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Website · by **denis_berthier** » Sat Aug 06, 2022 9:11 am

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Code: Select all: +-------+-------+-------+ ! . . . ! 4 . . ! 7 8 . ! ! 4 . . ! 1 . . ! . . 3 ! ! . . . ! . . . ! 1 . 4 ! +-------+-------+-------+ ! . 4 1 ! 3 . 7 ! 8 . 5 ! ! . . 5 ! . 1 8 ! . . . ! ! . . . ! . 4 . ! . 1 7 ! +-------+-------+-------+ ! 3 . 4 ! . . . ! 5 . 8 ! ! . 9 6 ! 8 . . ! . . . ! ! 8 . 2 ! 7 3 . ! . . . ! +-------+-------+-------+ ...4..78.4..1....3......1.4.413.78.5..5.18.......4..173.4...5.8.968.....8.273....;14022;367772 SER = 10.1

by **Cenoman** » Sun Aug 07, 2022 11:52 am

Tentative resolution. Not found any trick after step 3. No hard step, but a long path...

Code: Select all: +-------------------------+------------------------+------------------------+ | 12569 12356 39 | 4 2569# 23569 | 7 8 269# | | 4 2568 789 | 1 78 2569# | 269# 2569 3 | | 2569 23568 3789 | 2569# 78 23569 | 1 2569# 4 | +-------------------------+------------------------+------------------------+ | 269 4 1 | 3 269# 7 | 8 269# 5 | | 2679 267-3 5 | 269# 1 8 | 34-269 34-269 269# | | 269 2368 389 | 2569 4 2569# | 269-3# 1 7 | +-------------------------+------------------------+------------------------+ | 3 17 4 | 269 269 1269 | 5 2679 8 | | 157 9 6 | 8 25 4-125 | 234 2347 12 | | 8 15 2 | 7 3 14569 | 469 469 169 | +-------------------------+------------------------+------------------------+

1. (7)r5c2 = r5c1 - r8c1 = (7-3)r8c8 = (3)r5c8 => -3 r5c2; HP(34)r5c78, -3 r6c7
2. UR(34)r58c78 having a single external guardian => +4 r8c6

3. TH(269)b2356 (#) having five guardians:
(5)r1c5 - r8c5 = (51)r18c1
(5)r3c48 - r3c1 = (51)r18c1
(5)r6c6 - r6c4 = r3c4 - r3c1 = (51)r18c1
(5)r2c6 - r1c56 = (51)r18c1
=> +15 r18c1; 8 placements

Code: Select all: +-----------------------+------------------------+---------------------+ | 15 12356 39 | 4 2569 23569 | 7 8 269 | | 4 2568 789 | 1 78 2569 | 269 2569 3 | | 26 23568 3789 | 2569 78 23569 | 1 2569 4 | +-----------------------+------------------------+---------------------+ | 269 4 1 | 3 269 7 | 8 269 5 | | 7 26 5 | 269 1 8 | 4 3 269 | | 269 38 38 | 2569 4 2569 | 269 1 7 | +-----------------------+------------------------+---------------------+ | 3 7 4 | 269 269 1 | 5 269 8 | | 15 9 6 | 8 25 4 | 3 7 12 | | 8 15 2 | 7 3 569 | 69 4 169 | +-----------------------+------------------------+---------------------+

4. UR(38)r36c23 using externals (3)r3c6 == (87)r3c35 => -3 r3c3
5. UR(78)r23c35 using single external => +8 r6c3; 4 placements
6. X-Wing (5)c15\r18 => -5 r1c26
7. Group kite (9)r5c4 = r5c9 - r1c9 = r1c56 => -9 r3c4
8. (6=2)r3c1 - r4c1 = (r4c5|r4c8) - (r8c5&r7c8) = (2-1)r8c9 = r8c1 - r1c1 = (1)r1c2 => -6 r1c2
9. W-Wing: (2=1)r1c2 - r9c2 = r9c9 - (1=2)r8c9 => -2 r1c9
10. [(6)r1c56 = r1c9 - r5c9 = r5c4] = r5c2 - r46c1 = r3c1 => -6 r3c4

Code: Select all: +--------------------+-----------------------+---------------------+ | 15 12 3 | 4 2569 269 | 7 8 69 | | 4 2568 79 | 1 78 2569 | 269 2569 3 | | 26 2568 79 | 25 78 3 | 1 2569 4 | +--------------------+-----------------------+---------------------+ | 269 4 1 | 3 269 7 | 8 269 5 | | 7 26 5 | 269 1 8 | 4 3 269 | | 269 3 8 | 2569 4 2569 | 269 1 7 | +--------------------+-----------------------+---------------------+ | 3 7 4 | 269 269 1 | 5 269 8 | | 15 9 6 | 8 25 4 | 3 7 12 | | 8 15 2 | 7 3 569 | 69 4 169 | +--------------------+-----------------------+---------------------+

11. (6)r2c6 = r1c56 - (6=9)r1c9 - r1c56 = (9)r2c6 => -25 r2c6
12. (2)r6c6 = r12c6 - (2=5)r3c4 - r6c4 = (5)r6c6 => -69 r6c6, -2 r1c5
13. (6=2)r3c1 - r1c2 = (2-6|9)r1c6 = (69)r29c6 - (6)r1c56&r7c45 = (6)r1c9|r7c8 => -6 r3c8
14. (6=2)r3c1 - *r1c2 = (2-6|9)r1c6 = (69)r29c6 - (9)r1c56&r7c45 = (9)r1c9|r7c8 - (9=526)r3c148 =>-2 r3c2*, -6 r3c2; lcls, 2 placements
15. X-Chain (2)r6c6 = r1c6 - r3c4 = r3c8 - r2c7 = r6c7 => -2 r6c1; lcls, 2 placements

16. Kraken cell (569)r9c6
||(5)r9c6 - (5=2)r8c5 - r8c9 = (2-9)r5c9 = (9)r5c4
||(6)r9c6 - r2c6 = r1c56 - (6=9)r1c9 - r5c9 = (9)r5c4
||(9)r9c6 - r12c6 = (9)r1c5
= >-9 r4c5; ste

Website · by **denis_berthier** » Mon Aug 08, 2022 2:59 am

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Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 12569 12356 39 ! 4 2569 23569 ! 7 8 269 ! ! 4 2568 789 ! 1 256789 2569 ! 269 2569 3 ! ! 2569 23568 3789 ! 2569 256789 23569 ! 1 2569 4 ! +----------------------+----------------------+----------------------+ ! 269 4 1 ! 3 269 7 ! 8 269 5 ! ! 2679 2367 5 ! 269 1 8 ! 23469 23469 269 ! ! 269 2368 389 ! 2569 4 2569 ! 2369 1 7 ! +----------------------+----------------------+----------------------+ ! 3 17 4 ! 269 269 1269 ! 5 2679 8 ! ! 157 9 6 ! 8 25 1245 ! 234 2347 12 ! ! 8 15 2 ! 7 3 14569 ! 469 469 169 ! +----------------------+----------------------+----------------------+ 185 candidates.

After only Singles or after Singles + Subsets + Finned Fish, an anti-tridagon pattern with 6 guardians is found. This is currently too many guardians for SudoRules to deal with directly, so I had to allow some whips. As the puzzle will finally require whips[7], here's a direct solution found by activating W7+OR5FW7 since the start.
I chose this puzzle because:
- it has two short OR5 forcing whips;
- it has an OR5 forcing whip with five 0-length branches, i.e. a candidate directly linked to all the guardians. For 5 guardians, that must be quite rare.

Code: Select all: hidden-pairs-in-a-column: c5{n7 n8}{r2 r3} ==> r3c5≠9, r3c5≠6, r3c5≠5, r3c5≠2, r2c5≠9, r2c5≠6, r2c5≠5, r2c5≠2 finned-x-wing-in-columns: n5{c5 c1}{r8 r1} ==> r1c2≠5 biv-chain[3]: r5n7{c2 c1} - r8n7{c1 c8} - c8n3{r8 r5} ==> r5c2≠3 whip[1]: r5n3{c8 .} ==> r6c7≠3 hidden-pairs-in-a-block: b6{n3 n4}{r5c7 r5c8} ==> r5c8≠9, r5c8≠6, r5c8≠2, r5c7≠9, r5c7≠6, r5c7≠2 hidden-pairs-in-a-row: r6{n3 n8}{c2 c3} ==> r6c3≠9, r6c2≠6, r6c2≠2 whip[1]: c3n9{r3 .} ==> r1c1≠9, r3c1≠9 z-chain[4]: c2n6{r3 r5} - c9n6{r5 r9} - c9n1{r9 r8} - c1n1{r8 .} ==> r1c1≠6 t-whip[4]: c6n4{r8 r9} - r9n5{c6 c2} - r9n1{c2 c9} - r8c9{n1 .} ==> r8c6≠2 t-whip[5]: r8c5{n5 n2} - r7n2{c6 c8} - c8n7{r7 r8} - r8n3{c8 c7} - r8n4{c7 .} ==> r8c6≠5

Code: Select all: +-------------------+-------------------+-------------------+ ! 125 1236 39 ! 4 2569 23569 ! 7 8 269 ! ! 4 2568 789 ! 1 78 2569 ! 269 2569 3 ! ! 256 23568 3789 ! 2569 78 23569 ! 1 2569 4 ! +-------------------+-------------------+-------------------+ ! 269 4 1 ! 3 269 7 ! 8 269 5 ! ! 2679 267 5 ! 269 1 8 ! 34 34 269 ! ! 269 38 38 ! 2569 4 2569 ! 269 1 7 ! +-------------------+-------------------+-------------------+ ! 3 17 4 ! 269 269 1269 ! 5 2679 8 ! ! 157 9 6 ! 8 25 14 ! 234 2347 12 ! ! 8 15 2 ! 7 3 14569 ! 469 469 169 ! +-------------------+-------------------+-------------------+ OR5-anti-tridagon[12] (type antidiag) for digits 2, 6 and 9 in blocks: b2, with cells: r1c5, r2c6, r3c4 b3, with cells: r1c9, r2c7, r3c8 b5, with cells: r4c5, r6c6, r5c4 b6, with cells: r4c8, r6c7, r5c9 with 5 guardians: n5r1c5 n5r2c6 n5r3c4 n5r3c8 n5r6c6 OR5-forcing-whip-elim[1] based on OR5-anti-tridagon[12] for n5r1c5, n5r2c6, n5r3c4, n5r3c8 and n5r6c6: || n5r1c5 - || n5r2c6 - || n5r3c4 - || n5r3c8 - || n5r6c6 - ==> r3c6≠5 OR5-forcing-whip-elim[4] based on OR5-anti-tridagon[12] for n5r3c4, n5r3c8, n5r1c5, n5r2c6 and n5r6c6: || n5r3c4 - || n5r3c8 - || n5r1c5 - partial-whip[1]: r8n5{c5 c1} - || n5r2c6 - partial-whip[1]: c8n5{r2 r3} - || n5r6c6 - partial-whip[1]: c4n5{r6 r3} - ==> r3c1≠5

The end is like any puzzle in W7:

Code: Select all: hidden-pairs-in-a-column: c1{n1 n5}{r1 r8} ==> r8c1≠7, r1c1≠2 singles ==> r7c2=7, r5c1=7, r7c6=1, r8c6=4, r8c8=7, r8c7=3, r5c7=4, r5c8=3, r9c8=4 x-wing-in-columns: n5{c1 c5}{r1 r8} ==> r1c6≠5 t-whip[6]: c2n1{r1 r9} - c9n1{r9 r8} - b9n2{r8c9 r7c8} - b8n2{r7c4 r8c5} - r4n2{c5 c1} - r3c1{n2 .} ==> r1c2≠6 z-chain[3]: r1n6{c6 c9} - r5n6{c9 c2} - b1n6{r2c2 .} ==> r3c4≠6 t-whip[7]: b2n6{r3c6 r1c5} - c5n5{r1 r8} - r8n2{c5 c9} - r1c9{n2 n9} - r5n9{c9 c4} - c5n9{r4 r7} - r9c6{n9 .} ==> r6c6≠6 whip[7]: c7n2{r2 r6} - c8n2{r4 r7} - b8n2{r7c4 r8c5} - r4n2{c5 c1} - c1n9{r4 r6} - r6c6{n9 n5} - b8n5{r9c6 .} ==> r1c9≠2 z-chain[5]: c7n2{r2 r6} - c9n2{r5 r8} - r8n1{c9 c1} - r1n1{c1 c2} - r1n2{c2 .} ==> r2c6≠2 whip[6]: r5n9{c4 c9} - r1c9{n9 n6} - r9c9{n6 n1} - r9n9{c9 c7} - r9n6{c7 c6} - b2n6{r1c6 .} ==> r7c4≠9 whip[5]: r7c4{n2 n6} - b5n6{r5c4 r4c5} - c5n9{r4 r1} - r1c9{n9 n6} - c8n6{r2 .} ==> r7c5≠2 biv-chain[2]: c9n2{r5 r8} - b8n2{r8c5 r7c4} ==> r5c4≠2 biv-chain[2]: c7n2{r2 r6} - r5n2{c9 c2} ==> r2c2≠2 whip[1]: r2n2{c8 .} ==> r3c8≠2 biv-chain[3]: b8n2{r7c4 r8c5} - c5n5{r8 r1} - c4n5{r3 r6} ==> r6c4≠2 biv-chain[3]: c4n2{r3 r7} - r8n2{c5 c9} - r5n2{c9 c2} ==> r3c2≠2 biv-chain[3]: b5n2{r6c6 r4c5} - r8n2{c5 c9} - r5n2{c9 c2} ==> r6c1≠2 biv-chain[2]: b5n2{r6c6 r4c5} - c1n2{r4 r3} ==> r3c6≠2 biv-chain[4]: c4n5{r6 r3} - b3n5{r3c8 r2c8} - b3n2{r2c8 r2c7} - r6n2{c7 c6} ==> r6c6≠5 hidden-single-in-a-block ==> r6c4=5 biv-chain[3]: r3c4{n9 n2} - r7c4{n2 n6} - r7c5{n6 n9} ==> r1c5≠9 biv-chain[3]: c4n6{r5 r7} - r7n2{c4 c8} - c9n2{r8 r5} ==> r5c9≠6 finned-x-wing-in-columns: n6{c9 c6}{r9 r1} ==> r1c5≠6 whip[1]: b2n6{r3c6 .} ==> r9c6≠6 whip[1]: r9n6{c9 .} ==> r7c8≠6 naked-pairs-in-a-column: c5{r1 r8}{n2 n5} ==> r4c5≠2 singles ==> r6c6=2, r2c7=2 naked-triplets-in-a-row: r1{c3 c6 c9}{n9 n3 n6} ==> r1c2≠3 finned-swordfish-in-columns: n9{c1 c7 c5}{r4 r6 r9} ==> r9c6≠9 stte

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mith's TE3 min-expand #62886/63137

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Re: mith's TE3 min-expand #62886/63137

Re: mith's TE3 min-expand #62886/63137