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by **coloin** » Mon Oct 14, 2024 11:58 pm

Code: Select all: +---+---+---+ |94.|..7|.5.| |5..|6.9|.8.| |.6.|54.|...| +---+---+---+ |.76|8.4|...| |8.5|9..|...| |.9.|...|...| +---+---+---+ |...|...|27.| |78.|...|965| |...|...|..1| +---+---+---+ WD

by **yzfwsf** » Tue Oct 15, 2024 4:07 am

Basic,

Code: Select all: ,-----------------,------------------,--------------------, | 9 4 123 | 123 8 7 | 136 5 236 | | 5 123 7 | 6 123 9 | 134 8 234 | | 123 6 8 | 5 4 123 | 137 1239 2379 | :-----------------+------------------+--------------------: | 123 7 6 | 8 1235 4 | 135 1239 239 | | 8 123 5 | 9 12367 123 | 13467 1234 23467 | | 1234 9 1234 | 123 567 56 | 5678 123 678 | :-----------------+------------------+--------------------: | 1346 135 1349 | 134 569 568 | 2 7 348 | | 7 8 134 | 1234 123 123 | 9 6 5 | | 2346 235 2349 | 7 569 568 | 348 34 1 | '-----------------'------------------'--------------------'

Code: Select all: Impossible Pattern(123{r346c1,r8c356,r1c34,r2c25,r5c26,r3c6,r7c4}) Forcing Chain With Triplet Oddagon(123r16c34,r2c25,r3c16,r4c15,r5c26): Each true guardian of Impossible Pattern will all lead To: r4c5<>1,r4c5<>2,r4c5<>3,r679c5,r4c7,r6c6<>5 4r6c1 - 4r6c3 = 5r4c5 4r7c4 - 4r8c4 = 4r8c3 - 4r6c3 = 5r4c5 4r8c3 - 4r6c3 = 5r4c5

bte

Website · by **denis_berthier** » Tue Oct 15, 2024 4:29 am

.
BxB = 10, SER = 11.7

Code: Select all: Resolution state after Singles and whips[1]: +-------------------------+-------------------------+-------------------------+ ! 9 4 1238 ! 123 1238 7 ! 136 5 236 ! ! 5 123 1237 ! 6 123 9 ! 1347 8 2347 ! ! 123 6 12378 ! 5 4 1238 ! 137 1239 2379 ! +-------------------------+-------------------------+-------------------------+ ! 123 7 6 ! 8 1235 4 ! 135 1239 239 ! ! 8 123 5 ! 9 12367 1236 ! 13467 1234 23467 ! ! 1234 9 1234 ! 1237 123567 12356 ! 135678 123 23678 ! +-------------------------+-------------------------+-------------------------+ ! 1346 135 1349 ! 134 135689 13568 ! 2 7 348 ! ! 7 8 1234 ! 1234 123 123 ! 9 6 5 ! ! 2346 235 2349 ! 2347 2356789 23568 ! 348 34 1 ! +-------------------------+-------------------------+-------------------------+ 212 candidates.

At the start, we need only classical rules:

Code: Select all: naked-quads-in-a-block: b8{r7c4 r8c4 r8c6 r8c5}{n3 n4 n2 n1} ==> r9c6≠3, r9c6≠2, r9c5≠3, r9c5≠2, r9c4≠4, r9c4≠3, r9c4≠2, r7c6≠3, r7c6≠1, r7c5≠3, r7c5≠1 naked-single ==> r9c4=7 whip[1]: r9n2{c3 .} ==> r8c3≠2 naked-quads-in-a-row: r6{c1 c3 c4 c8}{n1 n4 n2 n3} ==> r6c9≠3, r6c9≠2, r6c7≠3, r6c7≠1, r6c6≠3, r6c6≠2, r6c6≠1, r6c5≠3, r6c5≠2, r6c5≠1 naked-triplets-in-a-column: c6{r6 r7 r9}{n5 n6 n8} ==> r5c6≠6, r3c6≠8 singles ==> r1c5=8, r3c3=8, r2c3=7 +-------------------+-------------------+-------------------+ ! 9 4 123 ! 123 8 7 ! 136 5 236 ! ! 5 123 7 ! 6 123 9 ! 134 8 234 ! ! 123 6 8 ! 5 4 123 ! 137 1239 2379 ! +-------------------+-------------------+-------------------+ ! 123 7 6 ! 8 1235 4 ! 135 1239 239 ! ! 8 123 5 ! 9 12367 123 ! 13467 1234 23467 ! ! 1234 9 1234 ! 123 567 56 ! 5678 123 678 ! +-------------------+-------------------+-------------------+ ! 1346 135 1349 ! 134 569 568 ! 2 7 348 ! ! 7 8 134 ! 1234 123 123 ! 9 6 5 ! ! 2346 235 2349 ! 7 569 568 ! 348 34 1 ! +-------------------+-------------------+-------------------+

We now have a tridagon with only two guardians

Code: Select all: Trid-OR2-relation for digits 1, 2 and 3 in blocks: b1, with cells (marked #): r1c3, r2c2, r3c1 b2, with cells (marked #): r1c4, r2c5, r3c6 b4, with cells (marked #): r6c3, r5c2, r4c1 b5, with cells (marked #): r6c4, r5c6, r4c5 with 2 guardians (in cells marked @): n5r4c5 n4r6c3 +----------------------+----------------------+----------------------+ ! 9 4 123# ! 123# 8 7 ! 136 5 236 ! ! 5 123# 7 ! 6 123# 9 ! 134 8 234 ! ! 123# 6 8 ! 5 4 123# ! 137 1239 2379 ! +----------------------+----------------------+----------------------+ ! 123# 7 6 ! 8 1235#@ 4 ! 135 1239 239 ! ! 8 123# 5 ! 9 12367 123# ! 13467 1234 23467 ! ! 1234 9 1234#@ ! 123# 567 56 ! 5678 123 678 ! +----------------------+----------------------+----------------------+ ! 1346 135 1349 ! 134 569 568 ! 2 7 348 ! ! 7 8 134 ! 1234 123 123 ! 9 6 5 ! ! 2346 235 2349 ! 7 569 568 ! 348 34 1 ! +----------------------+----------------------+----------------------+

Applying eleven's replacement technique based on this tridagon leads to an easy solution:

Code: Select all: ***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE ***** RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens. Trying in block 2 +----------------------+----------------------+----------------------+ ! 9 4 123 ! 3 8 7 ! 1236 5 1236 ! ! 5 123 7 ! 6 2 9 ! 1234 8 1234 ! ! 123 6 8 ! 5 4 1 ! 1237 1239 12379 ! +----------------------+----------------------+----------------------+ ! 123 7 6 ! 8 1235 4 ! 1235 1239 1239 ! ! 8 123 5 ! 9 12367 123 ! 123467 1234 123467 ! ! 1234 9 1234 ! 123 567 56 ! 5678 123 678 ! +----------------------+----------------------+----------------------+ ! 12346 1235 12349 ! 1234 569 568 ! 123 7 12348 ! ! 7 8 1234 ! 1234 123 123 ! 9 6 5 ! ! 12346 1235 12349 ! 7 569 568 ! 12348 1234 123 ! +----------------------+----------------------+----------------------+

There's now an easy solution in Z5:

Code: Select all: finned-x-wing-in-columns: n2{c6 c2}{r5 r8} ==> r8c3≠2 whip[1]: r8n2{c6 .} ==> r7c4≠2 biv-chain[3]: r6n4{c1 c3} - r8n4{c3 c4} - c4n2{r8 r6} ==> r6c1≠2 biv-chain[4]: r3c1{n3 n2} - r1c3{n2 n1} - r8c3{n1 n4} - b4n4{r6c3 r6c1} ==> r6c1≠3 z-chain[5]: r5c6{n2 n3} - r5c2{n3 n1} - b1n1{r2c2 r1c3} - b1n2{r1c3 r3c1} - r4n2{c1 .} ==> r5c7≠2, r5c9≠2, r5c8≠2 z-chain[5]: b1n1{r1c3 r2c2} - b1n3{r2c2 r3c1} - r4c1{n3 n2} - r5n2{c2 c6} - r6c4{n2 .} ==> r6c3≠1 z-chain[5]: r5n2{c2 c6} - r6c4{n2 n1} - b4n1{r6c1 r5c2} - b1n1{r2c2 r1c3} - b1n2{r1c3 .} ==> r4c1≠2 whip[1]: r4n2{c9 .} ==> r6c8≠2 biv-chain[3]: r6c8{n3 n1} - r6c4{n1 n2} - r5c6{n2 n3} ==> r5c7≠3, r5c8≠3, r5c9≠3 biv-chain[3]: b4n2{r5c2 r6c3} - r1c3{n2 n1} - r2c2{n1 n3} ==> r5c2≠3 whip[1]: r5n3{c6 .} ==> r4c5≠3 biv-chain[4]: b4n3{r6c3 r4c1} - b1n3{r3c1 r2c2} - b1n1{r2c2 r1c3} - r8c3{n1 n4} ==> r6c3≠4 hidden-single-in-a-block ==> r6c1=4 biv-chain[3]: r6n1{c4 c8} - r6n3{c8 c3} - r4c1{n3 n1} ==> r4c5≠1 S2 to the end.

Note that the tridagon has been used only to select the replacement cells and digits.
.

by **eleven** » Tue Oct 15, 2024 11:53 am

Same as yzfwsf, adding a proof for the impossible pattern:

Code: Select all: *----------------------------------- | . . 123 | 123 . . | | . 123 . | . 123 . | | 123 . . | . . 123 | |-----------------+----------------+ | 123 . . | . . . | | . 123 . | . . 123 | | 123 . . | . . . | |-----------------+----------------+ | . . . | 123 . . | | . . 123 | . 123 123 | | . . . | . . . | *-----------------------------------

Fixing c9 arbitrarely, here to 213, immediadetly leads to

Code: Select all: *----------------------------------- | . . ba12 | 13 . . | | . b12 . | . 13 . | | 3 . . | . . 2 | |-----------------+----------------+ | 12 . . | . . . | | . 3 . | . . 1 | | 12 . . | . . . | |-----------------+----------------+ | . . . |a12 . . | | . . ba12 | . ba12 3 | | . . . | . . . | *-----------------------------------

The kites a and b kill both 1's in box 3

So after basics (quads, triple) we have:

Code: Select all: +-----------------------+-----------------------+------------------------+ | 9 4 *123 | *123 8 7 | 136 5 236 | | 5 *123 7 | 6 *123 9 | 134 8 234 | | *123 6 8 | 5 4 *123 | 137 1239 2379 | +-----------------------+-----------------------+------------------------+ | *123 7 6 | 8 *123+5 4 | 135 1239 239 | | 8 *123 5 | 9 12367 *123 | 13467 1234 23467 | | 1234 9 *123+4 | *123 567 56 | 5678 123 678 | +-----------------------+-----------------------+------------------------+ | 1346 135 1349 | 134 569 568 | 2 7 348 | | 7 8 134 | 1234 123 123 | 9 6 5 | | 2346 235 2349 | 7 569 568 | 348 34 1 | +-----------------------+-----------------------+------------------------+

The tridagon 123 has externals 4r6c4 and 5r4c5

Code: Select all: +-----------------------+-----------------------+------------------------+ | 9 4 *123 | *123 8 7 | 136 5 236 | | 5 *123 7 | 6 *123 9 | 134 8 234 | | *123 6 8 | 5 4 *123 | 137 1239 2379 | +-----------------------+-----------------------+------------------------+ | *123 7 6 | 8 1235 4 | 135 1239 239 | | 8 *123 5 | 9 12367 *123 | 13467 1234 23467 | | *123+4 9 1234 | 123 567 56 | 5678 123 678 | +-----------------------+-----------------------+------------------------+ | 136-4 135 1349 | *13+4 569 568 | 2 7 348 | | 7 8 *13+4 | 1234 *123 *123 | 9 6 5 | | 2346 235 2349 | 7 569 568 | 348 34 1 | +-----------------------+-----------------------+------------------------+

The above pattern has the externals 4r6c1, r7c4, r8c3
All force -4r6c3 => (tridagon) 5r4c5, bte

by **Cenoman** » Tue Oct 15, 2024 3:41 pm

Using tridagon pattern with remote triple sub-pattern:

Code: Select all: +----------------------+-----------------------+-------------------------+ | 9 4 123* | 123* 8 7 | 136 5 236 | | 5 123* 7 | 6 123* 9 | 134 8 234 | | 123* 6 8 | 5 4 123* | 137 1239 2379 | +----------------------+-----------------------+-------------------------+ | 123* 7 6 | 8 1235* 4 | 135 1239 239 | | 8 123* 5 | 9 12367 123* | 13467 1234 23467 | | 1234 9 1234* | 123* 567 56 | 5678 123 678 | +----------------------+-----------------------+-------------------------+ | 1346 135 1349 | 134 569 568 | 2 7 348 | | 7 8 134 | 1234 123 123 | 9 6 5 | | 2346 235 2349 | 7 569 568 | 348 34 1 | +----------------------+-----------------------+-------------------------+

Tridagon (123)b1245 (*) having two guardians (5r4c5, 4r6c3)
r4c5 <> 5 => +4 r6c3 (single remaining guardian for the tridagon); Remote Triple (123)r1c34, r6c4 =>
(1)r1c3 == r16c4 - r7c4 = r8c456 }
(3)r1c3 == r16c4 - r7c4 = r8c456 } => -13 r7c3 (+4 r7c3): contradiction, so +5 r4c5; lclste

by **shye** » Tue Oct 15, 2024 4:43 pm

i had something similar to yzf & eleven:

Code: Select all: ,-----------------,------------------,--------------------, | 9 4 T123 |T123 8 7 | 136 5 236 | | 5 t123 7 | 6 t123 9 | 134 8 234 | |t123 6 8 | 5 4 t123 | 137 1239 2379 | :-----------------+------------------+--------------------: |t123 7 6 | 8 t123+5 4 | 135 1239 239 | | 8 t123 5 | 9 12367 t123 | 13467 1234 23467 | | 1234 9 T1234 |T123 567 56 | 5678 123 678 | :-----------------+------------------+--------------------: | 1346 135 1349 |X134 569 568 | 2 7 348 | | 7 8 X134 | 1234 123 123 | 9 6 5 | | 2346 235 2349 | 7 569 568 | 348 34 1 | '-----------------'------------------'--------------------'

T-marked cells form a tridagon, two internal guardians 5r3c5 & 4r6c3
r7c4 (X) in r8 is in c3
if X is 4 one guardian is eliminated and the other (5r3c5) must be true
if X is a tridagon candidate (123) it is removed from the uppercase T cells, and cannot be placed 4 times in the lowercase t cells
either case, +5r3c5 btte

XSudo input: Show

(XSudo finds eliminations for r6c1 using this pattern, which i found interesting but didnt see a nice way to prove it, if someone else does do let me know

)

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