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Website · by **denis_berthier** » Sat Dec 31, 2022 4:53 am

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Code: Select all: +-------+-------+-------+ ! . 2 . ! . 5 6 ! 7 . . ! ! . . . ! 1 8 . ! 2 . . ! ! . . . ! 2 . 7 ! . . 5 ! +-------+-------+-------+ ! 2 . . ! 7 . 5 ! 8 . 1 ! ! . . . ! 8 6 . ! . 7 2 ! ! . . . ! . . . ! 6 5 . ! +-------+-------+-------+ ! 3 . 5 ! . . . ! 1 . 8 ! ! . 1 2 ! 5 . 8 ! . . . ! ! 9 8 4 ! . . . ! 5 . 7 ! +-------+-------+-------+ .2..567.....18.2.....2.7..52..7.58.1...86..72......65.3.5...1.8.125.8...984...5.7;10932;206235 SER = 10.4

Code: Select all: Resolution state after Singles (and whips[1]): +----------------------+----------------------+----------------------+ ! 148 2 1389 ! 349 5 6 ! 7 13489 349 ! ! 4567 345679 3679 ! 1 8 349 ! 2 3469 3469 ! ! 1468 3469 13689 ! 2 349 7 ! 349 134689 5 ! +----------------------+----------------------+----------------------+ ! 2 3469 369 ! 7 349 5 ! 8 349 1 ! ! 145 3459 139 ! 8 6 1349 ! 349 7 2 ! ! 1478 3479 13789 ! 349 12349 12349 ! 6 5 349 ! +----------------------+----------------------+----------------------+ ! 3 67 5 ! 469 2479 249 ! 1 2469 8 ! ! 67 1 2 ! 5 3479 8 ! 349 3469 3469 ! ! 9 8 4 ! 36 123 123 ! 5 236 7 ! +----------------------+----------------------+----------------------+ 171 candidates, 1034 csp-links and 1034 links. Density = 7.11%

by **Cenoman** » Sun Jan 01, 2023 10:17 pm

Code: Select all: +-------------------------+------------------------+----------------------+ | 148 2 1389 | 349* 5 6 | 7 18 349* | | 457 34579 379 | 1 8 349* | 2 d3469* c3469 | | 1468 3469 13689 | 2 349* 7 | 349* 18 5 | +-------------------------+------------------------+----------------------+ | 2 3469 369 | 7 349* 5 | 8 349* 1 | | 145 3459 139 | 8 6 e1349* | 349* 7 2 | | 1478 f3479 13789 | f349* f12349 f12349 | 6 5 f349* | +-------------------------+------------------------+----------------------+ | 3 67 5 | 469 2479 249 | 1 2469 8 | | a67 1 2 | 5 3479 8 | 349 3469 b3469 | | 9 8 4 | 36 123 123 | 5 236 7 | +-------------------------+------------------------+----------------------+

1. TH (349)b2356 having two guardians (6r2c8, 1r5c6)
(7=6)r8c1 - r8c9 = r2c9 - (6)r2c8 == (1)r5c6 - (1=23497)r6c24569 => -7 r7c2; lcls, 6 placements

Code: Select all: +------------------------+----------------------+----------------------+ | 18-4^ 2 18+39 | 349 5 6 | 7 18*^ 349 | | 45 34579 379 | 1 8 349 | 2 3469 3469 | | 6 349 18*^ | 2 349 7 | 349 18*^ 5 | +------------------------+----------------------+----------------------+ | 2 349 6 | 7 349 5 | 8 349 1 | | 145 3459 139 | 8 6 349-1 | 349 7 2 | | 48-1 3479 3789-1 | 349 12# 12-349# | 6 5 349 | +------------------------+----------------------+----------------------+ | 3 6 5 | 49 7 249 | 1 249 8 | | 7 1 2 | 5 349 8 | 349 3469 3469 | | 9 8 4 | 6 12# 12+3# | 5 23 7 | +------------------------+----------------------+----------------------+

2. UR(18)r13c38 using internals +39 r1c3; NT(349)r1c349 => -4 r1c1
3. DP(18)r1c18, r3c38, r6c13 using externals => +1 r6c56 => -1 r5c6, r6c13; HP(12)r6c56
4. TH (349)b2356 has now a single guardian: +6 r2c8; 2 placements
5. UR(12)r69c56 using single internal => +3 r9c6; 6 placements

Code: Select all: +-----------------------+-------------------+--------------------+ | 18 2 39 | 349 5 6 | 7 18 349 | | 45 34579 379 | 1 8 49 | 2 6 349 | | 6 349 18 | 2 349 7 | 349 18 5 | +-----------------------+-------------------+--------------------+ | 2 349 6 | 7 349 5 | 8 349 1 | | 145 3459 139 | 8 6 49 | 349 7 2 | | 48 3479 3789 | 349 2 1 | 6 5 349 | +-----------------------+-------------------+--------------------+ | 3 6 5 | 49 7 2 | 1 49 8 | | 7 1 2 | 5 49 8 | 349 349 6 | | 9 8 4 | 6 1 3 | 5 2 7 | +-----------------------+-------------------+--------------------+

Almost Skyscrapers:
6. (4): [r7c4 = r7c8 - r4c8 = r4c5] = r4c2 - r3c2 = r2c12 - r2c6 = r5c6 => -4 r6c4
7. [(9)r5c6 = r2c6 - r2c9 = r6c9] = (9-4)r1c9 = (4-3)r1c4 = (3)r6c4 => -9 r6c4; 2 placements

Code: Select all: +----------------------+-----------------+--------------------+ | 18 2 39 | 49 5 6 | 7 18 349 | | 45 34579 379 | 1 8 49 | 2 6 349 | | 6 49 18 | 2 3 7 | 49 18 5 | +----------------------+-----------------+--------------------+ | 2 349 6 | 7 49 5 | 8 349 1 | | 145 3459 139 | 8 6 49 | 349 7 2 | | 48 479 789 | 3 2 1 | 6 5 49 | +----------------------+-----------------+--------------------+ | 3 6 5 | 49 7 2 | 1 49 8 | | 7 1 2 | 5 49 8 | 349 349 6 | | 9 8 4 | 6 1 3 | 5 2 7 | +----------------------+-----------------+--------------------+

8. Remote pair (4,9): r4c5 = r8c5 - r7c4 = r7c8 => -49 r4c8; lcls, 4 placements

Code: Select all: +-----------------+-----------------+------------------+ | 18 2 39 | 49 5 6 | 7 18 349 | | 45 35 7 | 1 8 49 | 2 6 349 | | 6 49 18 | 2 3 7 | 49 18 5 | +-----------------+-----------------+------------------+ | 2 49 6 | 7 49 5 | 8 3 1 | | 15 35 13 | 8 6 49 | 49 7 2 | | 48 7 89 | 3 2 1 | 6 5 49 | +-----------------+-----------------+------------------+ | 3 6 5 | 49 7 2 | 1 49 8 | | 7 1 2 | 5 49 8 | 3 49 6 | | 9 8 4 | 6 1 3 | 5 2 7 | +-----------------+-----------------+------------------+

9. Remote pair (4,9): r2c6 = r5c6 - r5c7 = r6c9 => -49 r2c9; ste

Website · by **denis_berthier** » Tue Jan 03, 2023 5:35 am

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Thanks for your solution.

In my series of examples, I intended this as one of ORk-g-whips.

hidden-pairs-in-a-column: c8{n1 n8}{r1 r3} ==> r3c8≠9, r3c8≠6, r3c8≠4, r3c8≠3, r1c8≠9, r1c8≠4, r1c8≠3
whip[1]: r3n6{c3 .} ==> r2c1≠6, r2c2≠6, r2c3≠6

Code: Select all: 130 g-candidates, 545 csp-glinks and 326 non-csp glinks +-------------------+-------------------+-------------------+ ! 148 2 1389 ! 349 5 6 ! 7 18 349 ! ! 457 34579 379 ! 1 8 349 ! 2 3469 3469 ! ! 1468 3469 13689 ! 2 349 7 ! 349 18 5 ! +-------------------+-------------------+-------------------+ ! 2 3469 369 ! 7 349 5 ! 8 349 1 ! ! 145 3459 139 ! 8 6 1349 ! 349 7 2 ! ! 1478 3479 13789 ! 349 12349 12349 ! 6 5 349 ! +-------------------+-------------------+-------------------+ ! 3 67 5 ! 469 2479 249 ! 1 2469 8 ! ! 67 1 2 ! 5 3479 8 ! 349 3469 3469 ! ! 9 8 4 ! 36 123 123 ! 5 236 7 ! +-------------------+-------------------+-------------------+ OR2-anti-tridagon[12] for digits 3, 4 and 9 in blocks: b2, with cells: r1c4, r2c6, r3c5 b3, with cells: r1c9, r2c8, r3c7 b5, with cells: r6c4, r5c6, r4c5 b6, with cells: r6c9, r5c7, r4c8 with 2 guardians: n6r2c8 n1r5c6

Trid-OR2-whip[5]: r8c1{n7 n6} - c9n6{r8 r2} - OR2{{n6r2c8 | n1r5c6}} - b4n1{r5c1 r6c3} - r6n8{c3 .} ==> r6c1≠7
biv-chain[4]: c1n6{r3 r8} - b7n7{r8c1 r7c2} - r6n7{c2 c3} - b4n8{r6c3 r6c1} ==> r3c1≠8
Trid-OR2-ctr-whip[6]: c5n1{r9 r6} - c5n2{r6 r7} - c5n7{r7 r8} - r8c1{n7 n6} - c9n6{r8 r2} - OR2{{n1r5c6 n6r2c8 | .}} ==> r9c5≠3
Trid-OR2-whip[6]: c1n6{r3 r8} - c9n6{r8 r2} - OR2{{n6r2c8 | n1r5c6}} - b4n1{r5c1 r6c3} - r6n7{c3 c2} - r7c2{n7 .} ==> r3c1≠1
hidden-pairs-in-a-row: r3{n1 n8}{c3 c8} ==> r3c3≠9, r3c3≠6, r3c3≠3
hidden-single-in-a-column ==> r4c3=6
Trid-OR2-gwhip[6]: c3n7{r2 r6} - r6n8{c3 c1} - r6n1{c1 c456} - OR2{{n1r5c6 | n6r2c8}} - b9n6{r7c8 r8c9} - r8c1{n6 .} ==> r2c1≠7
singles ==> r8c1=7, r7c2=6, r3c1=6, r9c4=6, r7c5=7
hidden-pairs-in-a-column: c5{n1 n2}{r6 r9} ==> r6c5≠9, r6c5≠4, r6c5≠3
t-whip[5]: r9n3{c6 c8} - r8n3{c9 c5} - r4n3{c5 c2} - r3n3{c2 c7} - r5n3{c7 .} ==> r6c6≠3, r2c6≠3
whip[5]: c4n3{r6 r1} - r3n3{c5 c7} - r5n3{c7 c6} - r4n3{c5 c8} - r9n3{c8 .} ==> r6c2≠3
Trid-OR2-whip[8]: OR2{{n6r2c8 | n1r5c6}} - c6n3{r5 r9} - r9n1{c6 c5} - b8n2{r9c5 r7c6} - r7n4{c6 c4} - r1n4{c4 c1} - c1n1{r1 r6} - c1n8{r6 .} ==> r2c8≠4
z-chain[3]: b3n4{r2c9 r3c7} - c5n4{r3 r4} - c8n4{r4 .} ==> r8c9≠4
g-whip[8]: c9n4{r6 r123} - r3n4{c7 c5} - b2n3{r3c5 r1c4} - r6c4{n3 n9} - r4c5{n9 n3} - r4c2{n3 n9} - b6n9{r4c8 r5c7} - r3n9{c7 .} ==> r6c2≠4
whip[7]: r8n9{c9 c5} - r4n9{c5 c2} - r3n9{c2 c7} - r5n9{c7 c6} - r2n9{c6 c3} - r2n7{c3 c2} - r6c2{n7 .} ==> r7c8≠9
whip[1]: b9n9{r8c9 .} ==> r8c5≠9
biv-chain[3]: b2n3{r1c4 r3c5} - r8c5{n3 n4} - r7c4{n4 n9} ==> r1c4≠9
t-whip[6]: b2n3{r1c4 r3c5} - r8c5{n3 n4} - b9n4{r8c7 r7c8} - r4n4{c8 c2} - r3c2{n4 n9} - r1n9{c3 .} ==> r1c9≠3
t-whip[6]: c6n3{r5 r9} - r8c5{n3 n4} - b9n4{r8c7 r7c8} - r4n4{c8 c2} - r3n4{c2 c7} - r5n4{c7 .} ==> r5c6≠9, r5c6≠1

At least one candidate of a previous Trid-OR2-relation has just been eliminated.
There remains a Trid-OR1-relation between candidates: n6r2c8

Code: Select all: +-------------------+-------------------+-------------------+ ! 148 2 1389 ! 34 5 6 ! 7 18 49 ! ! 45 34579 379 ! 1 8 49 ! 2 369 3469 ! ! 6 349 18 ! 2 349 7 ! 349 18 5 ! +-------------------+-------------------+-------------------+ ! 2 349 6 ! 7 349 5 ! 8 349 1 ! ! 145 3459 139 ! 8 6 34 ! 349 7 2 ! ! 148 79 13789 ! 349 12 1249 ! 6 5 349 ! +-------------------+-------------------+-------------------+ ! 3 6 5 ! 49 7 249 ! 1 24 8 ! ! 7 1 2 ! 5 34 8 ! 349 3469 369 ! ! 9 8 4 ! 6 12 123 ! 5 23 7 ! +-------------------+-------------------+-------------------+

Trid-ORk-relation with only one candidate => r2c8=6
Note that this is also the "classical" tridagon elimination rule. But as the ORk-relation has already been identified previously, what remains of it is used as such.

The end is easy:

Code: Select all: hidden-single-in-a-column ==> r8c9=6 whip[1]: b5n1{r6c6 .} ==> r6c1≠1, r6c3≠1 hidden-pairs-in-a-row: r6{n1 n2}{c5 c6} ==> r6c6≠9, r6c6≠4 biv-chain[3]: r5c6{n3 n4} - r2c6{n4 n9} - c5n9{r3 r4} ==> r4c5≠3 biv-chain[2]: c9n3{r2 r6} - r4n3{c8 c2} ==> r2c2≠3 biv-chain[2]: r1n3{c3 c4} - b5n3{r6c4 r5c6} ==> r5c3≠3 biv-chain[3]: r8c5{n3 n4} - r4c5{n4 n9} - c8n9{r4 r8} ==> r8c8≠3 finned-x-wing-in-columns: n3{c6 c8}{r9 r5} ==> r5c7≠3 x-wing-in-columns: n3{c5 c7}{r3 r8} ==> r3c2≠3 whip[1]: c2n3{r5 .} ==> r6c3≠3 biv-chain[3]: r5c7{n9 n4} - r5c6{n4 n3} - r6n3{c4 c9} ==> r6c9≠9 whip[1]: c9n9{r2 .} ==> r3c7≠9 biv-chain[2]: b5n9{r6c4 r4c5} - r3n9{c5 c2} ==> r6c2≠9 stte

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