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Website · by **denis_berthier** » Sun Mar 19, 2023 6:51 am

.
Same idea as my recent selections from mith's collection: use the most frequent impossible patterns.

Code: Select all: +-------+-------+-------+ ! 1 . . ! 4 . 6 ! . . . ! ! 4 . . ! 1 8 . ! . . . ! ! . . . ! . 2 7 ! . 1 . ! +-------+-------+-------+ ! . . . ! . 4 1 ! . 7 8 ! ! . 4 1 ! 7 . 8 ! 6 . 2 ! ! . 7 8 ! 2 . . ! 1 4 . ! +-------+-------+-------+ ! . . 4 ! . . . ! . . . ! ! . 9 5 ! . . . ! . 2 . ! ! . . . ! . . . ! . 6 7 ! +-------+-------+-------+ 1..4.6...4..18........27.1.....41.78.417.86.2.782..14...4.......95....2........67;3757;61558 SER = 11.1

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1 2358 2379 ! 4 359 6 ! 235789 3589 359 ! ! 4 2356 23679 ! 1 8 359 ! 23579 359 3569 ! ! 35689 3568 369 ! 359 2 7 ! 34589 1 34569 ! +----------------------+----------------------+----------------------+ ! 23569 2356 2369 ! 3569 4 1 ! 359 7 8 ! ! 359 4 1 ! 7 359 8 ! 6 359 2 ! ! 3569 7 8 ! 2 3569 359 ! 1 4 359 ! +----------------------+----------------------+----------------------+ ! 23678 12368 4 ! 35689 135679 2359 ! 3589 3589 1359 ! ! 3678 9 5 ! 368 1367 34 ! 348 2 134 ! ! 238 1238 23 ! 3589 1359 23459 ! 34589 6 7 ! +----------------------+----------------------+----------------------+ 199 candidates.

by **Cenoman** » Sun Mar 19, 2023 4:41 pm

Code: Select all: +--------------------------+---------------------------+-------------------------+ | 1 2358 2379 | 4 a359* 6 | 27 3589 359* | | 4 2356 23679 | 1 8 359* | 27 359* 3569 | | A68-359 3568 369 | 359*# 2 7 | B359+48* 1 C34569 | +--------------------------+---------------------------+-------------------------+ | 23569 2356 2369 | 359-6*# 4 1 | 359*# 7 8 | | 359 4 1 | 7 a359* 8 | 6 359* 2 | | 359-6 7 8 | 2 a359+6 359* | 1 4 359* | +--------------------------+---------------------------+-------------------------+ | e23678 d12368 4 | 68-359 13579-6 2359 | 3589 3589 1359 | | e378-6 9 5 | E68-3 137-6 E34 | E348 2 D134 | | 238 c1238 23 | 589-3 b1359 23459 | 34589 6 7 | +--------------------------+---------------------------+-------------------------+

1. (6=359)r156c5 - (3|5|9=1)r9c5 - r9c2 = (1-6)r7c2 = (6)r78c1 => -6 r6c1; 1 placement (+6r6c5, -6r4c4, -6r78c5)
2. TH(359)b2356 (*) having two guardians: 48r3c7 => -359 r3c7 and RT(359)r3c4, r4c4, r4c7 (#)
3. (3,5,9)r3c4 == r4c47 - r4c123 = r56c1 => -359 r3c1
4. (3)r34c4 == r4c7 - (3=84)r38c7 - r9c7 = r9c6 - (4=3)r8c6 => -3 r789c4
5. (5,9)r34c4 == r4c7 - r9c7 = r9c456 => -59 r7c4
6. (6=8)r3c1 - (8=4)r3c7 - r3c9 = r8c9 - (4=386)r8c467 => -6 r8c1; ste
Tags: first chain a to e; chains 3,4,5 untagged; last chain A to E

Website · by **denis_berthier** » Tue Mar 21, 2023 6:39 am

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A 100% pattern-based solution, using the following 3 easy to find "neighbours" of tridagon:
(EL15c97 is not really necessary for a solution in W5+(Trid+EL13c290+EL14c13+EL15c97)OR5W5.
Indeed, the solution is simpler without EL15c97, because finding it before it degenerates requires to allow up to 7 guardians; but I wanted to illustrate this pattern.)

Code: Select all: EL13c290: 2 cells missing, 3 added +-------+-------+-------+ ! . . . ! . . - ! . . X ! ! . . Z ! . X . ! . X . ! ! . . Z ! X . . ! X . . ! +-------+-------+-------+ ! . . Z ! . . - ! . . X ! ! . . . ! X . . ! . X . ! ! . . . ! . X . ! X . . ! +-------+-------+-------+ ! o o . ! . . o ! . . . ! ! o o . ! . . o ! . . . ! ! o o . ! . . o ! . . . ! +-------+-------+-------+ EL14c13: 1 cell missing, 3 added +-------+-------+-------+ ! . . Z ! . . X ! . . X ! ! . . . ! . X . ! . X . ! ! . . . ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! . . - ! . . X ! ! . . . ! X Z . ! . X . ! ! . . Z ! . X . ! X . . ! +-------+-------+-------+ ! o o . ! . . . ! . . . ! ! o o . ! . . . ! . . . ! ! o o . ! . . . ! . . . ! +-------+-------+-------+ El15c97: 1 cell missing, 4 added +-------+-------+-------+ ! . . . ! . . X ! . . X ! ! . . Z ! . - . ! . X . ! ! . Z . ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! X . . ! . . X ! ! . Z . ! . X . ! . X . ! ! . . Z ! . . X ! X . . ! +-------+-------+-------+ ! o . . ! . . . ! . . . ! ! o . . ! . . . ! . . . ! ! o . . ! . . . ! . . . ! +-------+-------+-------+ o = free cell - = cell missing wrt tridagon Z = cell added wrt tridagon

Starting from the resolution state after Singles and whips[1]:

Code: Select all: hidden-pairs-in-a-column: c7{n2 n7}{r1 r2} ==> r2c7≠9, r2c7≠5, r2c7≠3, r1c7≠9, r1c7≠8, r1c7≠5, r1c7≠3 finned-x-wing-in-rows: n6{r6 r8}{c1 c5} ==> r7c5≠6

3-digit patterns are detected now, after S3 has been applied;
corresponding ORk-relations are asserted;
ORk-splitting rules that allow to reduce their numbers of guardians are applied when available.

Code: Select all: OR3-anti-tridagon[12] for digits 5, 9 and 3 in blocks: b2, with cells (marked #): r1c5, r2c6, r3c4 b3, with cells (marked #): r1c9, r2c8, r3c7 b5, with cells (marked #): r5c5, r6c6, r4c4 b6, with cells (marked #): r5c8, r6c9, r4c7 with 3 guardians (in cells marked @): n4r3c7 n8r3c7 n6r4c4 +-------------------------+-------------------------+-------------------------+ ! 1 2358 2379 ! 4 359# 6 ! 27 3589 359# ! ! 4 2356 23679 ! 1 8 359# ! 27 359# 3569 ! ! 35689 3568 369 ! 359# 2 7 ! 34589#@ 1 34569 ! +-------------------------+-------------------------+-------------------------+ ! 23569 2356 2369 ! 3569#@ 4 1 ! 359# 7 8 ! ! 359 4 1 ! 7 359# 8 ! 6 359# 2 ! ! 3569 7 8 ! 2 3569 359# ! 1 4 359# ! +-------------------------+-------------------------+-------------------------+ ! 23678 12368 4 ! 35689 13579 2359 ! 3589 3589 1359 ! ! 3678 9 5 ! 368 1367 34 ! 348 2 134 ! ! 238 1238 23 ! 3589 1359 23459 ! 34589 6 7 ! +-------------------------+-------------------------+-------------------------+ EL13c290-OR4-relation for digits: 3, 5 and 9 in cells (marked #): (r4c4 r6c1 r6c9 r6c6 r5c1 r5c8 r5c5 r1c9 r1c5 r2c8 r2c6 r3c1 r3c4) with 4 guardians (in cells marked @) : n6r4c4 n6r6c1 n6r3c1 n8r3c1 +-------------------------+-------------------------+-------------------------+ ! 1 2358 2379 ! 4 359# 6 ! 27 3589 359# ! ! 4 2356 23679 ! 1 8 359# ! 27 359# 3569 ! ! 35689#@ 3568 369 ! 359# 2 7 ! 34589 1 34569 ! +-------------------------+-------------------------+-------------------------+ ! 23569 2356 2369 ! 3569#@ 4 1 ! 359 7 8 ! ! 359# 4 1 ! 7 359# 8 ! 6 359# 2 ! ! 3569#@ 7 8 ! 2 3569 359# ! 1 4 359# ! +-------------------------+-------------------------+-------------------------+ ! 23678 12368 4 ! 35689 13579 2359 ! 3589 3589 1359 ! ! 3678 9 5 ! 368 1367 34 ! 348 2 134 ! ! 238 1238 23 ! 3589 1359 23459 ! 34589 6 7 ! +-------------------------+-------------------------+-------------------------+ EL13c290-OR4-relation between candidates n6r4c4, n6r6c1, n6r3c1 and n8r3c1 + same valence for candidates n6r6c1 and n6r4c4 via c-chain[2]: n6r6c1,n6r6c5,n6r4c4 ==> EL13c290-OR4-relation can be split into two EL13c290-OR3-relations with respective lists of guardians: n6r4c4 n6r3c1 n8r3c1 and n6r6c1 n6r3c1 n8r3c1 . EL14c13s-OR5-relation for digits: 3, 5 and 9 in cells (marked #): (r7c9 r7c4 r3c4 r2c8 r2c6 r1c8 r1c9 r1c5 r6c9 r6c6 r5c8 r5c5 r4c7 r4c4) with 5 guardians (in cells marked @) : n1r7c9 n6r7c4 n8r7c4 n8r1c8 n6r4c4 +-------------------------+-------------------------+-------------------------+ ! 1 2358 2379 ! 4 359# 6 ! 27 3589#@ 359# ! ! 4 2356 23679 ! 1 8 359# ! 27 359# 3569 ! ! 35689 3568 369 ! 359# 2 7 ! 34589 1 34569 ! +-------------------------+-------------------------+-------------------------+ ! 23569 2356 2369 ! 3569#@ 4 1 ! 359# 7 8 ! ! 359 4 1 ! 7 359# 8 ! 6 359# 2 ! ! 3569 7 8 ! 2 3569 359# ! 1 4 359# ! +-------------------------+-------------------------+-------------------------+ ! 23678 12368 4 ! 35689#@ 13579 2359 ! 3589 3589 1359#@ ! ! 3678 9 5 ! 368 1367 34 ! 348 2 134 ! ! 238 1238 23 ! 3589 1359 23459 ! 34589 6 7 ! +-------------------------+-------------------------+-------------------------+ EL15c97s-OR7-relation for digits: 3, 5 and 9 in cells (marked #): (r7c8 r7c4 r9c7 r9c5 r3c4 r2c8 r2c6 r1c9 r1c5 r5c8 r5c5 r4c7 r4c4 r6c9 r6c6) with 7 guardians (in cells marked @) : n8r7c8 n6r7c4 n8r7c4 n4r9c7 n8r9c7 n1r9c5 n6r4c4 +-------------------------+-------------------------+-------------------------+ ! 1 2358 2379 ! 4 359# 6 ! 27 3589 359# ! ! 4 2356 23679 ! 1 8 359# ! 27 359# 3569 ! ! 35689 3568 369 ! 359# 2 7 ! 34589 1 34569 ! +-------------------------+-------------------------+-------------------------+ ! 23569 2356 2369 ! 3569#@ 4 1 ! 359# 7 8 ! ! 359 4 1 ! 7 359# 8 ! 6 359# 2 ! ! 3569 7 8 ! 2 3569 359# ! 1 4 359# ! +-------------------------+-------------------------+-------------------------+ ! 23678 12368 4 ! 35689#@ 13579 2359 ! 3589 3589#@ 1359 ! ! 3678 9 5 ! 368 1367 34 ! 348 2 134 ! ! 238 1238 23 ! 3589 1359#@ 23459 ! 34589#@ 6 7 ! +-------------------------+-------------------------+-------------------------+

The first eliminations + applications of ultra-persistency rules for ORk relations:

Code: Select all: z-chain[3]: c6n2{r7 r9} - r9n4{c6 c7} - r9n5{c7 .} ==> r7c6≠5 z-chain[3]: c6n2{r7 r9} - r9n4{c6 c7} - r9n9{c7 .} ==> r7c6≠9 z-chain[5]: c5n7{r8 r7} - c5n1{r7 r9} - c2n1{r9 r7} - r7n6{c2 c1} - r6n6{c1 .} ==> r8c5≠6 hidden-single-in-a-column ==> r6c5=6 +-------------------+-------------------+-------------------+ ! 1 2358 2379 ! 4 359 6 ! 27 3589 359 ! ! 4 2356 23679 ! 1 8 359 ! 27 359 3569 ! ! 35689 3568 369 ! 359 2 7 ! 34589 1 34569 ! +-------------------+-------------------+-------------------+ ! 23569 2356 2369 ! 359 4 1 ! 359 7 8 ! ! 359 4 1 ! 7 359 8 ! 6 359 2 ! ! 359 7 8 ! 2 6 359 ! 1 4 359 ! +-------------------+-------------------+-------------------+ ! 23678 12368 4 ! 35689 13579 23 ! 3589 3589 1359 ! ! 3678 9 5 ! 368 137 34 ! 348 2 134 ! ! 238 1238 23 ! 3589 1359 23459 ! 34589 6 7 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous Trid-OR3-relation between candidates n4r3c7 n8r3c7 n6r4c4 has just been eliminated. There remains a Trid-OR2-relation between candidates: n4r3c7 n8r3c7 At least one candidate of a previous EL13c290-OR3-relation between candidates n6r4c4 n6r3c1 n8r3c1 has just been eliminated. There remains an EL13c290-OR2-relation between candidates: n6r3c1 n8r3c1 At least one candidate of a previous EL13c290-OR3-relation between candidates n6r6c1 n6r3c1 n8r3c1 has just been eliminated. There remains an EL13c290-OR2-relation between candidates: n6r3c1 n8r3c1 At least one candidate of a previous EL14c13s-OR5-relation between candidates n1r7c9 n6r7c4 n8r7c4 n8r1c8 n6r4c4 has just been eliminated. There remains an EL14c13s-OR4-relation between candidates: n1r7c9 n6r7c4 n8r7c4 n8r1c8 At least one candidate of a previous EL14c13s-OR5-relation between candidates n8r7c8 n6r7c4 n8r7c4 n6r2c9 n6r4c4 has just been eliminated. There remains an EL14c13s-OR4-relation between candidates: n8r7c8 n6r7c4 n8r7c4 n6r2c9

(The following 3 OR2-whips[1] will appear as a single one, when I've coded "blocked" behaviour of ORk-whips[1].)
EL13c290-OR2-whip[1]: OR2{{n8r3c1 n6r3c1 | .}} ==> r3c1≠9
whip[1]: c1n9{r6 .} ==> r4c3≠9
EL13c290-OR2-whip[1]: OR2{{n8r3c1 n6r3c1 | .}} ==> r3c1≠5
whip[1]: c1n5{r6 .} ==> r4c2≠5
EL13c290-OR2-whip[1]: OR2{{n8r3c1 n6r3c1 | .}} ==> r3c1≠3
Trid-OR2-whip[1]: OR2{{n8r3c7 n4r3c7 | .}} ==> r3c7≠9, r3c7≠5, r3c7≠3

Code: Select all: z-chain[4]: r8c6{n3 n4} - r8c7{n4 n8} - r3c7{n8 n4} - c9n4{r3 .} ==> r8c9≠3 z-chain[4]: r3c7{n8 n4} - r8c7{n4 n3} - r8c6{n3 n4} - r9n4{c6 .} ==> r9c7≠8 +-------------------+-------------------+-------------------+ ! 1 2358 2379 ! 4 359 6 ! 27 3589 359 ! ! 4 2356 23679 ! 1 8 359 ! 27 359 3569 ! ! 68 3568 369 ! 359 2 7 ! 48 1 34569 ! +-------------------+-------------------+-------------------+ ! 23569 236 236 ! 359 4 1 ! 359 7 8 ! ! 359 4 1 ! 7 359 8 ! 6 359 2 ! ! 359 7 8 ! 2 6 359 ! 1 4 359 ! +-------------------+-------------------+-------------------+ ! 23678 12368 4 ! 35689 13579 23 ! 3589 3589 1359 ! ! 3678 9 5 ! 368 137 34 ! 348 2 14 ! ! 238 1238 23 ! 3589 1359 23459 ! 3459 6 7 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL15c97s-OR6-relation between candidates n8r7c8 n6r7c4 n8r7c4 n4r9c7 n8r9c7 n1r9c5 has just been eliminated. There remains an EL15c97s-OR5-relation between candidates: n8r7c8 n6r7c4 n8r7c4 n4r9c7 n1r9c5 t-whip[4]: r3c7{n8 n4} - c9n4{r3 r8} - r8c6{n4 n3} - r8c7{n3 .} ==> r7c7≠8 biv-chain[3]: c7n8{r8 r3} - r3c1{n8 n6} - r8n6{c1 c4} ==> r8c4≠8 biv-chain[4]: r8n8{c1 c7} - r3c7{n8 n4} - c9n4{r3 r8} - r8c6{n4 n3} ==> r8c1≠3 z-chain[3]: r8n3{c6 c7} - b9n8{r8c7 r7c8} - c4n8{r7 .} ==> r9c4≠3 z-chain[4]: r8n3{c6 c7} - r8n8{c7 c1} - r9c1{n8 n2} - r9c3{n2 .} ==> r9c5≠3, r9c6≠3 t-whip[4]: r7c6{n2 n3} - r8n3{c6 c7} - r8n8{c7 c1} - c1n7{r8 .} ==> r7c1≠2 t-whip[4]: r8c6{n3 n4} - r9n4{c6 c7} - r3c7{n4 n8} - r8c7{n8 .} ==> r8c4≠3, r8c5≠3 naked-single ==> r8c4=6 +-------------------+-------------------+-------------------+ ! 1 2358 2379 ! 4 359 6 ! 27 3589 359 ! ! 4 2356 23679 ! 1 8 359 ! 27 359 3569 ! ! 68 3568 369 ! 359 2 7 ! 48 1 34569 ! +-------------------+-------------------+-------------------+ ! 23569 236 236 ! 359 4 1 ! 359 7 8 ! ! 359 4 1 ! 7 359 8 ! 6 359 2 ! ! 359 7 8 ! 2 6 359 ! 1 4 359 ! +-------------------+-------------------+-------------------+ ! 3678 12368 4 ! 3589 13579 23 ! 359 3589 1359 ! ! 78 9 5 ! 6 17 34 ! 348 2 14 ! ! 238 1238 23 ! 589 159 2459 ! 3459 6 7 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL14c13s-OR4-relation between candidates n1r7c9 n6r7c4 n8r7c4 n8r1c8 has just been eliminated. There remains an EL14c13s-OR3-relation between candidates: n1r7c9 n8r7c4 n8r1c8 At least one candidate of a previous EL15c97s-OR5-relation between candidates n8r7c8 n6r7c4 n8r7c4 n4r9c7 n1r9c5 has just been eliminated. There remains an EL15c97s-OR4-relation between candidates: n8r7c8 n8r7c4 n4r9c7 n1r9c5 biv-chain[3]: r9n1{c2 c5} - r8c5{n1 n7} - r8c1{n7 n8} ==> r9c2≠8 t-whip[3]: r3c1{n6 n8} - c2n8{r3 r7} - r7n6{c2 .} ==> r4c1≠6 t-whip[3]: r7n6{c2 c1} - r3c1{n6 n8} - c2n8{r3 .} ==> r7c2≠3, r7c2≠2, r7c2≠1 singles ==> r9c2=1, r7c6=2 +-------------------+-------------------+-------------------+ ! 1 2358 2379 ! 4 359 6 ! 27 3589 359 ! ! 4 2356 23679 ! 1 8 359 ! 27 359 3569 ! ! 68 3568 369 ! 359 2 7 ! 48 1 34569 ! +-------------------+-------------------+-------------------+ ! 2359 236 236 ! 359 4 1 ! 359 7 8 ! ! 359 4 1 ! 7 359 8 ! 6 359 2 ! ! 359 7 8 ! 2 6 359 ! 1 4 359 ! +-------------------+-------------------+-------------------+ ! 3678 68 4 ! 3589 13579 2 ! 359 3589 1359 ! ! 78 9 5 ! 6 17 34 ! 348 2 14 ! ! 238 1 23 ! 589 59 459 ! 3459 6 7 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL15c97s-OR4-relation between candidates n8r7c8 n8r7c4 n4r9c7 n1r9c5 has just been eliminated. There remains an EL15c97s-OR3-relation between candidates: n8r7c8 n8r7c4 n4r9c7 hidden-pairs-in-a-block: b8{n1 n7}{r7c5 r8c5} ==> r7c5≠9, r7c5≠5, r7c5≠3

EL15c97s-OR3-whip[3]: OR3{{n8r7c4 n8r7c8 | n4r9c7}} - r3c7{n4 n8} - r8n8{c7 .} ==> r7c1≠8, r7c2≠8

Code: Select all: singles ==> r7c2=6, r4c3=6, r3c1=6, r2c9=6 +----------------+----------------+----------------+ ! 1 2358 2379 ! 4 359 6 ! 27 3589 359 ! ! 4 235 2379 ! 1 8 359 ! 27 359 6 ! ! 6 358 39 ! 359 2 7 ! 48 1 3459 ! +----------------+----------------+----------------+ ! 2359 23 6 ! 359 4 1 ! 359 7 8 ! ! 359 4 1 ! 7 359 8 ! 6 359 2 ! ! 359 7 8 ! 2 6 359 ! 1 4 359 ! +----------------+----------------+----------------+ ! 37 6 4 ! 3589 17 2 ! 359 3589 1359 ! ! 78 9 5 ! 6 17 34 ! 348 2 14 ! ! 238 1 23 ! 589 59 459 ! 3459 6 7 ! +----------------+----------------+----------------+ EL14c13s-OR3-relation between candidates n1r7c9, n8r7c4 and n8r1c8 + same valence for candidates n8r1c8 and n8r7c4 via c-chain[2]: n8r1c8,n8r7c8,n8r7c4 ==> EL14c13s-OR3-relation can be split into two EL14c13s-OR2-relations with respective lists of guardians: n1r7c9 n8r7c4 and n1r7c9 n8r1c8 . biv-chain[3]: r7c1{n3 n7} - r8c1{n7 n8} - b9n8{r8c7 r7c8} ==> r7c8≠3 finned-x-wing-in-columns: n3{c5 c8}{r5 r1} ==> r1c9≠3

EL14c13s-OR2-whip[3]: OR2{{n1r7c9 | n8r1c8}} - r3c7{n8 n4} - b9n4{r8c7 .} ==> r8c9≠1
stte

The New Sudoku Players' Forum

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