6 | 3 a4-28 7 | 289 b2489 128 | | 79 8 2 | 49 1 5 | c34 6 d37 | +---------------------+--------------------+-------------------------+

9. (4)r8c5 = r8c8 - (4=3)r9c7 - r9c9 = (35-4)r46c9 = r4c6 => -4 r5c5

10. Kraken cell (248)r1c8 (untagged)
(2)r1c8 - (r7c8 | r1c4) = (2)r7c46* & W-Wing [(8=2)r5c5 - r6c4*=*r7c4 - (2=8)r7c6]^
||
(8)r1c8 - (r7c8 | r1c4) = (8)r7c46^ & W-Wing [(2=8)r5c5 - r6c4*=*r7c4 - (8=2)r7c6]*
||
(4)r1c8 - r8c8 = (4)r8c5*^
=> -2*, -8^ r8c5; 13 placements
Presented as a net:

Hidden Text: Show

Code: Select all: +--------------------+-------------------+----------------------+ | 1 234 c38 | y248 5 6 | 7 z248 9 | |Ab4-28 5 7 | 1 Df28 9 |vf238 u34-28 6 | |Bb248 6 9 | 7 3 C248 | 1 5 C248 | +--------------------+-------------------+----------------------+ |wa28 7 d358 | 6 9 w248 |we238 1 2458 | | 6 24 1 | 5 C28 3 | 9 7 C248 | | 9 234 358 | x248 7 1 | 6 2348 2458 | +--------------------+-------------------+----------------------+ | 3 1 4 | 28 6 28 | 5 9 7 | | 5 9 6 | 3 4 7 | 28 z28 1 | | 7 8 2 | 9 1 5 | 4 6 3 | +--------------------+-------------------+----------------------+

11. (8)r4c1 = r23c1 - (8=3)r1c3 - r4c3 = r4c7 - (3=28)r2c57 => -8 r2c1

12. (4)r2c1 = (4-8)r3c1 = [r3c6 = r3c9 - r5c9 = r5c5] - (8=2)r2c5 => -2 r2c1

13. (3)r2c8 = r2c7 - (3=284)r4c167 - r6c4 = r1c4 - (4=28)r18c8 =>-28r2c8; 3 placements

Code: Select all: +-------------------+-------------------+--------------------+ | 1 a23 a38 | d4-28 5 6 | 7 248 9 | | 4 5 7 | 1 28* 9 | 28 3 6 | | a28* 6 9 | 7 3 c248* | 1 5 248 | +-------------------+-------------------+--------------------+ | a28* 7 58 | 6 9 ba248 | 3 1 45-28 | | 6 24 1 | 5 28* 3 | 9 7 248* | | 9 234 358 | 248 7 1 | 6 248 2458 | +-------------------+-------------------+--------------------+ | 3 1 4 | a28 6 a28 | 5 9 7 | | 5 9 6 | 3 4 7 | 28 28 1 | | 7 8 2 | 9 1 5 | 4 6 3 | +-------------------+-------------------+--------------------+

14. [RP(28):r7c4 = r7c6 - r4c6* = r4c1 - r3c1 = r1c23] = (4)r4c6 - r3c6 = (4)r1c4 => -28 r1c4; 5 placements

15. RP(28):r5c9 = r5c5 - r2c5 = r3c6 - r3c1 = r4c1 => -28 r4c9; ste

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