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Website · by **denis_berthier** » Fri Dec 02, 2022 6:30 am

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Code: Select all: +-------+-------+-------+ ! . . . ! . . 6 ! . . . ! ! . . 7 ! 1 8 . ! . . 6 ! ! . . 8 ! . 7 2 ! . 4 1 ! +-------+-------+-------+ ! . . . ! 7 4 . ! 1 . 8 ! ! . . . ! . 6 . ! 4 2 . ! ! . 4 . ! . . . ! . 6 7 ! +-------+-------+-------+ ! 3 . 9 ! 6 . . ! . . . ! ! 5 1 . ! . . . ! . . . ! ! . . 4 ! . . . ! . 1 2 ! +-------+-------+-------+ .....6.....718...6..8.72.41...74.1.8....6.42..4.....673.96.....51.........4....12;11610;297118 SER = 10.4

I think you will all find the same start:

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1249 2359 1235 ! 3459 359 6 ! 235789 35789 359 ! ! 249 2359 7 ! 1 8 3459 ! 2359 359 6 ! ! 69 3569 8 ! 359 7 2 ! 359 4 1 ! +----------------------+----------------------+----------------------+ ! 269 23569 2356 ! 7 4 359 ! 1 359 8 ! ! 1789 35789 135 ! 3589 6 13589 ! 4 2 359 ! ! 189 4 135 ! 23589 12359 13589 ! 359 6 7 ! +----------------------+----------------------+----------------------+ ! 3 278 9 ! 6 125 14578 ! 578 578 45 ! ! 5 1 26 ! 23489 239 34789 ! 36789 3789 349 ! ! 678 678 4 ! 3589 359 35789 ! 356789 1 2 ! +----------------------+----------------------+----------------------+ 199 candidates. hidden-pairs-in-a-row: r1{n7 n8}{c7 c8} ==> r1c8≠9, r1c8≠5, r1c8≠3, r1c7≠9, r1c7≠5, r1c7≠3, r1c7≠2 hidden-single-in-a-block ==> r2c7=2 +----------------------+----------------------+----------------------+ ! 1249 2359 1235 ! 3459 359 6 ! 78 78 359 ! ! 49 359 7 ! 1 8 3459 ! 2 359 6 ! ! 69 3569 8 ! 359 7 2 ! 359 4 1 ! +----------------------+----------------------+----------------------+ ! 269 23569 2356 ! 7 4 359 ! 1 359 8 ! ! 1789 35789 135 ! 3589 6 13589 ! 4 2 359 ! ! 189 4 135 ! 23589 12359 13589 ! 359 6 7 ! +----------------------+----------------------+----------------------+ ! 3 278 9 ! 6 125 14578 ! 578 578 45 ! ! 5 1 26 ! 23489 239 34789 ! 36789 3789 349 ! ! 678 678 4 ! 3589 359 35789 ! 356789 1 2 ! +----------------------+----------------------+----------------------+ OR4-anti-tridagon[12] for digits 3, 5 and 9 in blocks: b2, with cells: r1c5, r2c6, r3c4 b3, with cells: r1c9, r2c8, r3c7 b5, with cells: r6c5, r4c6, r5c4 b6, with cells: r6c7, r4c8, r5c9 with 4 guardians: n4r2c6 n8r5c4 n1r6c5 n2r6c5

Can be solved without replacement.

by **eleven** » Fri Dec 02, 2022 12:04 pm

It's noticable, how many UR's, especially of type 3, can be found in those trigagon puzzles. You wil hardly find one of them in thousands of other very hard puzzles.

Code: Select all: +----------------------+----------------------+-----------------------+ ! 1249 2359 1235 ! 3459 359 6 !#78 #78 359 ! ! 49 359 7 ! 1 8 3459 ! 2 359 6 ! ! 69 3569 8 ! 359 7 2 ! 359 4 1 ! +----------------------+----------------------+-----------------------+ ! 269 23569 2356 ! 7 4 359 ! 1 359 8 ! ! 1789 35789 135 ! 3589 6 13589 ! 4 2 359 ! ! 189 4 135 ! 23589 12359 13589 ! 359 6 7 ! +----------------------+----------------------+-----------------------+ ! 3 278 9 ! 6 12-5 1478-5 !#78+5 #78+5 4-5 ! ! 5 1 26 ! 23489 239 34789 ! 36789 3789 349 ! ! 678 678 4 ! 3589 359 35789 ! 36789-5 1 2 ! +----------------------+----------------------+-----------------------+

UR 78r17c79 => -5r7c569,r9c7

Code: Select all: +----------------------+-----------------------+----------------------+ | 1249 2359 1235 | 3459 359 6 |#78 #78 359 | | 49 359 7 | 1 8 3459 | 2 359 6 | | 69 3569 8 | 359 7 2 | 359 4 1 | +----------------------+-----------------------+-----------------------+ | 269 23569 2356 | 7 4 359 | 1 359 8 | | 1789 35789 135 | 3589 6 13589 | 4 2 359 | | 189 4 135 | 23589 12359 13589 | 359 6 7 | +----------------------+-----------------------+-----------------------+ | 3 278 9 | 6 12 178 | 578 578 4 | | 5 1 *26 | 48-239 *239 478-39 |#78+369 #78+39 *39 | | 678 678 4 | 3589 359 35789 | 36789 1 2 | +----------------------+-----------------------+-----------------------+

UR 78r18c79 (type 3) => -2369r8c46 (quad with * cells) or 7/8 must be in r8c46, and 4 in the other cell.

Code: Select all: +----------------------+----------------------+----------------------+ |b1249 2359 c1235 |w3459 359 6 | 78 78 359 | |b49 359 7 | 1 8 a359+4 | 2 359 6 | |b69 3569 8 | 359 7 2 | 359 4 1 | +----------------------+----------------------+----------------------+ |b269 23569 d2356 | 7 4 359 | 1 359 8 | | 1789 35789 d135 |u359+8 6 13589 | 4 2 359 | | 189 4 d135 | 2 A359+1 13589 | 359 6 7 | +----------------------+----------------------+----------------------+ | 3 D278 9 | 6 B12 178 | 578 578 4 | | 5 1 D26 |v48 239 478 | 36789 3789 39 | | 678 678 4 | 3589 359 35789 | 36789 1 2 | +----------------------+----------------------+----------------------+

3 guardians 8r5c4, 4r2c6, 1r6c5
8r5c4 - (8=4)r8c4 - r1c4 = 4r2c6
4r2c6 - (4=96*24)r2341c1 = (4-1)r1c1 = r1c3 - (1|*2=356)r564c3
1r6c5 - (1=2)r7c5 - r7c2 = (2-6)r8c3 = 6r4c3
=> 6r4c3

Code: Select all: *-------------------------------------------------------------------* | 1249 2359 135 | 3459 359 6 | 78 78 359 | | a49 a359 7 | 1 8 3459 | 2 d39 6 | | a69 a3569 8 | 359 7 2 | 359 4 1 | |-----------------------+-----------------------+-------------------| | a29 b2359 6 | 7 4 359 | 1 c39 8 | | 1789 35789 135 | 3589 6 3589 | 4 2 359 | | 89 4 35 | 2 1 3589 | 359 6 7 | |-----------------------+-----------------------+-------------------| | 3 78 9 | 6 2 1 | 78 5 4 | | 5 1 2 | 48 39 478 | 6 78 39 | | 678 678 4 | 3589 359 35789 | 39 1 2 | *-------------------------------------------------------------------*

4r2c1 = 96235 r234c1,r23c2 - (2|3|5=9)r4c2 - r4c8 = 9r2c8 => -9r2c1

Code: Select all: +-------------------+-------------------+-------------------+ | 1 2 b35 | 4 359 6 | 7 8 c359 | | 4 359 7 | 1 8 359 | 2 39 6 | | 69 3569 8 | 359 7 2 |d359 4 1 | +-------------------+-------------------+-------------------+ | 2 359 6 | 7 4 359 | 1 39 8 | | 7 3589 1 | 359 6 3589 | 4 2 359 | | 89 4 a35 | 2 1 3589 | 39-5 6 7 | +-------------------+-------------------+-------------------+ | 3 7 9 | 6 2 1 | 8 5 4 | | 5 1 2 | 8 39 4 | 6 7 39 | | 68 68 4 | 359 359 7 | 39 1 2 | +-------------------+-------------------+-------------------+

kite 5: 5r6c3 = r1c3 - r1c9 = r3c7 => -5r6c7, stte

by **DEFISE** » Fri Dec 02, 2022 2:04 pm

Code: Select all: |--------------------------------------------------------------------| | 1249 2359 1235 | 3459 359 6 | 78 78 359 | | 49 359 7 | 1 8 3459 | 2 359 6 | | 69 3569 8 | 359 7 2 | 359 4 1 | |--------------------------------------------------------------------| | 269 23569 2356 | 7 4 359 | 1 359 8 | | 1789 35789 135 | 3589 6 13589 | 4 2 359 | | 189 4 135 | 23589 12359 13589 | 359 6 7 | |--------------------------------------------------------------------| | 3 278 9 | 6 125 14578 | 578 578 45 | | 5 1 26 | 23489 239 34789 | 36789 3789 349 | | 678 678 4 | 3589 359 35789 | 356789 1 2 | |--------------------------------------------------------------------|

g-whip[6]: r7n2{c5 c2}- r8c3{n2 n6}- r9n6{c1 c7}- r9n3{c7 c456}- r8c5{n3 n9}- r9n9{c5 .} => -2r6c5
Single: 2r6c4
=> 3 guardians remaining : 4r2c6,8r5c4,1r6c5

Trid-OR3-ctr-S2-braid[10]: b7n7{r7c2 r9c12}- r1{c2n2 HP:c13n12}- c1n4{r1 r2}- c6{r2n4 HP:r78n47}- r7n1{c6 c5}-
r7n8{c6 c78}- r8n8{c7 c4}- OR3{{all guardians |.}} => -2r7c2

Single(s): 2r7c5, 1r7c6, 4r7c9, 2r8c3, 6r8c7, 6r4c3, 1r6c5
Box/Line: 5r7b9 => -5r9c7
Naked pairs: 39r8c59 => -3r8c4 -9r8c4 -3r8c6 -9r8c6 -3r8c8 -9r8c8
Naked pairs: 78c8r18 => -7r7c8 -8r7c8
Single(s): 5r7c8
Naked pairs: 78c7r17 => -7r9c7 -8r9c7
Xwing in rows: 5r24c26 => -5r1c2 -5r3c2 -5r5c2 -5r5c6 -5r6c6 -5r9c6

whip[2]: r3n5{c4 c7}- c9n5{r1 .} => -5r5c4
Single(s): 5r4c6, 5r2c2

whip[4]: r2c8{n9 n3}- r4n3{c8 c2}- r3c2{n3 n6}- r3c1{n6 .} => -9r2c1
STTE

by **Cenoman** » Fri Dec 02, 2022 11:20 pm

Code: Select all: +------------------------+--------------------------+------------------------+ | 1249 2359 1235 | 3459 359 6 | 78 78 359 | | 49 359 7 | 1 8 3459 | 2 359 6 | | 69 3569 8 | 359 7 2 | 359 4 1 | +------------------------+--------------------------+------------------------+ | 269 23569 2356 | 7 4 359 | 1 359 8 | | 1789 35789 135 | 3589 6 13589 | 4 2 359 | | 189 4 135 | 23589 12359 13589 | 359 6 7 | +------------------------+--------------------------+------------------------+ | 3 278 9 | 6 125 14578 | 578 578 45 | | 5 1 26 | 23489 239 34789 | 36789 3789 349 | | 678 678 4 | 3589 359 35789 | 356789 1 2 | +------------------------+--------------------------+------------------------+

1. UR(78)r17c78 using internals => -5 r7c569, r9c7; lcls, 1 placement

Code: Select all: +------------------------+--------------------------+-----------------------+ |zD129-4 y2359 z1235 | 3459 359 6 | 78 78 359 | | b49 359 7 | 1 8 3459 | 2 a359 6 | | 69 3569 8 | 359 7 2 | 359 4 1 | +------------------------+--------------------------+-----------------------+ | B269 B23569 2356 | 7 4 359 | 1 A359 8 | | C1789 C35789 135 | 3589 6 13589 | 4 2 359 | | C189 4 135 | 23589 12359 13589 | 359 6 7 | +------------------------+--------------------------+-----------------------+ | 3 x278 9 | 6 12 178 | 578 578 4 | | 5 1 w26 | 23489 v239 34789 | 36789 u3789 v39 | | 678 678 4 | 3589 359 35789 | 36789 1 2 | +------------------------+--------------------------+-----------------------+

2. Kraken column (9)r248c8
(9)r2c8 - (9=4)r2c1
(9)r4c8 - r4c12 = (978-1)b4p457 = (1)r1c1
(9)r8c8 - (9=32)r8c59 - r8c3 = r7c2 - r1c2 = (21)r1c13
=> -4 r1c1; lcls, 4 placements [elimination of three guardians of TH(359)b2356...]

Code: Select all: +------------------------+-----------------------+---------------------+ | 129 2359 1235 | 4 359* 6 | 78 78 359* | | 4 359 7 | 1 8 359* | 2 359* 6 | | 69 3569 8 | 359* 7 2 | 359* 4 1 | +------------------------+-----------------------+---------------------+ | 269 23569 2356 | 7 4 359* | 1 359* 8 | | 1789 35789 135 | 359* 6 13589 | 4 2 359* | | 189 4 135 | 2 359+1* 13589 | 359* 6 7 | +------------------------+-----------------------+---------------------+ | 3 27 9 | 6 12 17 | 58 58 4 | | 5 1 26 | 8 239 4 | 3679 379 39 | | 678 678 4 | 359 359 3579 | 369 1 2 | +------------------------+-----------------------+---------------------+

3. TH(359)b2356 having a single guardian => +1 r6c5; 18 placements

Code: Select all: +-------------------+---------------------+-------------------+ | 1 2 35* | 4 359 6 | 7 8 39-5 | | 4 359 7 | 1 8 359 | 2 39 6 | | 69 3569 8 | 359 7 2 | 359* 4 1 | +-------------------+---------------------+-------------------+ | 2 359 6 | 7 4 359 | 1 39 8 | | 7 3589 1 | 359 6 3589 | 4 2 359 | | 89 4 35* | 2 1 3589 | 359* 6 7 | +-------------------+---------------------+-------------------+ | 3 7 9 | 6 2 1 | 8 5 4 | | 5 1 2 | 8 39 4 | 6 7 39 | | 68 68 4 | 359 359 7 | 39 1 2 | +-------------------+---------------------+-------------------+

4. Skyscraper: (5)r3c7 = r6c7 - r6c3 = r1c3 => -5 r1c9; ste

Website · by **denis_berthier** » Sun Dec 04, 2022 5:10 am

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Without using uniqueness or g-whips, the solution is quite hard.
Here's the simplest-first solution in gW6+OR3W6 (starting from the RS at the end of the 1st post):

Code: Select all: OR4-anti-tridagon[12] for digits 3, 5 and 9 in blocks: b2, with cells: r1c5, r2c6, r3c4 b3, with cells: r1c9, r2c8, r3c7 b5, with cells: r6c5, r4c6, r5c4 b6, with cells: r6c7, r4c8, r5c9 with 4 guardians: n4r2c6 n8r5c4 n1r6c5 n2r6c5 z-chain[4]: r7n2{c5 c2} - r8c3{n2 n6} - c7n6{r8 r9} - r9n5{c7 .} ==> r7c5≠5 z-chain[5]: r7n5{c9 c6} - r7n1{c6 c5} - r7n2{c5 c2} - r8c3{n2 n6} - c7n6{r8 .} ==> r9c7≠5 whip[1]: r9n5{c6 .} ==> r7c6≠5 whip[5]: r9n3{c6 c7} - c7n6{r9 r8} - r8c3{n6 n2} - r8c5{n2 n9} - b9n9{r8c7 .} ==> r8c4≠3 whip[5]: r9n9{c6 c7} - c7n6{r9 r8} - r8c3{n6 n2} - r8c5{n2 n3} - b9n3{r8c7 .} ==> r8c4≠9 whip[5]: r9n3{c6 c7} - c7n6{r9 r8} - r8c3{n6 n2} - r8c5{n2 n9} - b9n9{r8c7 .} ==> r8c6≠3 whip[5]: r9n9{c6 c7} - c7n6{r9 r8} - r8c3{n6 n2} - r8c5{n2 n3} - b9n3{r8c7 .} ==> r8c6≠9

g-whip[5]: r8c3{n2 n6} - r9n6{c2 c7} - b9n9{r9c7 r8c789} - r8c5{n9 n3} - b9n3{r8c7 .} ==> r8c4≠2

Code: Select all: hidden-single-in-a-column ==> r6c4=2 At least one candidate of a previous Trid-OR4-relation has just been eliminated. There remains a Trid-OR3-relation between candidates: n4r2c6 n8r5c4 n1r6c5 +-------------------+-------------------+-------------------+ ! 1249 2359 1235 ! 3459 359 6 ! 78 78 359 ! ! 49 359 7 ! 1 8 3459 ! 2 359 6 ! ! 69 3569 8 ! 359 7 2 ! 359 4 1 ! +-------------------+-------------------+-------------------+ ! 269 23569 2356 ! 7 4 359 ! 1 359 8 ! ! 1789 35789 135 ! 3589 6 13589 ! 4 2 359 ! ! 189 4 135 ! 23589 1359 13589 ! 359 6 7 ! +-------------------+-------------------+-------------------+ ! 3 278 9 ! 6 12 1478 ! 578 578 45 ! ! 5 1 26 ! 48 239 478 ! 36789 3789 349 ! ! 678 678 4 ! 3589 359 35789 ! 36789 1 2 ! +-------------------+-------------------+-------------------+

Trid-OR3-whip[3]: r8c4{n4 n8} - OR3{{n8r5c4 n4r2c6 | n1r6c5}} - r7n1{c5 .} ==> r7c6≠4
hidden-single-in-a-row ==> r7c9=4
finned-x-wing-in-columns: n5{c9 c3}{r1 r5} ==> r5c2≠5
Trid-OR3-whip[4]: c1n1{r6 r1} - r1n4{c1 c4} - OR3{{n4r2c6 n1r6c5 | n8r5c4}} - r8c4{n8 .} ==> r6c3≠1
g-whip[5]: r8n7{c8 c6} - r8n4{c6 c4} - r8n8{c4 c789} - r7c7{n8 n5} - r7c8{n5 .} ==> r9c7≠7
Trid-OR3-ctr-whip[5]: r6n8{c1 c6} - b5n1{r6c6 r5c6} - r7c6{n1 n7} - r8c6{n7 n4} - OR3{{n1r6c5 n8r5c4 n4r2c6 | .}} ==> r6c1≠1
whip[1]: r6n1{c6 .} ==> r5c6≠1
t-whip[6]: r8c4{n8 n4} - r1n4{c4 c1} - r1n1{c1 c3} - r1n2{c3 c2} - r7n2{c2 c5} - r7n1{c5 .} ==> r7c6≠8
z-chain[3]: r7n8{c8 c2} - b7n2{r7c2 r8c3} - r8n6{c3 .} ==> r8c7≠8
t-whip[4]: r8n7{c8 c6} - r8n4{c6 c4} - r8n8{c4 c8} - r1c8{n8 .} ==> r7c8≠7
z-chain[3]: r7c8{n8 n5} - r7c7{n5 n7} - r1c7{n7 .} ==> r9c7≠8
Trid-OR3-whip[6]: r1n1{c3 c1} - r1n4{c1 c4} - r8c4{n4 n8} - OR3{{n8r5c4 n4r2c6 | n1r6c5}} - r7c5{n1 n2} - b7n2{r7c2 .} ==> r1c3≠2
hidden-pairs-in-a-column: c3{n2 n6}{r4 r8} ==> r4c3≠5, r4c3≠3
Trid-OR3-whip[5]: r1n2{c2 c1} - r1n4{c1 c4} - r8c4{n4 n8} - OR3{{n8r5c4 n4r2c6 | n1r6c5}} - r7c5{n1 .} ==> r7c2≠2

The end is easy, in S3Fin+W3

Code: Select all: singles ==> r8c3=2, r4c3=6, r8c7=6, r7c5=2, r7c6=1, r6c5=1 naked-pairs-in-a-block: b9{r8c9 r9c7}{n3 n9} ==> r8c8≠9, r8c8≠3 naked-pairs-in-a-column: c8{r1 r8}{n7 n8} ==> r7c8≠8 naked-single ==> r7c8=5 x-wing-in-rows: n5{r2 r4}{c2 c6} ==> r9c6≠5, r6c6≠5, r5c6≠5, r3c2≠5, r1c2≠5 biv-chain[2]: c9n5{r1 r5} - r6n5{c7 c3} ==> r1c3≠5 singles ==> r2c2=5, r4c6=5 biv-chain[3]: b6n5{r5c9 r6c7} - r6c3{n5 n3} - r4n3{c2 c8} ==> r5c9≠3 biv-chain[2]: b6n3{r6c7 r4c8} - r2n3{c8 c6} ==> r6c6≠3 whip[1]: b5n3{r5c6 .} ==> r5c2≠3, r5c3≠3 naked-pairs-in-a-row: r6{c1 c6}{n8 n9} ==> r6c7≠9 finned-swordfish-in-rows: n9{r2 r6 r4}{c8 c6 c1} ==> r5c1≠9 biv-chain[3]: c7n5{r3 r6} - r5c9{n5 n9} - b9n9{r8c9 r9c7} ==> r3c7≠9 singles ==> r9c7=9, r8c9=3, r8c5=9 finned-x-wing-in-columns: n3{c7 c3}{r6 r3} ==> r3c2≠3 whip[1]: b1n3{r1c3 .} ==> r1c4≠3, r1c5≠3 stte

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