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Website · by **denis_berthier** » Thu Apr 17, 2025 1:25 pm

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Code: Select all: +-------+-------+-------+ ! . . 3 ! 4 . . ! 7 . . ! ! 4 5 . ! . . . ! . 2 3 ! ! 7 . 8 ! . . . ! . . . ! +-------+-------+-------+ ! 2 7 . ! . . 4 ! 8 5 . ! ! 8 . . ! . . . ! . . . ! ! . 3 5 ! . . . ! . 4 . ! +-------+-------+-------+ ! . . . ! 8 6 . ! . 7 . ! ! . . . ! 1 9 . ! . . 8 ! ! . . . ! . . 2 ! . . 5 ! +-------+-------+-------+ ..34..7..45.....237.8......27...485.8.........35....4....86..7....19...8.....2..5

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 169 2 3 ! 4 5 169 ! 7 8 169 ! ! 4 5 169 ! 679 178 16789 ! 169 2 3 ! ! 7 169 8 ! 2369 123 1369 ! 5 169 4 ! +-------------------+-------------------+-------------------+ ! 2 7 169 ! 369 13 4 ! 8 5 169 ! ! 8 1469 1469 ! 5 127 1679 ! 12369 1369 12679 ! ! 169 3 5 ! 2679 1278 16789 ! 1269 4 12679 ! +-------------------+-------------------+-------------------+ ! 1359 149 1249 ! 8 6 35 ! 12349 7 129 ! ! 356 46 2467 ! 1 9 357 ! 2346 36 8 ! ! 1369 8 1679 ! 37 4 2 ! 1369 1369 5 ! +-------------------+-------------------+-------------------+ 165 candidates.

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by **marek stefanik** » Mon Apr 21, 2025 1:36 pm

Code: Select all: ,------------------,-------------------,--------------------, |#169 2 3 | 4 5 169 | 7 8 #169 | | 4 5 #169 | 679 178 16789 |#169 2 3 | | 7 d169 8 | 2369 123 1369 | 5 d169 4 | :------------------+-------------------+--------------------: | 2 7 #169 | 369 13 4 | 8 5 #169 | | 8 d1469 1469 | 5 127 1679 | 12369 d1369 b12679 | |#169 3 5 | 2679 1278 16789 |c#1269 4 b12679 | :------------------+-------------------+--------------------: | 1359 e149 1249 | 8 6 35 | 12349 7 a2–19 | | 356 46 2467 | 1 9 357 | 2346 36 8 | | 1369 8 1679 | 37 4 2 | 1369 e1369 5 | '------------------'-------------------'--------------------'

2r7c9 = 2r56c9 – 2r6c7 = 169# – (11|99)r35c28 = 19r7c2|r9c8 => –19r7c9

Code: Select all: ,-----------------,------------------,-------------------, |x169 2 3 | 4 5 169 | 7 8 169 | | 4 5 169 | 679 178 1689 |x169 2 3 | | 7 169 8 |x269 xa12 3 | 5 169 4 | :-----------------+------------------+-------------------: | 2 7 169 | 69 3 4 | 8 5 169 | | 8 1469 1469 | 5 b127 169 |c236–19 1369 1679 | | 169 3 5 | 2679 1278 1689 |d26–19 4 1679 | :-----------------+------------------+-------------------: | 3 149 149 | 8 6 5 | 149 7 2 | | 5 46 2 | 1 9 7 | 346 36 8 | | 169 8 7 | 3 4 2 | 169 x169 5 | '-----------------'------------------'-------------------'

Let x be the 1|6|9 in r3c45.
If x appears in r1c9 and r2c3, it cannot appear in r4, i.e. contra. So xr1c1, xr2c7, xr9c8.
All 1s and 9s in b9 see r56c7 due to the r2c7 r9c8 equivalence. –19r56c7
1x = 2r3c5 – 2r5c5 = 2r5c7 – (2=6)r6c7 => –6x, 7.3 skfr

I needed quite a few steps to finish the job from here.

Hidden Text: Show

Code: Select all: ,-----------------,------------------,-----------------, |cx19 2 3 | 4 5 c169 | 7 8 b69–1 | | 4 5 169 | 679 78–1 689–1|x19 2 3 | | 7 69–1 8 |x29 x12 3 | 5 169 4 | :-----------------+------------------+-----------------: | 2 7 e169 | 69 3 4 | 8 5 af169 | | 8 d1469 d1469 | 5 127 c169 |236 b369–1 a1679 | |d169 3 5 | 2679 1278 1689 | 26 4 a1679 | :-----------------+------------------+-----------------: | 3 149 149 | 8 6 5 | 149 7 2 | | 5 46 2 | 1 9 7 | 346 36 8 | | 169 8 7 | 3 4 2 | 169 19 5 | '-----------------'------------------'-----------------'

1r456c9 = 1r1c9&r5c8 – (1=969)r1c16&r5c6 – 9b4p567 = (9–1)r4c3 = 1r4c9 => –1r1c9, –1r5c8
Due to the r1c1 r2c7 equivalence, all 1s in r1 see r2c56 and all 1s in b3 see r3c2. –1r2c56, –1r3c2

Code: Select all: ,-----------------,------------------,----------------, | 19 2 3 | 4 5 169 | 7 8 69 | | 4 5 169 | 679 78 689 | 19 2 3 | | 7 C69 8 |δ2–9 12 3 | 5 169 4 | :-----------------+------------------+----------------: | 2 7 a169 |b69 3 4 | 8 5 169 | | 8 AB1469 A1469 | 5 127 169 | 236 369 1679 | |α169 3 5 |γ2679 1278 1689 |β26 4 1679 | :-----------------+------------------+----------------: | 3 149 149 | 8 6 5 | 149 7 2 | | 5 B46 2 | 1 9 7 | 346 36 8 | | 169 8 7 | 3 4 2 | 169 19 5 | '-----------------'------------------'----------------'

Kraken 6b4:
6r4c3 – (6=9)r4c4
(46–1|9)r5c23 = 46r58c2 – (6=9)r3c2
6r6c1 – (6=2)r6c7 – 2r6c4 = 2r3c4
=> –9r3c4

Code: Select all: ,----------------,----------------,----------------, | 1 2 3 | 4 5 69 | 7 8 69 | | 4 5 69 | 679 78 689 | 1 2 3 | | 7 69 8 | 2 1 3 | 5 69 4 | :----------------+----------------+----------------: | 2 7 169 | 69 3 4 | 8 5 169 | | 8 1469 1469 | 5 a27 169 | 36–2 369 1679 | |b69 3 5 |b679 78–2 1689 |c26 4 1679 | :----------------+----------------+----------------: | 3 149 149 | 8 6 5 | 49 7 2 | | 5 46 2 | 1 9 7 | 346 36 8 | | 69 8 7 | 3 4 2 | 69 1 5 | '----------------'----------------'----------------'

(2=7)r5c5 – (7=69)r6c14 – (6=2)r6c7 => –2r5c7, –2r6c5

Code: Select all: ,----------------,---------------,---------------, | 1 2 3 | 4 5 *69 | 7 8 *69 | | 4 5 *69 |*679 78 *689 | 1 2 3 | | 7 69 8 | 2 1 3 | 5 69 4 | :----------------+---------------+---------------: | 2 7 169 | 69 3 4 | 8 5 *169 | | 8 1469 14–69| 5 2 1–69 |*36 *369 7 | | 69 3 5 | 679 78 1689 | 2 4 *169 | :----------------+---------------+---------------: | 3 149 149 | 8 6 5 | 49 7 2 | | 5 46 2 | 1 9 7 | 346 36 8 | | 69 8 7 | 3 4 2 | 69 1 5 | '----------------'---------------'---------------'

69r1b6\r5c69 => –69r5c6
69r12b6\r5c39b2 => –69r5c3, stte

Marek

Website · by **denis_berthier** » Tue Apr 22, 2025 7:50 am

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Thanks for your solutions.
This B22 can be solved with short chains, by using a tridagon plus two of the most frequent patterns close to it.

Code: Select all: Trid-OR3-relation for digits 1, 6 and 9 in blocks: b1, with cells (marked #): r1c1, r2c3, r3c2 b3, with cells (marked #): r1c9, r2c7, r3c8 b4, with cells (marked #): r6c1, r4c3, r5c2 b6, with cells (marked #): r6c7, r4c9, r5c8 with 3 guardians (in cells marked @): n4r5c2 n3r5c8 n2r6c7 +----------------------+----------------------+----------------------+ ! 169# 2 3 ! 4 5 169 ! 7 8 169# ! ! 4 5 169# ! 679 178 16789 ! 169# 2 3 ! ! 7 169# 8 ! 2369 123 1369 ! 5 169# 4 ! +----------------------+----------------------+----------------------+ ! 2 7 169# ! 369 13 4 ! 8 5 169# ! ! 8 1469#@ 1469 ! 5 127 1679 ! 12369 1369#@ 12679 ! ! 169# 3 5 ! 2679 1278 16789 ! 1269#@ 4 12679 ! +----------------------+----------------------+----------------------+ ! 1359 149 1249 ! 8 6 35 ! 12349 7 129 ! ! 356 46 2467 ! 1 9 357 ! 2346 36 8 ! ! 1369 8 1679 ! 37 4 2 ! 1369 1369 5 ! +----------------------+----------------------+----------------------+ EL13c290s-OR4-relation for digits: 1, 6 and 9 in cells (marked #): (r9c3 r9c1 r9c8 r6c1 r6c7 r4c3 r4c9 r1c1 r1c9 r2c3 r2c7 r3c2 r3c8) with 4 guardians (in cells marked @) : n7r9c3 n3r9c1 n3r9c8 n2r6c7 +----------------------+----------------------+----------------------+ ! 169# 2 3 ! 4 5 169 ! 7 8 169# ! ! 4 5 169# ! 679 178 16789 ! 169# 2 3 ! ! 7 169# 8 ! 2369 123 1369 ! 5 169# 4 ! +----------------------+----------------------+----------------------+ ! 2 7 169# ! 369 13 4 ! 8 5 169# ! ! 8 1469 1469 ! 5 127 1679 ! 12369 1369 12679 ! ! 169# 3 5 ! 2679 1278 16789 ! 1269#@ 4 12679 ! +----------------------+----------------------+----------------------+ ! 1359 149 1249 ! 8 6 35 ! 12349 7 129 ! ! 356 46 2467 ! 1 9 357 ! 2346 36 8 ! ! 1369#@ 8 1679#@ ! 37 4 2 ! 1369 1369#@ 5 ! +----------------------+----------------------+----------------------+ EL14c1s-OR5-relation for digits: 1, 6 and 9 in cells (marked #): (r9c1 r9c7 r9c8 r5c8 r6c1 r6c7 r4c3 r4c9 r1c1 r1c9 r2c3 r2c7 r3c2 r3c8) with 5 guardians (in cells marked @) : n3r9c1 n3r9c7 n3r9c8 n3r5c8 n2r6c7 +----------------------+----------------------+----------------------+ ! 169# 2 3 ! 4 5 169 ! 7 8 169# ! ! 4 5 169# ! 679 178 16789 ! 169# 2 3 ! ! 7 169# 8 ! 2369 123 1369 ! 5 169# 4 ! +----------------------+----------------------+----------------------+ ! 2 7 169# ! 369 13 4 ! 8 5 169# ! ! 8 1469 1469 ! 5 127 1679 ! 12369 1369#@ 12679 ! ! 169# 3 5 ! 2679 1278 16789 ! 1269#@ 4 12679 ! +----------------------+----------------------+----------------------+ ! 1359 149 1249 ! 8 6 35 ! 12349 7 129 ! ! 356 46 2467 ! 1 9 357 ! 2346 36 8 ! ! 1369#@ 8 1679 ! 37 4 2 ! 1369#@ 1369#@ 5 ! +----------------------+----------------------+----------------------+

biv-chain[3]: r8c8{n3 n6} - r8c2{n6 n4} - b9n4{r8c7 r7c7} ==> r7c7≠3
hidden-pairs-in-a-row: r7{n3 n5}{c1 c6} ==> r7c1≠9, r7c1≠1
Trid-OR3-whip[4]: r8c8{n3 n6} - r8c2{n6 n4} - OR3{{n4r5c2 n3r5c8 | n2r6c7}} - r8c7{n2 .} ==> r9c8≠3

At least one candidate of a previous EL13c290s-OR4-relation between candidates n7r9c3 n3r9c1 n3r9c8 n2r6c7 has just been eliminated.
There remains an EL13c290s-OR3-relation between candidates: n7r9c3 n3r9c1 n2r6c7

At least one candidate of a previous EL14c1s-OR5-relation between candidates n3r9c1 n3r9c7 n3r9c8 n3r5c8 n2r6c7 has just been eliminated.
There remains an EL14c1s-OR4-relation between candidates: n3r9c1 n3r9c7 n3r5c8 n2r6c7

EL13c290s-OR3-ctr-whip[4]: r8n7{c6 c3} - r8n2{c3 c7} - b9n3{r8c7 r9c7} - OR3{{n7r9c3 n3r9c1 n2r6c7 | .}} ==> r8c6≠3
EL13c290s-OR3-whip[4]: r9c4{n3 n7} - OR3{{n7r9c3 n3r9c1 | n2r6c7}} - r8n2{c7 c3} - r8n7{c3 .} ==> r9c7≠3

At least one candidate of a previous EL14c1s-OR4-relation between candidates n3r9c1 n3r9c7 n3r5c8 n2r6c7 has just been eliminated.
There remains an EL14c1s-OR3-relation between candidates: n3r9c1 n3r5c8 n2r6c7

whip[1]: b9n3{r8c8 .} ==> r8c1≠3
biv-chain[3]: r8c1{n6 n5} - r8c6{n5 n7} - b7n7{r8c3 r9c3} ==> r9c3≠6
Trid-OR3-whip[4]: r8n2{c3 c7} - c7n3{r8 r5} - OR3{{n3r5c8 n2r6c7 | n4r5c2}} - r8n4{c2 .} ==> r8c3≠7
singles ==> r9c3=7, r9c4=3, r7c6=5, r7c1=3, r8c6=7, r8c1=5, r4c5=3, r3c6=3

At least one candidate of a previous EL14c1s-OR3-relation between candidates n3r9c1 n3r5c8 n2r6c7 has just been eliminated.
There remains an EL14c1s-OR2-relation between candidates: n3r5c8 n2r6c7

EL14c1s-OR2-whip[2]: OR2{{n2r6c7 | n3r5c8}} - r8n3{c8 .} ==> r8c7≠2
hidden-single-in-a-row ==> r8c3=2
EL14c1s-OR2-whip[3]: OR2{{n2r6c7 | n3r5c8}} - r8n3{c8 c7} - c7n4{r8 .} ==> r7c7≠2

easy end in W4.

by **eleven** » Tue Apr 22, 2025 1:58 pm

denis_berthier wrote:easy end in W4.

Maybe for you. I had some easy steps to get there, but i did not see a good way to finish, the ER is still 8.9.

Code: Select all: +----------------------+-----------------------+-----------------------+ | *169 2 3 | 4 5 169 | 7 8 *169 | | 4 5 *169 | 679 178 16789 | *169 2 3 | | 7 *169 8 | 2369 123 1369 | 5 *169 4 | +----------------------+-----------------------+-----------------------+ | 2 7 *169 | 369 13 4 | 8 5 *169 | | 8 *169+4 1469 | 5 127 1679 | 12369 *169+3 12679 | | *169 3 5 | 2679 1278 16789 | *169+2 4 12679 | +----------------------+-----------------------+-----------------------+ | #1359 149 1249 | 8 6 #35 | 12349 7 129 | | #5-36 a46 247-6 | 1 9 #357 | b2346 a36 8 | | 1369 8 1679 | 37 4 2 | 1369 1369 5 | +----------------------+-----------------------+-----------------------+

UR 35r78c16 => -3r8c1 (3r8c1 -> 5r7c1,r8c3 -> 3r7c6)
Tridagon 169 (*): 6r8c13 => 4r8c2,3r8c8,2r8c7 kills all extra candidates => -6r8c13

Code: Select all: +----------------------+----------------------+----------------------+ | 169 2 3 | 4 5 169 | 7 8 169 | | 4 5 169 | 679 178 16789 | 169 2 3 | | 7 169 8 | 2369 123 1369 | 5 169 4 | +----------------------+----------------------+----------------------+ | 2 7 169 | 369 13 4 | 8 5 169 | | 8 1469 1469 | 5 127 1679 | 12369 1369 12679 | | 169 3 5 | 2679 1278 16789 | 1269 4 12679 | +----------------------+----------------------+----------------------+ | 139 149 1249 | 8 6 5 |d1249-3 7 129 | | 5 b46 247 | 1 9 37 |c2346 a36 8 | | 1369 8 1679 | 37 4 2 | 1369 1369 5 | +----------------------+----------------------+----------------------+

(3=6)r8c8 - (6=4)r8c2 - r8c7 = 4r7c7 => -3r7c7

Code: Select all: +----------------------+----------------------+----------------------+ | 169 2 3 | 4 5 169 | 7 8 169 | | 4 5 169 | 679 178 16789 | 169 2 3 | | 7 169 8 | 2369 123 1369 | 5 169 4 | +----------------------+----------------------+----------------------+ | 2 7 169 | 369 13 4 | 8 5 169 | | 8 1469 1469 | 5 127 1679 | 12369 1369 12679 | | 169 3 5 | 2679 1278 16789 | 1269 4 12679 | +----------------------+----------------------+----------------------+ |*3 149 1249 | 8 6 *5 | 1249 7 129 | |*5 46 247 | 1 9 *7-3 | 2346 36 8 | | 169 8 1679 | 37 4 2 | 1369 1369 5 | +----------------------+----------------------+----------------------+

UR1.1 35r78c18 => -3r6c6

Code: Select all: +---------------------+----------------------+-----------------------+ | 169 2 3 | 4 5 169 | 7 8 169 | | 4 5 169 | 679 178 1689 | 169 2 3 | | 7 169 8 | 269 12 3 | 5 169 4 | +---------------------+----------------------+-----------------------+ | 2 7 169 | 69 3 4 | 8 5 169 | | 8 *1469 1469 | 5 127 169 | 12369 *1369 12679 | | 169 3 5 | 2679 1278 1689 | *1269 4 12679 | +---------------------+----------------------+-----------------------+ | 3 a149 #1249 | 8 6 5 | #1249 7 b2-19 | | 5 a46 #246 | 1 9 7 | #2346 b36 8 | | 169 8 7 | 3 4 2 | b169 b169 5 | +---------------------+----------------------+-----------------------+

Tridagon (4r5c2|3r5c8|2r6c7) + UR24 (4r78c2=2r7c9):
4r5c2 - r78c2 == 2r7c9
3r5c8 - (3=6192)b9p5783
2r6c7 - r78c7 = 2r7c9
=> -19r7c9

Website · by **denis_berthier** » Wed Apr 23, 2025 3:05 am

eleven wrote:
denis_berthier wrote:easy end in W4.

Maybe for you.

OK, "easy" is relative. But W4 is certainly easy for a puzzle originally in B22.
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