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Website · by **denis_berthier** » Tue Jan 03, 2023 5:42 am

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Happy New Year.

Medium difficulty.

Code: Select all: +-------+-------+-------+ ! . . . ! 4 . 6 ! 7 . . ! ! . . . ! . . . ! . 3 6 ! ! 6 8 . ! . . . ! 4 5 1 ! +-------+-------+-------+ ! 2 . . ! 7 6 . ! . 9 . ! ! . . . ! . . . ! . . . ! ! 9 7 . ! . 4 8 ! . 6 2 ! +-------+-------+-------+ ! . 9 . ! 8 . 7 ! . . . ! ! 7 . . ! . 2 4 ! . . . ! ! 8 4 . ! 6 9 . ! . . . ! +-------+-------+-------+ ...4.67.........3668....4512..76..9..........97..48.62.9.8.7...7...24...84.69....;157;145927 SER = 10.9

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 135 1235 1359 ! 4 1358 6 ! 7 28 89 ! ! 145 125 14579 ! 159 1578 159 ! 289 3 6 ! ! 6 8 379 ! 239 37 239 ! 4 5 1 ! +----------------------+----------------------+----------------------+ ! 2 135 13458 ! 7 6 135 ! 1358 9 3458 ! ! 1345 1356 134568 ! 12359 135 12359 ! 1358 1478 34578 ! ! 9 7 135 ! 135 4 8 ! 135 6 2 ! +----------------------+----------------------+----------------------+ ! 135 9 12356 ! 8 135 7 ! 12356 124 345 ! ! 7 1356 1356 ! 135 2 4 ! 135689 18 3589 ! ! 8 4 1235 ! 6 9 135 ! 1235 127 357 ! +----------------------+----------------------+----------------------+ 184 candidates.

by **Cenoman** » Tue Jan 03, 2023 10:19 am

Code: Select all: +-------------------------+---------------------+--------------------------+ | 135 1235 1359 | 4 1358 6 | 7 28 89 | | 145 125 14579 | 159 1578 159 | 289 3 6 | | 6 8 37+9 | 239# 37 239# | 4 5 1 | +-------------------------+---------------------+--------------------------+ | 2 135* 13458 | 7 6 135* | 1358 9 3458 | | 1345* 1356 134568 | 29# 135* 29# | 1358 1478 34578 | | 9 7 135* | 135* 4 8 | 135 6 2 | +-------------------------+---------------------+--------------------------+ | 135* 9 12356 | 8 135* 7 | 12356 124 345 | | 7 1356* 1356 | 135* 2 4 | 135689 18 3589 | | 8 4 1235* | 6 9 135* | 1235 127 357 | +-------------------------+---------------------+--------------------------+

Note TH(135)b4578, having three guardians (4r5c1, 6r8c2, 2r9c3)
But let's start with 1. UR(29)r35c46, using single external => +9r3c3; lcls, 9 placements

Code: Select all: +----------------------+--------------------+------------------------+ | 135 1235 135 | 4 158 6 | 7 28 9 | | 4 125 7 | 159 158 159 | 28 3 6 | | 6 8 9 | 23 7 23 | 4 5 1 | +----------------------+--------------------+------------------------+ | 2 135 48 | 7 6 135 | 1358 9 3458 | | 135 6 48 | 29 135 29 | 1358 1478 34578 | | 9 7 135 | 135 4 8 | 135 6 2 | +----------------------+--------------------+------------------------+ | 135 9 1235 | 8 135 7 | 6 124 345 | | 7 135 6 | 135 2 4 | 9 18 358 | | 8 4 135+2 | 6 9 135 | 1235 127 357 | +----------------------+--------------------+------------------------+

2. TH(135)b4578 has now a single guardian => +2 r9c3; ste

Note that, without uniqueness, a first AIC step yields the same guardian elimination:
1. (4=159)r2c146 - r3c46 = (9-7)r3c3 = (74)r2c13 => +4 r2c1; NT(135)b4p249 => +6 r5c2; TH single guardian +2 r9c3
Resolution state:

Hidden Text: Show

Website · by **denis_berthier** » Thu Jan 05, 2023 5:10 am

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Yes, this is a case where a Tridagon with 3 guardians appears at the start, but two of them can easily be discarded:

Code: Select all: hidden-pairs-in-a-row: r5{n2 n9}{c4 c6} ==> r5c6≠5, r5c6≠3, r5c6≠1, r5c4≠5, r5c4≠3, r5c4≠1 +----------------------+----------------------+----------------------+ ! 135 1235 1359 ! 4 1358 6 ! 7 28 89 ! ! 145 125 14579 ! 159 1578 159 ! 289 3 6 ! ! 6 8 379 ! 239 37 239 ! 4 5 1 ! +----------------------+----------------------+----------------------+ ! 2 135 13458 ! 7 6 135 ! 1358 9 3458 ! ! 1345 1356 134568 ! 29 135 29 ! 1358 1478 34578 ! ! 9 7 135 ! 135 4 8 ! 135 6 2 ! +----------------------+----------------------+----------------------+ ! 135 9 12356 ! 8 135 7 ! 12356 124 345 ! ! 7 1356 1356 ! 135 2 4 ! 135689 18 3589 ! ! 8 4 1235 ! 6 9 135 ! 1235 127 357 ! +----------------------+----------------------+----------------------+ OR3-anti-tridagon[12] for digits 1, 3 and 5 in blocks: b4, with cells: r4c2, r5c1, r6c3 b5, with cells: r4c6, r5c5, r6c4 b7, with cells: r8c2, r7c1, r9c3 b8, with cells: r8c4, r7c5, r9c6 with 3 guardians: n4r5c1 n6r8c2 n2r9c3

Code: Select all: z-chain[3]: r2n7{c3 c5} - c5n8{r2 r1} - r1n1{c5 .} ==> r2c3≠1 z-chain[3]: r2n7{c3 c5} - c5n8{r2 r1} - r1n5{c5 .} ==> r2c3≠5 z-chain[4]: r2n4{c1 c3} - r2n7{c3 c5} - c5n8{r2 r1} - r1n1{c5 .} ==> r2c1≠1 z-chain[4]: r2n4{c1 c3} - r2n7{c3 c5} - c5n8{r2 r1} - r1n5{c5 .} ==> r2c1≠5 naked-single ==> r2c1=4 At least one candidate of a previous Trid-OR3-relation has just been eliminated. There remains a Trid-OR2-relation between candidates: n6r8c2 n2r9c3 +----------------------+----------------------+----------------------+ ! 135 1235 1359 ! 4 1358 6 ! 7 28 89 ! ! 4 125 479 ! 159 1578 159 ! 289 3 6 ! ! 6 8 379 ! 239 37 239 ! 4 5 1 ! +----------------------+----------------------+----------------------+ ! 2 135 13458 ! 7 6 135 ! 1358 9 3458 ! ! 135 1356 134568 ! 29 135 29 ! 1358 1478 34578 ! ! 9 7 135 ! 135 4 8 ! 135 6 2 ! +----------------------+----------------------+----------------------+ ! 135 9 12356 ! 8 135 7 ! 12356 124 345 ! ! 7 1356 1356 ! 135 2 4 ! 135689 18 3589 ! ! 8 4 1235 ! 6 9 135 ! 1235 127 357 ! +----------------------+----------------------+----------------------+ hidden-pairs-in-a-column: c3{n4 n8}{r4 r5} ==> r5c3≠6, r5c3≠5, r5c3≠3, r5c3≠1, r4c3≠5, r4c3≠3, r4c3≠1 hidden-single-in-a-block ==> r5c2=6 At least one candidate of a previous Trid-OR2-relation has just been eliminated. There remains a Trid-OR1-relation between candidates: n2r9c3 +----------------------+----------------------+----------------------+ ! 135 1235 1359 ! 4 1358 6 ! 7 28 89 ! ! 4 125 79 ! 159 1578 159 ! 289 3 6 ! ! 6 8 379 ! 239 37 239 ! 4 5 1 ! +----------------------+----------------------+----------------------+ ! 2 135 48 ! 7 6 135 ! 1358 9 3458 ! ! 135 1356 48 ! 29 135 29 ! 1358 1478 34578 ! ! 9 7 135 ! 135 4 8 ! 135 6 2 ! +----------------------+----------------------+----------------------+ ! 135 9 12356 ! 8 135 7 ! 12356 124 345 ! ! 7 135 1356 ! 135 2 4 ! 135689 18 3589 ! ! 8 4 1235 ! 6 9 135 ! 1235 127 357 ! +----------------------+----------------------+----------------------+

Trid-ORk-relation with only one candidate => r9c3=2
Notice that the full non-degenerate anti-tridagon pattern was still present and could have been used instead of the inherited ORk-relation.

The end is in gW8:

Code: Select all: hidden-triplets-in-a-column: c7{n2 n6 n9}{r2 r7 r8} ==> r8c7≠8, r8c7≠5, r8c7≠3, r8c7≠1, r7c7≠5, r7c7≠3, r7c7≠1, r2c7≠8 singles ==> r2c5=8, r3c5=7, r2c3=7 whip[1]: c7n8{r5 .} ==> r4c9≠8, r5c8≠8, r5c9≠8 hidden-pairs-in-a-column: c9{n8 n9}{r1 r8} ==> r8c9≠5, r8c9≠3 biv-chain[4]: r8n6{c3 c7} - b9n9{r8c7 r8c9} - r1n9{c9 c3} - r3c3{n9 n3} ==> r8c3≠3 biv-chain[4]: r1n2{c2 c8} - b3n8{r1c8 r1c9} - r1n9{c9 c3} - r3c3{n9 n3} ==> r1c2≠3 whip[8]: r8n3{c2 c4} - r8n5{c4 c3} - r8n6{c3 c7} - c7n9{r8 r2} - r1n9{c9 c3} - r3c3{n9 n3} - c6n3{r3 r4} - c2n3{r4 .} ==> r8c2≠1 g-whip[8]: r9n1{c6 c789} - r8c8{n1 n8} - r8c9{n8 n9} - r1n9{c9 c3} - r3c3{n9 n3} - c6n3{r3 r4} - c4n3{r6 r8} - c2n3{r8 .} ==> r9c6≠5 whip[1]: r9n5{c9 .} ==> r7c9≠5 biv-chain[3]: r7c9{n3 n4} - c8n4{r7 r5} - b6n7{r5c8 r5c9} ==> r5c9≠3 t-whip[6]: r7c9{n4 n3} - b7n3{r7c3 r8c2} - b8n3{r8c4 r9c6} - r4n3{c6 c7} - r4n8{c7 c3} - r5c3{n8 .} ==> r5c9≠4 t-whip[6]: r4n8{c7 c3} - r4n4{c3 c9} - r7c9{n4 n3} - b7n3{r7c1 r8c2} - b8n3{r8c4 r9c6} - r4n3{c6 .} ==> r4c7≠5, r4c7≠1 biv-chain[3]: c2n3{r8 r4} - r4n1{c2 c6} - r9c6{n1 n3} ==> r8c4≠3 hidden-single-in-a-row ==> r8c2=3 z-chain[5]: r8c4{n5 n1} - r2c4{n1 n9} - c7n9{r2 r8} - r8n6{c7 c3} - r8n5{c3 .} ==> r6c4≠5 biv-chain[3]: r4n1{c2 c6} - r6c4{n1 n3} - b4n3{r6c3 r5c1} ==> r5c1≠1 z-chain[3]: c1n1{r1 r7} - c5n1{r7 r5} - r4n1{c6 .} ==> r1c2≠1 z-chain[3]: r5c1{n3 n5} - b5n5{r5c5 r4c6} - r4n3{c6 .} ==> r5c7≠3 z-chain[4]: c6n5{r4 r2} - c4n5{r2 r8} - c4n1{r8 r2} - c2n1{r2 .} ==> r4c6≠1 hidden-single-in-a-row ==> r4c2=1 whip[1]: r2n1{c6 .} ==> r1c5≠1 whip[1]: c2n5{r2 .} ==> r1c1≠5, r1c3≠5 biv-chain[3]: c1n5{r7 r5} - c1n3{r5 r1} - r1c5{n3 n5} ==> r7c5≠5 hidden-single-in-a-block ==> r8c4=5 biv-chain[3]: b2n5{r1c5 r2c6} - r4c6{n5 n3} - b8n3{r9c6 r7c5} ==> r1c5≠3 stte

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