The New Sudoku Players' Forum

Website · by **denis_berthier** » Thu Jan 05, 2023 12:41 pm

.
In spite of a low SER (in the collection), this is a really hard one (but still solvable without uniqueness or replacement).

Code: Select all: +-------+-------+-------+ ! 1 . 3 ! . . . ! . . . ! ! . 5 . ! . 8 9 ! . . . ! ! 7 9 8 ! . . . ! 5 . 4 ! +-------+-------+-------+ ! . . . ! 6 4 . ! . . . ! ! . 8 . ! . 9 5 ! 2 . 6 ! ! . . . ! . . 2 ! 4 . . ! +-------+-------+-------+ ! 6 . . ! 9 . . ! . 4 2 ! ! . . 2 ! . 6 4 ! 9 . 5 ! ! . . . ! . . . ! 6 . . ! +-------+-------+-------+ 1.3.......5..89...798...5.4...64.....8..952.6.....24..6..9...42..2.649.5......6..;10557;289152 SER = 10.4

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1 246 3 ! 2457 257 67 ! 78 26789 789 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +----------------------+----------------------+----------------------+ ! 2359 1237 1579 ! 6 4 1378 ! 1378 135789 13789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 1367 15679 ! 1378 137 2 ! 4 135789 13789 ! +----------------------+----------------------+----------------------+ ! 6 137 157 ! 9 1357 1378 ! 1378 4 2 ! ! 38 137 2 ! 1378 6 4 ! 9 1378 5 ! ! 34589 1347 14579 ! 123578 12357 1378 ! 6 1378 1378 ! +----------------------+----------------------+----------------------+ 195 candidates.

by **Cenoman** » Fri Jan 06, 2023 5:11 pm

Code: Select all: +-------------------------+--------------------------+--------------------------+ | 1 246 3 | 2457 257 67 | 78 26789 789 | | 24 5 46 | 12347 8 9 | 137 12367 137 | | 7 9 8 | 123 123 136 | 5 1236 4 | +-------------------------+--------------------------+--------------------------+ | 2359 1237 1579 | 6 4 1378* | 1378* 135789 13789 | | 34 8 147 | 137* 9 5 | 2 137* 6 | | 359 1367 15679 | 1378 137* 2 | 4 135789 13789* | +-------------------------+--------------------------+--------------------------+ | 6 137 157 | 9 1357* 1378 | 1378* 4 2 | | 38 137 2 | 1378* 6 4 | 9 1378* 5 | | 34589 1347 14579 | 123578 12357 1378* | 6 1378 1378* | +-------------------------+--------------------------+--------------------------+

TH(137)b5689. At the starting resolution state, the TH has ten guardians (8r4c67, 89r6c9, 5r7c5, 8r7c7, 8r8c48, 8r9c69!)
First steps eliminating four of these (5r7c5, 8r4c6, 8r6c9, 8r8c4):
1. (5=1374)b7p2358 - r1c2 = (4-5)r1c4 = (5)r1c5 => -5 r7c5; lcls, 1 placement

2. Kraken column (4)r259c3
(4-6)r2c3 = r1c2 - (6=78)r1c67 - r7c7 = (8)r7c6
(4)r5c3 - (4=3)r5c1 - (3=8)r8c1
(4-9)r9c3 = (9-8)r9c1 = (8)r8c1
=> -8 r8c4; 1 placement (+8 r6c4)

Code: Select all: +-----------------------+-----------------------+--------------------------+ | 1 246 3 | 24-7+5 25-7 67 | 78 26789 789 | | 24 5 46 | 12347 8 9 | 137 12367 137 | | 7 9 8 | 123 123 136 | 5 1236 4 | +-----------------------+-----------------------+--------------------------+ | 2359 1237 179 | 6 4 137 | 1378 135789 13789 | | 34 8 147 | 137 9 5 | 2 137 6 | | 359 1367 1679 | 8 137 2 | 4 13579 1379 | +-----------------------+-----------------------+--------------------------+ | 6 137 5 | 9 137 1378 | 1378 4 2 | | 38 137 2 | 137 6 4 | 9 1378 5 | | 3489 1347 1479 | 25 25 1378 | 6 1378 1378 | +-----------------------+-----------------------+--------------------------+

3. TH(137)b5689 has now six guardians, two of which have the same parity (8r7c7 and 8r9c6, both conjugates of 8r7c6)
(8)r8c8 - r8c1 = (89-4)r9c13 = (45)r1c45
(8)r47c7 - (8=7)r1c7
(9)r6c9 - (9=87)r1c79
(8)r9c9 - (8=137)r589c8 - (1|3|7=26)r23c8 - (2|6=897)r1c789
=> -7 r1c45

4. UR(25)r19c45 using single internal => +5 r1c4; 5 placements

Code: Select all: +-----------------------+----------------------+--------------------------+ | 1 26 3 | 4 5 67 | 78 289-67 789 | | 24 5 46 | 1237 8 9 | 137 26-137 137 | | 7 9 8 | 123 13 136 | 5 1236 4 | +-----------------------+----------------------+--------------------------+ | 259-3 127-3 179 | 6 4 137 | 1378 59-1378 13789 | | 34 8 147 | 137 9 5 | 2 137 6 | | 59-3 1367 1679 | 8 137 2 | 4 13579 1379 | +-----------------------+----------------------+--------------------------+ | 6 137 5 | 9 137 1378 | 1378 4 2 | | 38 137 2 | 137 6 4 | 9 1378 5 | | 89-3 4 179 | 5 2 1378 | 6 1378 1378 | +-----------------------+----------------------+--------------------------+

5. (2)r4c2 = r4c1 - (2=43)r25c1 => -3 r4c2
6. (5)r4c8 = (5-2)r4c1 = r4c2 - r1c2 = (259)r146c8 => -1378 r4c8

7. 8. 9. Krakens-TH
(8)r8c8 - (8=342)r258c1 - r1c2 = (2)r1c8
(8)r47c7 - (8=67)r1c67
(9)r6c9 - r1c9 = (9)r1c8
(8)r9c9 - (8=137)r589c8 - (1|3|7=26)r23c8
=> -67 r1c8

(8)r8c8 - (8=342)r258c1 - (2=137)r2c479
(8)r47c7 - (8=137)b3p146
(9)r6c9 - (9=8137)r12c79
(8)r9c9 - (8=137)r589c8
=> -137 r2c8; lcls, 1 placement

(8)r8c8 - (8=3)r8c1
(8)47c7 - (8=672)r1c267 - (2=43)r25c1
(9)r6c9 - (9=6782)r1c2679 - (2=43)r25c1
(8)r9c9 - r89c8 = (859-2)r146c8 = r2c1 - (2=43)r25c1
=> -3 r469c1

Code: Select all: +---------------------+---------------------+-------------------------+ | 1 26 3 | 4 5 67 | 78 289 789 | | 24 5 46 | 137 8 9 | 137 26 137 | | 7 9 8 | 2 13 136 | 5 16-3 4 | +---------------------+---------------------+-------------------------+ | 259 127 179 | 6 4 137 | 1378 59 13789 | | 34 8 147 | 137 9 5 | 2 17-3 6 | | 59 36 1679 | 8 137 2 | 4 13579 1379 | +---------------------+---------------------+-------------------------+ | 6 137 5 | 9 137 178-3 | 1378 4 2 | | 38 17-3 2 | 137 6 4 | 9 178-3 5 | | 89 4 179 | 5 2 1378 | 6 1378 1378 | +---------------------+---------------------+-------------------------+

10. [(3)r2c79 = r2c4 - r5c4 = r5c8] = (3-4)r5c1 = (4-2)r2c1 = (2-6)r2c8 = (6)r3c8 => -3 r3c8; lcls (-3 r2c4)

11. (3)r8c4 = r5c4 - r5c1 = (3-8)r8c1 = r8c8 - r7c7 = (8)r7c6 => -3 r7c6
12. X-Wing (3)c14\r58 => -3 r5c8, r8c28

Code: Select all: +----------------------+---------------------+-------------------------+ | 1 26 3 | 4 5 67 | 78 289 789 | | 24 5 46 | 17 8 9 | 137 26 137 | | 7 9 8 | 2 13 136 | 5 16 4 | +----------------------+---------------------+-------------------------+ | 259 127 179 | 6 4 137 | 1378 59 13789 | | 34 8 147 | 137 9 5 | 2 17 6 | | 59 1367 1679 | 8 137 2 | 4 13579 1379 | +----------------------+---------------------+-------------------------+ | 6 137 5 | 9 17-3 178 | 178-3 4 2 | | 38 17 2 | 137 6 4 | 9 178 5 | | 89 4 179 | 5 2 1378 | 6 1378 1378 | +----------------------+---------------------+-------------------------+

13. (3)r7c2 = r6c2 - r6c8 = r9c8 => -3 r7c7
14. (3)r7c2 = (3-6)r6c2 = r1c2 - r1c6 = (6-3)r3c6 = (3)r3c5 => -3 r7c5; ste

by **totuan** » Sat Jan 07, 2023 11:41 am

Code: Select all: *-----------------------------------------------------------------------------* | 1 246 3 | 457-2 57-2 67 | 78 26789 789 | | 24 5 46 | 12347 8 9 | 137 12367 137 | | 7 9 8 | 123 123 136 | 5 1236 4 | |-------------------------+-------------------------+-------------------------| | 2359 1237 1579 | 6 4 1378 | 1378 135789 13789 | | 34 8 147 | 137 9 5 | 2 137 6 | | 359 1367 15679 | 1378 137 2 | 4 135789 13789 | |-------------------------+-------------------------+-------------------------| | 6 137 157 | 9 137-5 1378 | 1378 4 2 | | 38 137 2 | 1378 6 4 | 9 1378 5 | | 34589 1347 14579 | 123578 12357 1378 | 6 1378 1378 | *-----------------------------------------------------------------------------*

My path for this one:
DP (137)r789c23 => (5)r7c3=(4)r9c2
01: (5)r7c3==(4)r9c2-r1c2=(4-5)r1c4=r1c5 => r7c5<>5, r7c3=5
02: UR (25)r19c45 => r1c45<>2

Code: Select all: *-----------------------------------------------------------------------------* | 1 d246 3 | 457 57 67 | 78 c26789 789 | |e24 5 46 | 12347 8 9 | 137 12367 137 | | 7 9 8 | 123 123 136 | 5 1236 4 | |-------------------------+-------------------------+-------------------------| |*2359 1237 179 | 6 4 1378 | 1378 *135789 13789 | |f34 8 147 | 137 9 5 | 2 137 6 | |*359 1367 1679 | 1378 137 2 | 4 *135789 13789 | |-------------------------+-------------------------+-------------------------| | 6 137 5 | 9 137 1378 | 1378 4 2 | |a38 137 2 | 137-8 6 4 | 9 137-8 5 | |b3489 1347 1479 | 25 25 1378 | 6 1378 1378 | *-----------------------------------------------------------------------------*

UR (59)r46c18 => (9)r9c1=(9)r1c8
03: (8)r8c1=(8-9)r9c1==(9-2)r1c8=r1c2-(2=4)r2c1-(34=8)r58c1 => r8c48<>8, some singles

Code: Select all: *-----------------------------------------------------------------------------* | 1 246 3 | 457 57 67 | 78 789-26 789 | | 24 5 46 | 12347 8 9 | 137 12367 137 | | 7 9 8 | 123 123 136 | 5 1236 4 | |-------------------------+-------------------------+-------------------------| | 2359 1237 179 | 6 4 *137 |*1378 135789 13789 | | 34 8 147 |*137 9 5 | 2 *137 6 | | 359 1367 1679 | 8 *137 2 | 4 13579 *1379 | |-------------------------+-------------------------+-------------------------| | 6 137 5 | 9 *137 1378 |*1378 4 2 | | 8 137 2 |*137 6 4 | 9 *137 5 | | 349 1347 1479 | 25 25 *1378 | 6 1378 *1378 | *-----------------------------------------------------------------------------*

Tridagon (137) * marked cells => (8)r47c7=(8)r9c1=(8)r9c9=(9)r9c6
04: Present as diagram: => r1c8<>26, stte

Code: Select all: (9)r6c9-r1c9=r1c8* || (8)r47c7-----------(8=7)r1c7-(457=26)r1c2456* || | (8)r9c6-r7c6=r7c7- || (8)r9c9-(8=137)r589c8-(137=26)r23c8*

Thanks for the puzzle!
totuan

Website · by **denis_berthier** » Sun Jan 08, 2023 10:38 am

.
First solution, in gW13+OR4gW7:

Code: Select all: OR10-anti-tridagon[12] for digits 1, 3 and 7 in blocks: b5, with cells: r4c6, r5c4, r6c5 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r9c6, r8c4, r7c5 b9, with cells: r9c9, r8c8, r7c7 with 10 guardians: n8r4c6 n8r4c7 n8r6c9 n9r6c9 n5r7c5 n8r7c7 n8r8c4 n8r8c8 n8r9c6 n8r9c9 biv-chain[3]: r5c1{n3 n4} - r2c1{n4 n2} - b4n2{r4c1 r4c2} ==> r4c2≠3 z-chain[4]: r9n9{c1 c3} - r9n4{c3 c2} - r1n4{c2 c4} - c4n5{r1 .} ==> r9c1≠5 whip[1]: b7n5{r9c3 .} ==> r4c3≠5, r6c3≠5 z-chain[4]: r9n9{c3 c1} - r9n4{c1 c2} - r1n4{c2 c4} - c4n5{r1 .} ==> r9c3≠5 hidden-single-in-a-block ==> r7c3=5 At least one candidate of a previous Trid-OR10-relation has just been eliminated. There remains a Trid-OR9-relation between candidates: n8r4c6 n8r4c7 n8r6c9 n9r6c9 n8r7c7 n8r8c4 n8r8c8 n8r9c6 n8r9c9 +----------------------+----------------------+----------------------+ ! 1 246 3 ! 2457 257 67 ! 78 26789 789 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +----------------------+----------------------+----------------------+ ! 2359 127 179 ! 6 4 1378 ! 1378 135789 13789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 1367 1679 ! 1378 137 2 ! 4 135789 13789 ! +----------------------+----------------------+----------------------+ ! 6 137 157 ! 9 137 1378 ! 1378 4 2 ! ! 38 137 2 ! 1378 6 4 ! 9 1378 5 ! ! 3489 1347 1479 ! 123578 12357 1378 ! 6 1378 1378 ! +----------------------+----------------------+----------------------+ hidden-pairs-in-a-block: b8{n2 n5}{r9c4 r9c5} ==> r9c5≠7, r9c5≠3, r9c5≠1, r9c4≠8, r9c4≠7, r9c4≠3, r9c4≠1 t-whip[3]: c4n8{r6 r8} - r8c1{n8 n3} - c2n3{r9 .} ==> r6c4≠3 t-whip[4]: c4n8{r6 r8} - r7n8{c6 c7} - r1c7{n8 n7} - r2n7{c9 .} ==> r6c4≠7 t-whip[6]: r2n7{c9 c4} - r1c6{n7 n6} - b1n6{r1c2 r2c3} - r2n4{c3 c1} - r5n4{c1 c3} - r5n7{c3 .} ==> r1c8≠7 t-whip[6]: r8c1{n8 n3} - c2n3{r9 r6} - c2n6{r6 r1} - r1c6{n6 n7} - r1c7{n7 n8} - r7n8{c7 .} ==> r8c4≠8 At least one candidate of a previous Trid-OR9-relation has just been eliminated. There remains a Trid-OR8-relation between candidates: n8r4c6 n8r4c7 n8r6c9 n9r6c9 n8r7c7 n8r8c8 n8r9c6 n8r9c9 +----------------------+----------------------+----------------------+ ! 1 246 3 ! 2457 257 67 ! 78 2689 789 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +----------------------+----------------------+----------------------+ ! 2359 127 179 ! 6 4 1378 ! 1378 135789 13789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 1367 1679 ! 18 137 2 ! 4 135789 13789 ! +----------------------+----------------------+----------------------+ ! 6 137 5 ! 9 137 1378 ! 1378 4 2 ! ! 38 137 2 ! 137 6 4 ! 9 1378 5 ! ! 3489 1347 1479 ! 25 25 1378 ! 6 1378 1378 ! +----------------------+----------------------+----------------------+ hidden-single-in-a-column ==> r6c4=8 At least one candidate of a previous Trid-OR8-relation has just been eliminated. There remains a Trid-OR7-relation between candidates: n8r4c6 n8r4c7 n9r6c9 n8r7c7 n8r8c8 n8r9c6 n8r9c9 +----------------------+----------------------+----------------------+ ! 1 246 3 ! 2457 257 67 ! 78 2689 789 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +----------------------+----------------------+----------------------+ ! 2359 127 179 ! 6 4 1378 ! 1378 135789 13789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 1367 1679 ! 18 137 2 ! 4 135789 1379 ! +----------------------+----------------------+----------------------+ ! 6 137 5 ! 9 137 1378 ! 1378 4 2 ! ! 38 137 2 ! 137 6 4 ! 9 1378 5 ! ! 3489 1347 1479 ! 25 25 1378 ! 6 1378 1378 ! +----------------------+----------------------+----------------------+ At least one candidate of a previous Trid-OR7-relation has just been eliminated. There remains a Trid-OR6-relation between candidates: n8r4c7 n9r6c9 n8r7c7 n8r8c8 n8r9c6 n8r9c9 Resolution state RS1: +----------------------+----------------------+----------------------+ ! 1 246 3 ! 2457 257 67 ! 78 2689 789 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +----------------------+----------------------+----------------------+ ! 2359 127 179 ! 6 4 137 ! 1378 135789 13789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 1367 1679 ! 8 137 2 ! 4 13579 1379 ! +----------------------+----------------------+----------------------+ ! 6 137 5 ! 9 137 1378 ! 1378 4 2 ! ! 38 137 2 ! 137 6 4 ! 9 1378 5 ! ! 3489 1347 1479 ! 25 25 1378 ! 6 1378 1378 ! +----------------------+----------------------+----------------------+

Code: Select all: t-whip[9]: r7n8{c6 c7} - r1c7{n8 n7} - r2n7{c9 c4} - r1c6{n7 n6} - b1n6{r1c2 r2c3} - r2n4{c3 c1} - r5c1{n4 n3} - r5c4{n3 n1} - r8c4{n1 .} ==> r7c6≠3 g-whip[12]: c2n3{r9 r6} - c2n6{r6 r1} - r1c6{n6 n7} - r1c7{n7 n8} - r7n8{c7 c6} - r9c6{n8 n1} - r4c6{n1 n3} - c9n3{r4 r2} - c9n1{r2 r456} - r4c7{n1 n7} - r2n7{c7 c8} - r2n6{c8 .} ==> r9c1≠3 whip[13]: r6n6{c2 c3} - r2c3{n6 n4} - r5c3{n4 n7} - r4c2{n7 n2} - r1c2{n2 n6} - r1c6{n6 n7} - b5n7{r4c6 r6c5} - c4n7{r5 r8} - r8n1{c4 c8} - r5c8{n1 n3} - r6c9{n3 n9} - r1c9{n9 n8} - r1c7{n8 .} ==> r6c2≠1 whip[13]: r7n8{c6 c7} - r1c7{n8 n7} - r2n7{c9 c4} - r1c6{n7 n6} - b1n6{r1c2 r2c3} - r2n4{c3 c1} - c1n2{r2 r4} - r4n5{c1 c8} - r4n8{c8 c9} - r4n9{c9 c3} - r4n7{c3 c2} - r8n7{c2 c8} - r5n7{c8 .} ==> r7c6≠7 whip[13]: r1c7{n8 n7} - r2n7{c9 c4} - r1c6{n7 n6} - b1n6{r1c2 r2c3} - r2n4{c3 c1} - r5c1{n4 n3} - r5c4{n3 n1} - r5c8{n1 n7} - r8n7{c8 c2} - r8n1{c2 c8} - r7c7{n1 n3} - c2n3{r7 r9} - c2n4{r9 .} ==> r4c7≠8 At least one candidate of a previous Trid-OR6-relation has just been eliminated. There remains a Trid-OR5-relation between candidates: n9r6c9 n8r7c7 n8r8c8 n8r9c6 n8r9c9 +----------------------+----------------------+----------------------+ ! 1 246 3 ! 2457 257 67 ! 78 2689 789 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +----------------------+----------------------+----------------------+ ! 2359 127 179 ! 6 4 137 ! 137 135789 13789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 367 1679 ! 8 137 2 ! 4 13579 1379 ! +----------------------+----------------------+----------------------+ ! 6 137 5 ! 9 137 18 ! 1378 4 2 ! ! 38 137 2 ! 137 6 4 ! 9 1378 5 ! ! 489 1347 1479 ! 25 25 1378 ! 6 1378 1378 ! +----------------------+----------------------+----------------------+ whip[6]: c8n5{r6 r4} - r4n8{c8 c9} - b6n9{r4c9 r6c9} - r1c9{n9 n7} - b9n7{r9c9 r7c7} - c5n7{r7 .} ==> r6c8≠7 t-whip[8]: r1c7{n8 n7} - r2n7{c9 c4} - r1c6{n7 n6} - b1n6{r1c2 r2c3} - r2n4{c3 c1} - c1n2{r2 r4} - r4n5{c1 c8} - r4n8{c8 .} ==> r1c9≠8 t-whip[4]: r4n8{c9 c8} - c8n5{r4 r6} - c8n9{r6 r1} - r1c9{n9 .} ==> r4c9≠7 t-whip[8]: c8n9{r6 r1} - r1c9{n9 n7} - r1c6{n7 n6} - r3n6{c6 c8} - c8n2{r3 r2} - c1n2{r2 r4} - r4n5{c1 c8} - r4n8{c8 .} ==> r4c9≠9 whip[8]: c8n5{r4 r6} - c8n9{r6 r1} - r1c9{n9 n7} - r1c6{n7 n6} - r3n6{c6 c8} - c8n2{r3 r2} - c1n2{r2 r4} - r4n5{c1 .} ==> r4c8≠8 hidden-single-in-a-block ==> r4c9=8 At least one candidate of a previous Trid-OR5-relation has just been eliminated. There remains a Trid-OR4-relation between candidates: n9r6c9 n8r7c7 n8r8c8 n8r9c6 +-------------------+-------------------+-------------------+ ! 1 246 3 ! 2457 257 67 ! 78 2689 79 ! ! 24 5 46 ! 12347 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 1236 4 ! +-------------------+-------------------+-------------------+ ! 2359 127 179 ! 6 4 137 ! 137 13579 138 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 367 1679 ! 8 137 2 ! 4 1359 1379 ! +-------------------+-------------------+-------------------+ ! 6 137 5 ! 9 137 18 ! 1378 4 2 ! ! 38 137 2 ! 137 6 4 ! 9 1378 5 ! ! 489 1347 1479 ! 25 25 1378 ! 6 1378 137 ! +-------------------+-------------------+-------------------+ Trid-OR4-gwhip[6]: c2n6{r1 r6} - c2n3{r6 r789} - r8c1{n3 n8} - c8n8{r8 r9} - OR4{{n8r9c6 n8r7c7 n8r8c8 | n9r6c9}} - r1n9{c9 .} ==> r1c8≠6 z-chain[4]: b7n7{r9c2 r9c3} - c6n7{r9 r1} - r1n6{c6 c2} - c2n2{r1 .} ==> r4c2≠7 z-chain[4]: r3n2{c5 c8} - r3n6{c8 c6} - r1n6{c6 c2} - b1n2{r1c2 .} ==> r2c4≠2 z-chain[4]: r3n2{c5 c8} - r3n6{c8 c6} - r1n6{c6 c2} - r1n4{c2 .} ==> r1c4≠2 biv-chain[5]: c4n2{r3 r9} - c4n5{r9 r1} - b2n4{r1c4 r2c4} - r2c3{n4 n6} - b3n6{r2c8 r3c8} ==> r3c8≠2 whip[1]: r3n2{c5 .} ==> r1c5≠2 z-chain[4]: c8n5{r4 r6} - c8n9{r6 r1} - r1n2{c8 c2} - r4c2{n2 .} ==> r4c8≠1 z-chain[5]: c8n5{r4 r6} - c8n9{r6 r1} - r1n2{c8 c2} - r4n2{c2 c1} - r4n5{c1 .} ==> r4c8≠3, r4c8≠7 z-chain[5]: c8n6{r3 r2} - r2c3{n6 n4} - r5n4{c3 c1} - r5n3{c1 c4} - r2n3{c4 .} ==> r3c8≠3 whip[1]: r3n3{c6 .} ==> r2c4≠3 whip[6]: c8n5{r6 r4} - c8n9{r4 r1} - r1n2{c8 c2} - r4c2{n2 n1} - r8n1{c2 c4} - r5n1{c4 .} ==> r6c8≠1 Trid-OR4-whip[7]: r2c1{n4 n2} - r1n2{c2 c8} - r1n8{c8 c7} - r7n8{c7 c6} - r9n8{c6 c8} - OR4{{n8r9c6 n8r7c7 n8r8c8 | n9r6c9}} - r1n9{c9 .} ==> r9c1≠4 whip[8]: c9n9{r1 r6} - c8n9{r4 r1} - r1n2{c8 c2} - r4c2{n2 n1} - r6n1{c3 c5} - c5n7{r6 r7} - c6n7{r9 r4} - c7n7{r4 .} ==> r1c9≠7 naked-single ==> r1c9=9 At least one candidate of a previous Trid-OR4-relation has just been eliminated. There remains a Trid-OR3-relation between candidates: n8r7c7 n8r8c8 n8r9c6 +-------------------+-------------------+-------------------+ ! 1 246 3 ! 457 57 67 ! 78 289 9 ! ! 24 5 46 ! 147 8 9 ! 137 12367 137 ! ! 7 9 8 ! 123 123 136 ! 5 16 4 ! +-------------------+-------------------+-------------------+ ! 2359 12 179 ! 6 4 137 ! 137 59 8 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 367 1679 ! 8 137 2 ! 4 359 137 ! +-------------------+-------------------+-------------------+ ! 6 137 5 ! 9 137 18 ! 1378 4 2 ! ! 38 137 2 ! 137 6 4 ! 9 1378 5 ! ! 89 1347 1479 ! 25 25 1378 ! 6 1378 137 ! +-------------------+-------------------+-------------------+ Trid-OR3-whip[1]: OR3{{n8r9c6 n8r8c8 n8r7c7 | .}} ==> r9c8≠8 hidden-pairs-in-a-column: c8{n5 n9}{r4 r6} ==> r6c8≠3 biv-chain[4]: r4n2{c1 c2} - r1n2{c2 c8} - c8n8{r1 r8} - r8c1{n8 n3} ==> r4c1≠3 biv-chain[4]: c2n4{r9 r1} - r1n2{c2 c8} - c8n8{r1 r8} - r8c1{n8 n3} ==> r9c2≠3 finned-x-wing-in-rows: n3{r4 r9}{c6 c7} ==> r7c7≠3 biv-chain[3]: r7n3{c5 c2} - r8c1{n3 n8} - r9n8{c1 c6} ==> r9c6≠3 whip[1]: r9n3{c9 .} ==> r8c8≠3 biv-chain[4]: b8n3{r7c5 r8c4} - r8c1{n3 n8} - r9n8{c1 c6} - r7c6{n8 n1} ==> r7c5≠1 biv-chain[5]: b4n5{r6c1 r4c1} - r4n2{c1 c2} - r1n2{c2 c8} - c8n8{r1 r8} - r8c1{n8 n3} ==> r6c1≠3 naked-pairs-in-a-row: r6{c1 c8}{n5 n9} ==> r6c3≠9 biv-chain[3]: c1n4{r2 r5} - b4n3{r5c1 r6c2} - c2n6{r6 r1} ==> r1c2≠4 singles ==> r9c2=4, r1c4=4, r1c5=5, r9c5=2, r9c4=5, r3c4=2 x-wing-in-columns: n3{c1 c4}{r5 r8} ==> r8c2≠3, r5c8≠3 naked-triplets-in-a-row: r2{c4 c7 c9}{n7 n1 n3} ==> r2c8≠7, r2c8≠3, r2c8≠1 hidden-single-in-a-column ==> r9c8=3 finned-swordfish-in-columns: n7{c2 c5 c8}{r8 r7 r6} ==> r6c9≠7 biv-chain[2]: c9n7{r9 r2} - b2n7{r2c4 r1c6} ==> r9c6≠7 naked-pairs-in-a-block: b8{r7c6 r9c6}{n1 n8} ==> r8c4≠1 whip[1]: b8n1{r9c6 .} ==> r3c6≠1, r4c6≠1 biv-chain[2]: c4n1{r5 r2} - r3n1{c5 c8} ==> r5c8≠1 naked-single ==> r5c8=7 biv-chain[2]: c5n7{r6 r7} - r8n7{c4 c2} ==> r6c2≠7 whip[1]: c2n7{r8 .} ==> r9c3≠7 hidden-single-in-a-row ==> r9c9=7 naked-pairs-in-a-row: r7{c6 c7}{n1 n8} ==> r7c2≠1 finned-x-wing-in-columns: n1{c9 c5}{r6 r2} ==> r2c4≠1 stte

Website · by **denis_berthier** » Sun Jan 08, 2023 10:43 am

.
Second solution, in gW12+OR5gW12, using my newly coded ORk-splitting rules (see http://forum.enjoysudoku.com/ork-forcing-whips-ork-contrad-whips-and-ork-whips-t40189-8.html)

Same start as before, until resolution state RS1. Then the splitting rule (syntax not fixed):

Code: Select all: Trid-OR6-relation between candidates n8r4c7, n9r6c9, n8r7c7, n8r8c8, n8r9c6 and n8r9c9 + same valence for candidates n8r9c6 and n8r7c7, via c-chain[2]: n8r9c6,n8r7c6,n8r7c7 ==> Trid-OR6-relation can be split into two Trid-OR5-relations with respective lists of guardians: n8r4c7 n9r6c9 n8r7c7 n8r8c8 n8r9c9 and n8r4c7 n9r6c9 n8r8c8 n8r9c6 n8r9c9 .

Then comes a sequence of hard steps (but simpler than in first solution):

t-whip[9]: r7n8{c6 c7} - r1c7{n8 n7} - r2n7{c9 c4} - r1c6{n7 n6} - b1n6{r1c2 r2c3} - r2n4{c3 c1} - r5c1{n4 n3} - r5c4{n3 n1} - r8c4{n1 .} ==> r7c6≠3
Trid-OR5-whip[12]: c2n6{r6 r1} - r1c6{n6 n7} - r1c7{n7 n8} - r1c9{n8 n9} - r1c8{n9 n2} - r1c5{n2 n5} - c5n7{r1 r7} - r8n7{c4 c8} - OR5{{n8r8c8 n8r4c7 n8r7c7 n9r6c9 | n8r9c9}} - c8n8{r9 r4} - c8n5{r4 r6} - c8n9{r6 .} ==> r6c2≠7
g-whip[12]: c2n3{r9 r6} - c2n6{r6 r1} - r1c6{n6 n7} - r1c7{n7 n8} - r7n8{c7 c6} - r9c6{n8 n1} - r4c6{n1 n3} - c9n3{r4 r2} - c9n1{r2 r456} - r4c7{n1 n7} - r2n7{c7 c8} - r2n6{c8 .} ==> r9c1≠3
Trid-OR5-gwhip[12]: r3n6{c8 c6} - r1c6{n6 n7} - r1c7{n7 n8} - r1c9{n8 n9} - r1c8{n9 n6} - c2n6{r1 r6} - c2n3{r6 r789} - r8c1{n3 n8} - OR5{{n8r8c8 n8r4c7 n8r7c7 n9r6c9 | n8r9c9}} - c8n8{r9 r4} - c8n5{r4 r6} - c8n9{r6 .} ==> r3c8≠2
whip[1]: r3n2{c5 .} ==> r1c4≠2, r1c5≠2, r2c4≠2
z-chain[5]: c8n5{r4 r6} - c8n9{r6 r1} - r1n2{c8 c2} - r4n2{c2 c1} - r4n5{c1 .} ==> r4c8≠1, r4c8≠8, r4c8≠7, r4c8≠3
Trid-OR5-whip[7]: r1n9{c8 c9} - r1n8{c9 c7} - r4n8{c7 c9} - OR5{{n8r9c9 n8r4c7 n8r7c7 n9r6c9 | n8r8c8}} - r8c1{n8 n3} - b4n3{r4c1 r6c2} - c2n6{r6 .} ==> r1c8≠6
z-chain[5]: c8n6{r3 r2} - r2c3{n6 n4} - r5n4{c3 c1} - r5n3{c1 c4} - r2n3{c4 .} ==> r3c8≠3
whip[1]: r3n3{c6 .} ==> r2c4≠3
t-whip[5]: c2n6{r6 r1} - c6n6{r1 r3} - r3c8{n6 n1} - b2n1{r3c4 r2c4} - r5n1{c4 .} ==> r6c2≠1
Trid-OR5-whip[7]: c1n2{r4 r2} - r1n2{c2 c8} - r1n9{c8 c9} - r1n8{c9 c7} - r4n8{c7 c9} - OR5{{n8r9c9 n8r4c7 n8r7c7 n9r6c9 | n8r8c8}} - r8c1{n8 .} ==> r4c1≠3
Trid-OR5-whip[7]: c2n4{r9 r1} - r1n2{c2 c8} - r1n9{c8 c9} - r1n8{c9 c7} - r4n8{c7 c9} - OR5{{n8r9c9 n8r4c7 n8r7c7 n9r6c9 | n8r8c8}} - r8c1{n8 .} ==> r9c2≠3

The end is more routine (wrt to tridagon puzzles)

Code: Select all: whip[6]: r4n8{c9 c7} - r1c7{n8 n7} - r2c9{n7 n3} - r4n3{c9 c6} - r9n3{c6 c8} - b6n3{r5c8 .} ==> r4c9≠1 whip[6]: r9n3{c9 c6} - r4n3{c6 c9} - r4n8{c9 c7} - c7n1{r4 r2} - r2c9{n1 n7} - r1c7{n7 .} ==> r7c7≠3 whip[7]: r1n6{c6 c2} - r1n4{c2 c4} - c4n5{r1 r9} - c4n2{r9 r3} - r3n3{c4 c5} - r7n3{c5 c2} - r6c2{n3 .} ==> r3c6≠6 singles ==> r1c6=6, r2c3=6, r3c8=6, r6c2=6 whip[1]: c2n3{r8 .} ==> r8c1≠3 naked-single ==> r8c1=8 At least one candidate of a previous Trid-OR5-relation has just been eliminated. There remains a Trid-OR4-relation between candidates: n8r4c7 n9r6c9 n8r7c7 n8r9c9 +-------------------+-------------------+-------------------+ ! 1 24 3 ! 457 57 6 ! 78 289 789 ! ! 24 5 6 ! 147 8 9 ! 137 1237 137 ! ! 7 9 8 ! 123 123 13 ! 5 6 4 ! +-------------------+-------------------+-------------------+ ! 259 127 179 ! 6 4 137 ! 1378 59 3789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 6 179 ! 8 137 2 ! 4 13579 1379 ! +-------------------+-------------------+-------------------+ ! 6 137 5 ! 9 137 178 ! 178 4 2 ! ! 8 137 2 ! 137 6 4 ! 9 137 5 ! ! 49 147 1479 ! 25 25 1378 ! 6 1378 1378 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous Trid-OR5-relation has just been eliminated. There remains a Trid-OR4-relation between candidates: n8r4c7 n9r6c9 n8r9c6 n8r9c9 +-------------------+-------------------+-------------------+ ! 1 24 3 ! 457 57 6 ! 78 289 789 ! ! 24 5 6 ! 147 8 9 ! 137 1237 137 ! ! 7 9 8 ! 123 123 13 ! 5 6 4 ! +-------------------+-------------------+-------------------+ ! 259 127 179 ! 6 4 137 ! 1378 59 3789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 6 179 ! 8 137 2 ! 4 13579 1379 ! +-------------------+-------------------+-------------------+ ! 6 137 5 ! 9 137 178 ! 178 4 2 ! ! 8 137 2 ! 137 6 4 ! 9 137 5 ! ! 49 147 1479 ! 25 25 1378 ! 6 1378 1378 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous Trid-OR6-relation has just been eliminated. There remains a Trid-OR5-relation between candidates: n8r4c7 n9r6c9 n8r7c7 n8r9c6 n8r9c9 +-------------------+-------------------+-------------------+ ! 1 24 3 ! 457 57 6 ! 78 289 789 ! ! 24 5 6 ! 147 8 9 ! 137 1237 137 ! ! 7 9 8 ! 123 123 13 ! 5 6 4 ! +-------------------+-------------------+-------------------+ ! 259 127 179 ! 6 4 137 ! 1378 59 3789 ! ! 34 8 147 ! 137 9 5 ! 2 137 6 ! ! 359 6 179 ! 8 137 2 ! 4 13579 1379 ! +-------------------+-------------------+-------------------+ ! 6 137 5 ! 9 137 178 ! 178 4 2 ! ! 8 137 2 ! 137 6 4 ! 9 137 5 ! ! 49 147 1479 ! 25 25 1378 ! 6 1378 1378 ! +-------------------+-------------------+-------------------+ whip[1]: r3n1{c6 .} ==> r2c4≠1 Trid-OR4-ctr-whip[4]: r1n9{c8 c9} - r1n8{c9 c7} - c8n8{r1 r9} - OR4{{n8r4c7 n9r6c9 n8r7c7 n8r9c9 | .}} ==> r1c8≠2 stte

The New Sudoku Players' Forum

#51007 in 63137 T&E(3) .min-expands

#51007 in 63137 T&E(3) .min-expands

Re: #51007 in 63137 T&E(3) .min-expands

Re: #51007 in 63137 T&E(3) .min-expands

Re: #51007 in 63137 T&E(3) .min-expands

Re: #51007 in 63137 T&E(3) .min-expands