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Website · by **denis_berthier** » Thu Apr 13, 2023 4:20 am

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Code: Select all: +-------+-------+-------+ ! . . . ! 4 5 . ! 7 . 9 ! ! . 5 . ! 1 . . ! . . . ! ! . . . ! . 2 . ! . 1 5 ! +-------+-------+-------+ ! 2 . 4 ! 5 7 . ! 1 9 . ! ! . . . ! 9 . 2 ! 5 . 4 ! ! . . . ! . 1 4 ! . 7 2 ! +-------+-------+-------+ ! 3 1 . ! . 9 5 ! . . . ! ! . 7 . ! . . . ! 9 . 1 ! ! 8 . . ! . . 1 ! . . . ! +-------+-------+-------+ ...45.7.9.5.1.........2..152.457.19....9.25.4....14.7231..95....7....9.18....1...;2997;38329 SER = 11.6

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 16 2368 12368 ! 4 5 368 ! 7 2368 9 ! ! 4679 5 236789 ! 1 368 36789 ! 23468 23468 368 ! ! 4679 34689 36789 ! 368 2 36789 ! 3468 1 5 ! +----------------------+----------------------+----------------------+ ! 2 368 4 ! 5 7 368 ! 1 9 368 ! ! 167 368 13678 ! 9 368 2 ! 5 368 4 ! ! 569 3689 35689 ! 368 1 4 ! 368 7 2 ! +----------------------+----------------------+----------------------+ ! 3 1 26 ! 2678 9 5 ! 2468 2468 678 ! ! 456 7 256 ! 2368 3468 368 ! 9 23568 1 ! ! 8 2469 2569 ! 2367 346 1 ! 236 2356 367 ! +----------------------+----------------------+----------------------+ 183 candidates.

by **marek stefanik** » Thu Apr 13, 2023 8:50 am

Code: Select all: ,---------------------,-----------------,-------------------, | 16 2368 12368 | 4 5 #368 | 7 368–2 9 | | 4679 5 236789 | 1 #368 79 | 23468 #368+24#368 | | 4679 34689 36789 | 368 2 79 |#368+4 1 5 | :---------------------+-----------------+-------------------: | 2 368 4 | 5 7 #368 | 1 9 #368 | | 17 368 17 | 9 #368 2 | 5 #368 4 | | 569 3689 35689 | 368 1 4 |#368 7 2 | :---------------------+-----------------+-------------------: | 3 1 26 | 2678 9 5 | 2468 2468 678 | | 456 7 256 | 2368 #368+4#368 | 9 #368+25 1 | | 8 2469 2569 | 2367 346 1 | 236 2356 367 | '---------------------'-----------------'-------------------'

Impossibility proof: Show

Code: Select all: ,--------------------,-----------------,-------------------, | 16 2368 12368 | 4 5 A368 | 7 #368 9 | | 4679 5 36789 | 1 B368 79 | 23468 23468 #368 | | 4679 49–368 36789 |C368 2 79 |#4–368 1 5 | :--------------------+-----------------+-------------------: | 2 *368 4 | 5 7 A368 | 1 9 A368 | | 17 *368 17 | 9 B368 2 | 5 B368 4 | |*569 *3689 *35689 |C368 1 4 |C368 7 2 | :--------------------+-----------------+-------------------: | 3 1 26 | 2678 9 5 | 2468 2468 678 | | 456 7 256 | 2368 3468 368 | 9 23568 1 | | 8 2469 2569 | 2367 346 1 | 236 2356 367 | '--------------------'-----------------'-------------------'

TH 368ABC# => –368r3c7, A-, B-, and C-marked cells are RTs
368b4C \ r36c2 => –368r3c2

Code: Select all: ,------------,----------------,-----------------, | 1 2 38 | 4 5 368 | 7 #368 9 | | 4 5 7 | 1 #368 9 |#2368 #2368 #368 | | 6 9 38 | 38 2 7 | 4 1 5 | :------------+----------------+-----------------: | 2 368 4 | 5 7 368 | 1 9 368 | | 7 368 1 | 9 #368 2 | 5 6–38 4 | | 9 368 5 | 368 1 4 | 368 7 2 | :------------+----------------+-----------------: | 3 1 26 | 2678 9 5 | 268 4 678 | | 5 7 26 | 2368 4 368 | 9 *2368 1 | | 8 4 9 | 2367 *36 1 |*236 5 *367 | '------------'----------------'-----------------'

(#) 8c5b3 \ r25c8 => –8r5c8
(#*) 3c5b39 \ r259c8 => –3r5c8

Code: Select all: ,-----------,-------------,----------------, | 1 2 *38 | 4 5 6 | 7 *38 9 | | 4 5 7 | 1 38 9 | 2368 238 368 | | 6 9 *38 |*38 2 7 | 4 1 5 | :-----------+-------------+----------------: | 2 6 4 | 5 7 38 | 1 9 38 | | 7 38 1 | 9 38 2 | 5 6 4 | | 9 38 5 | 6 1 4 | 38 7 2 | :-----------+-------------+----------------: | 3 1 2 |*78 9 5 | 68 4 678 | | 5 7 6 |*238 4 *38 | 9 2–38 1 | | 8 4 9 |*237 6 1 | 23 5 37 | '-----------'-------------'----------------'

38r13b8 \ r8c348 => –38r8c8, stte

Marek

Website · by **denis_berthier** » Mon Apr 17, 2023 4:00 am

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Here is a solution using tridagon plus three different impossible patterns (all in Select1 or Select2) with very short chains (all lengths ≤ 3)

Code: Select all: hidden-pairs-in-a-row: r5{n1 n7}{c1 c3} ==> r5c3≠8, r5c3≠6, r5c3≠3, r5c1≠6 hidden-pairs-in-a-column: c6{n7 n9}{r2 r3} ==> r3c6≠8, r3c6≠6, r3c6≠3, r2c6≠8, r2c6≠6, r2c6≠3 Trid-OR2-relation for digits 3, 8 and 6 in blocks: b2, with cells (marked #): r1c6, r2c5, r3c4 b3, with cells (marked #): r1c8, r2c9, r3c7 b5, with cells (marked #): r4c6, r5c5, r6c4 b6, with cells (marked #): r4c9, r5c8, r6c7 with 2 guardians (in cells marked @): n2r1c8 n4r3c7 +----------------------+----------------------+----------------------+ ! 16 2368 12368 ! 4 5 368# ! 7 2368#@ 9 ! ! 4679 5 236789 ! 1 368# 79 ! 23468 23468 368# ! ! 4679 34689 36789 ! 368# 2 79 ! 3468#@ 1 5 ! +----------------------+----------------------+----------------------+ ! 2 368 4 ! 5 7 368# ! 1 9 368# ! ! 17 368 17 ! 9 368# 2 ! 5 368# 4 ! ! 569 3689 35689 ! 368# 1 4 ! 368# 7 2 ! +----------------------+----------------------+----------------------+ ! 3 1 26 ! 2678 9 5 ! 2468 2468 678 ! ! 456 7 256 ! 2368 3468 368 ! 9 23568 1 ! ! 8 2469 2569 ! 2367 346 1 ! 236 2356 367 ! +----------------------+----------------------+----------------------+

The three 3-digit impossible patterns that will be used, two in Imp630-Select1 and one in Imp630-Select2:

Code: Select all: EL13c290-OR3-relation for digits: 3, 6 and 8 in cells (marked #): (r6c4 r5c2 r5c8 r5c5 r4c2 r4c9 r4c6 r1c8 r1c6 r2c9 r2c5 r3c2 r3c4) with 3 guardians (in cells marked @) : n2r1c8 n4r3c2 n9r3c2 +-------------------------+-------------------------+-------------------------+ ! 16 2368 12368 ! 4 5 368# ! 7 2368#@ 9 ! ! 4679 5 236789 ! 1 368# 79 ! 23468 23468 368# ! ! 4679 34689#@ 36789 ! 368# 2 79 ! 3468 1 5 ! +-------------------------+-------------------------+-------------------------+ ! 2 368# 4 ! 5 7 368# ! 1 9 368# ! ! 17 368# 17 ! 9 368# 2 ! 5 368# 4 ! ! 569 3689 35689 ! 368# 1 4 ! 368 7 2 ! +-------------------------+-------------------------+-------------------------+ ! 3 1 26 ! 2678 9 5 ! 2468 2468 678 ! ! 456 7 256 ! 2368 3468 368 ! 9 23568 1 ! ! 8 2469 2569 ! 2367 346 1 ! 236 2356 367 ! +-------------------------+-------------------------+-------------------------+ EL14c1s-OR4-relation for digits: 3, 6 and 8 in cells (marked #): (r8c8 r8c6 r8c4 r3c4 r1c8 r1c6 r2c9 r2c5 r5c8 r5c5 r4c9 r4c6 r6c7 r6c4) with 4 guardians (in cells marked @) : n2r8c8 n5r8c8 n2r8c4 n2r1c8 +-------------------------+-------------------------+-------------------------+ ! 16 2368 12368 ! 4 5 368# ! 7 2368#@ 9 ! ! 4679 5 236789 ! 1 368# 79 ! 23468 23468 368# ! ! 4679 34689 36789 ! 368# 2 79 ! 3468 1 5 ! +-------------------------+-------------------------+-------------------------+ ! 2 368 4 ! 5 7 368# ! 1 9 368# ! ! 17 368 17 ! 9 368# 2 ! 5 368# 4 ! ! 569 3689 35689 ! 368# 1 4 ! 368# 7 2 ! +-------------------------+-------------------------+-------------------------+ ! 3 1 26 ! 2678 9 5 ! 2468 2468 678 ! ! 456 7 256 ! 2368#@ 3468 368# ! 9 23568#@ 1 ! ! 8 2469 2569 ! 2367 346 1 ! 236 2356 367 ! +-------------------------+-------------------------+-------------------------+ Note that there is another (and simpler) EL14c1 relation, but it will nor be used: EL14c1-OR3-relation for digits: 3, 6 and 8 in cells (marked #): (r3c4 r2c9 r2c5 r1c2 r1c8 r1c6 r4c9 r4c6 r5c2 r5c8 r5c5 r6c2 r6c7 r6c4) with 3 guardians (in cells marked @) : n2r1c2 n2r1c8 n9r6c2 +----------------------+----------------------+----------------------+ ! 16 2368#@ 12368 ! 4 5 368# ! 7 2368#@ 9 ! ! 4679 5 236789 ! 1 368# 79 ! 23468 23468 368# ! ! 4679 34689 36789 ! 368# 2 79 ! 3468 1 5 ! +----------------------+----------------------+----------------------+ ! 2 368 4 ! 5 7 368# ! 1 9 368# ! ! 17 368# 17 ! 9 368# 2 ! 5 368# 4 ! ! 569 3689#@ 35689 ! 368# 1 4 ! 368# 7 2 ! +----------------------+----------------------+----------------------+ ! 3 1 26 ! 2678 9 5 ! 2468 2468 678 ! ! 456 7 256 ! 2368 3468 368 ! 9 23568 1 ! ! 8 2469 2569 ! 2367 346 1 ! 236 2356 367 ! +----------------------+----------------------+----------------------+ EL13c176s-OR6-relation for digits: 3, 6 and 8 in cells (marked #): (r8c5 r8c6 r8c8 r3c7 r1c6 r2c5 r2c9 r2c8 r6c7 r4c6 r4c9 r5c5 r5c8) with 6 guardians (in cells marked @) : n4r8c5 n2r8c8 n5r8c8 n4r3c7 n2r2c8 n4r2c8 +-------------------------+-------------------------+-------------------------+ ! 16 2368 12368 ! 4 5 368# ! 7 2368 9 ! ! 4679 5 236789 ! 1 368# 79 ! 23468 23468#@ 368# ! ! 4679 34689 36789 ! 368 2 79 ! 3468#@ 1 5 ! +-------------------------+-------------------------+-------------------------+ ! 2 368 4 ! 5 7 368# ! 1 9 368# ! ! 17 368 17 ! 9 368# 2 ! 5 368# 4 ! ! 569 3689 35689 ! 368 1 4 ! 368# 7 2 ! +-------------------------+-------------------------+-------------------------+ ! 3 1 26 ! 2678 9 5 ! 2468 2468 678 ! ! 456 7 256 ! 2368 3468#@ 368# ! 9 23568#@ 1 ! ! 8 2469 2569 ! 2367 346 1 ! 236 2356 367 ! +-------------------------+-------------------------+-------------------------+

We first have a few Trid-OR2-whips[2 or 3] using the same Trid-OR3-relation at different places in the chain:
Trid-OR2-whip[2]: OR2{{n2r1c8 | n4r3c7}} - r7n4{c7 .} ==> r7c8≠2
Trid-OR2-whip[3]: OR2{{n2r1c8 | n4r3c7}} - c2n4{r3 r9} - c2n2{r9 .} ==> r1c3≠2
Trid-OR2-whip[3]: OR2{{n4r3c7 | n2r1c8}} - b9n2{r9c8 r9c7} - c2n2{r9 .} ==> r7c7≠4
hidden-single-in-a-block ==> r7c8=4
Trid-OR2-whip[3]: OR2{{n4r3c7 | n2r1c8}} - c2n2{r1 r9} - c2n4{r9 .} ==> r3c1≠4
Trid-OR2-whip[3]: c2n4{r9 r3} - OR2{{n4r3c7 | n2r1c8}} - c2n2{r1 .} ==> r9c2≠6, r9c2≠9
singles ==> r9c3=9, r9c8=5

Code: Select all: +-------------------+-------------------+-------------------+ ! 16 2368 1368 ! 4 5 368 ! 7 2368 9 ! ! 4679 5 23678 ! 1 368 79 ! 23468 2368 368 ! ! 679 34689 3678 ! 368 2 79 ! 3468 1 5 ! +-------------------+-------------------+-------------------+ ! 2 368 4 ! 5 7 368 ! 1 9 368 ! ! 17 368 17 ! 9 368 2 ! 5 368 4 ! ! 569 3689 3568 ! 368 1 4 ! 368 7 2 ! +-------------------+-------------------+-------------------+ ! 3 1 26 ! 2678 9 5 ! 268 4 678 ! ! 456 7 256 ! 2368 3468 368 ! 9 2368 1 ! ! 8 24 9 ! 2367 346 1 ! 236 5 367 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL13c176s-OR6-relation between candidates n4r8c5 n2r8c8 n5r8c8 n4r3c7 n2r2c8 n4r2c8 has been eliminated. There remains an EL13c176s-OR4-relation between candidates: n4r8c5 n2r8c8 n4r3c7 n2r2c8 At least one candidate of a previous EL14c1s-OR4-relation between candidates n2r8c8 n5r8c8 n2r8c4 n2r1c8 has just been eliminated. There remains an EL14c1s-OR3-relation between candidates: n2r8c8 n2r8c4 n2r1c8

EL14c1s-OR3-whip[2]: OR3{{n2r8c4 n2r8c8 | n2r1c8}} - c2n2{r1 .} ==> r8c3≠2
biv-chain[3]: r8n2{c4 c8} - r1n2{c8 c2} - b7n2{r9c2 r7c3} ==> r7c4≠2
z-chain[3]: c4n2{r8 r9} - b7n2{r9c2 r7c3} - b7n6{r7c3 .} ==> r8c4≠6
EL13c290-OR3-whip[3]: OR3{{n9r3c2 n4r3c2 | n2r1c8}} - c2n2{r1 r9} - c2n4{r9 .} ==> r3c2≠3, r3c2≠6
z-chain[3]: c2n6{r6 r1} - b1n2{r1c2 r2c3} - r7c3{n2 .} ==> r6c3≠6
EL13c290-OR3-whip[3]: OR3{{n9r3c2 n4r3c2 | n2r1c8}} - c2n2{r1 r9} - c2n4{r9 .} ==> r3c2≠8

Code: Select all: biv-chain[4]: c1n4{r8 r2} - r3c2{n4 n9} - b4n9{r6c2 r6c1} - c1n5{r6 r8} ==> r8c1≠6 whip[1]: b7n6{r8c3 .} ==> r1c3≠6, r2c3≠6, r3c3≠6 biv-chain[4]: b4n9{r6c1 r6c2} - r3c2{n9 n4} - b7n4{r9c2 r8c1} - c1n5{r8 r6} ==> r6c1≠6 whip[1]: b4n6{r6c2 .} ==> r1c2≠6 +-------------------+-------------------+-------------------+ ! 16 238 138 ! 4 5 368 ! 7 2368 9 ! ! 4679 5 2378 ! 1 368 79 ! 23468 2368 368 ! ! 679 49 378 ! 368 2 79 ! 3468 1 5 ! +-------------------+-------------------+-------------------+ ! 2 368 4 ! 5 7 368 ! 1 9 368 ! ! 17 368 17 ! 9 368 2 ! 5 368 4 ! ! 59 3689 358 ! 368 1 4 ! 368 7 2 ! +-------------------+-------------------+-------------------+ ! 3 1 26 ! 678 9 5 ! 268 4 678 ! ! 45 7 56 ! 238 3468 368 ! 9 2368 1 ! ! 8 24 9 ! 2367 346 1 ! 236 5 367 ! +-------------------+-------------------+-------------------+ EL13c176s-OR4-relation between candidates n4r8c5, n2r8c8, n4r3c7 and n2r2c8 + same valence for candidates n4r8c5 and n4r3c7 via c-chain[4]: n4r8c5,n4r9c5,n4r9c2,n4r3c2,n4r3c7 ==> EL13c176s-OR4-relation can be split into two EL13c176s-OR3-relations with respective lists of guardians: n2r8c8 n4r3c7 n2r2c8 and n4r8c5 n2r8c8 n2r2c8 .

EL13c176s-OR3-whip[3]: OR3{{n2r2c8 n2r8c8 | n4r8c5}} - b7n4{r8c1 r9c2} - c2n2{r9 .} ==> r1c8≠2

The end is easy, in S3+BC3:

Code: Select all: singles ==> r1c2=2, r9c2=4, r3c2=9, r3c6=7, r2c6=9, r3c1=6, r1c1=1, r5c1=7, r2c1=4, r5c3=1, r8c1=5, r6c1=9, r8c3=6, r7c3=2, r6c3=5, r3c7=4, r2c3=7, r8c5=4 t-whip[2]: c5n8{r5 r2} - b3n8{r2c9 .} ==> r5c8≠8 biv-chain[3]: r8c6{n8 n3} - r9c5{n3 n6} - b2n6{r2c5 r1c6} ==> r1c6≠8 finned-swordfish-in-rows: n8{r1 r3 r8}{c8 c3 c4} ==> r7c4≠8 whip[1]: r7n8{c9 .} ==> r8c8≠8 whip[1]: c8n8{r2 .} ==> r2c7≠8, r2c9≠8 biv-chain[3]: r8c8{n3 n2} - r9n2{c7 c4} - r9n7{c4 c9} ==> r9c9≠3 hidden-pairs-in-a-block: b9{n2 n3}{r8c8 r9c7} ==> r9c7≠6 biv-chain[3]: r5c8{n6 n3} - b9n3{r8c8 r9c7} - r9c5{n3 n6} ==> r5c5≠6 biv-chain[3]: r5c8{n6 n3} - r5c5{n3 n8} - r2n8{c5 c8} ==> r2c8≠6 finned-x-wing-in-columns: n6{c6 c8}{r1 r4} ==> r4c9≠6 swordfish-in-rows: n6{r1 r4 r5}{c8 c6 c2} ==> r6c2≠6 biv-chain[3]: r4n6{c2 c6} - r6n6{c4 c7} - b6n8{r6c7 r4c9} ==> r4c2≠8 biv-chain[3]: r5c5{n3 n8} - r4n8{c6 c9} - c9n3{r4 r2} ==> r2c5≠3 whip[1]: r2n3{c9 .} ==> r1c8≠3 biv-chain[2]: b9n3{r8c8 r9c7} - c5n3{r9 r5} ==> r5c8≠3 stte

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13252 in 158,276 T&E(3) min-expands

13252 in 158,276 T&E(3) min-expands

Re: 13252 in 158,276 T&E(3) min-expands

Re: 13252 in 158,276 T&E(3) min-expands