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Website · by **denis_berthier** » Tue Mar 07, 2023 4:34 pm

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Code: Select all: +-------+-------+-------+ ! 1 . . ! . 5 6 ! . . . ! ! . 5 7 ! . . 9 ! . . 6 ! ! . . . ! 7 2 . ! . . 5 ! +-------+-------+-------+ ! 2 . 9 ! . 6 5 ! 1 . . ! ! . 7 . ! . . . ! . . . ! ! . 1 6 ! . 7 . ! 5 . . ! +-------+-------+-------+ ! . . . ! 5 . 2 ! . 4 . ! ! . . . ! . . . ! 9 . 1 ! ! . . . ! 6 . 7 ! . 5 8 ! +-------+-------+-------+ 1...56....57..9..6...72...52.9.651...7........16.7.5.....5.2.4.......9.1...6.7.58;376;63161 SER = 11.0

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1 23489 2348 ! 348 5 6 ! 3478 3789 3479 ! ! 348 5 7 ! 1348 348 9 ! 2348 1238 6 ! ! 34689 34689 348 ! 7 2 1348 ! 348 1389 5 ! +----------------------+----------------------+----------------------+ ! 2 348 9 ! 348 6 5 ! 1 378 347 ! ! 3458 7 3458 ! 123489 348 1348 ! 3468 3689 2349 ! ! 348 1 6 ! 23489 7 348 ! 5 389 2349 ! +----------------------+----------------------+----------------------+ ! 36789 3689 138 ! 5 1389 2 ! 367 4 37 ! ! 345678 23468 23458 ! 348 348 348 ! 9 2367 1 ! ! 349 2349 1234 ! 6 1349 7 ! 23 5 8 ! +----------------------+----------------------+----------------------+ 196 candidates.

by **Leren** » Wed Mar 08, 2023 8:38 am

Withdrawn. Thanks to marek stefanik for the clarification. Leren

by **marek stefanik** » Wed Mar 08, 2023 10:48 am

I should have made it clearer in the other thread, the RTs exist if all other TH cells are restricted to the TH digits. This is not enough to prove them here.
See this placement of 3458b1245:

Code: Select all: .-----------------.---------------.-----------------. | 1 2 *3 |#4 5 6 | 7 389 349 | |#4 5 7 | 1 #8 9 | 2348 238 6 | | 6 9 #8 | 7 2 #3 | 348 1 5 | :-----------------+---------------+-----------------: | 2 #4 9 |#3 6 5 | 1 378 347 | |*8 7 #5 | 29 #4 1 | 348 6 29 | |#3 1 6 | 29 7 #8 | 5 389 2349 | :-----------------+---------------+-----------------: | 3789 38 138 | 5 19 2 | 6 4 37 | | 57 6 25 | 348 348 348 | 9 27 1 | | 349 34 1234 | 6 19 7 | 23 5 8 | '-----------------'---------------'-----------------'

If all #-marked cells were restricted to 348, you'd get the RTs, but here you have to eliminate 5r5c3 first.

This can be done for example using this impossible pattern:

Code: Select all: .-----------------.---------------.-----------------. | 1 2 348 | 348 5 6 | 7 389 349 | |#348 5 7 | 1 #348 9 |b#348+2 238 6 | | 6 9 #348 | 7 2 #348 |#348 1 5 | :-----------------+---------------+-----------------: | 2 #348 9 |#348 6 5 | 1 378 347 | |a#348+5 7 348–5| 29 #348 1 |#348 6 29 | |#348 1 6 | 29 7 #348 | 5 389 2349 | :-----------------+---------------+-----------------: | 3789 38 138 | 5 19 2 | 6 4 37 | | 7–5 6 e25 | 348 348 348 | 9 d27 1 | | 349 34 1234 | 6 19 7 |c23 5 8 | '-----------------'---------------'-----------------'

impossible pattern 348# => 2r2c7 = 5r5c1
5r5c1 = 2r2c7 – 2r9c7 = 2r8c8 – (2=5)r8c3 => –5r5c3, –5r8c1

Impossibility proof: Show

Now you can use the RTs:

Code: Select all: .--------------.---------------.---------------. | 1 2 348 |#348 5 6 | 7 389 349 | |#348 5 7 | 1 aA#348 9 | 2 3–8 6 | | 6 9 #348 | 7 2 #3–48|d48 1 5 | :--------------+---------------+---------------: | 2 #38 9 |#348 6 5 | 1 7 34 | | 5 7 bA#348 |29 bA#348 1 |c48 6 29 | |#348 1 6 | 29 7 #348 | 5 389 2349 | :--------------+---------------+---------------: | 38 38 1 | 5 9 2 | 6 4 7 | | 7 6 5 | 348 348 348 | 9 2 1 | | 9 4 2 | 6 1 7 | 3 5 8 | '--------------'---------------'---------------'

TH without a rectangle cell => RT 348A...
48: r2c5 = r5c35 – r5c7 = r3c7 => –8r2c8, –48r3c6
Combined with TH 348 b2p159 b3p357 b5p159 b6p348 => 9r1c9 = 9r6c8 => –9r1c8, –9r56c9 gives stte.

Marek

by **Cenoman** » Wed Mar 08, 2023 6:12 pm

Code: Select all: +------------------------+--------------------+-----------------------+ | 1 a2348* 2348 | 348* 5 6 | 3478 3789 3479 | | 348* 5 7 | 1 348* 9 | 2348 238 6 | | 69 69 348* | 7 2 348* | 348 1 5 | +------------------------+--------------------+-----------------------+ | 2 348* 9 | 348* 6 5 | 1 378 347 | | 3458 7 b3458* | 29 348* 1 | 3468 3689 2349 | | 348* 1 6 | 29 7 348* | 5 389 2349 | +------------------------+--------------------+-----------------------+ | 36789 3689 138 | 5 19 2 | 367 4 37 | | 567 6-2 c25 | 348 348 348 | 9 267 1 | | 349 2349 1234 | 6 19 7 | 23 5 8 | +------------------------+--------------------+-----------------------+

1. TH(348)b1245 having two guardians 2r1c2, 5r5c3
(2)r1c2 == (5)r5c3 - (5=2)r8c3 => -2 r89c2; lcls, 7 placements

Borrowing Marek's Impossible Pattern:

Code: Select all: +----------------------+--------------------+----------------------+ | 1 2 348 | 348 5 6 | 7 389 349 | | 348# 5 7 | 1 348# 9 | 2348# 238 6 | | 6 9 348# | 7 2 348# | 348# 1 5 | +----------------------+--------------------+----------------------+ | 2 348# 9 | 348# 6 5 | 1 378 347 | | 3458# 7 3458 | 29 348# 1 | 348# 6 29 | | 348# 1 6 | 29 7 348# | 5 389 2349 | +----------------------+--------------------+----------------------+ | 3789 38 138 | 5 19 2 | 6 4 37 | | 57 6 25 | 348 348 348 | 9 27 1 | | 349 34 1234 | 6 19 7 | 23 5 8 | +----------------------+--------------------+----------------------+

2. Impossible pattern in (#) cells, having two gardians: 2r2c7, 5r5c1
(5)r5c1 == (2)r2c7 - r9c7 = r8c8 - (2=5)r8c3 => -5 r5c3; 14 placements
This Impossible Pattern is #176 in eleven's 13 cells, 3digits, 6boxes IP

My initial solution

Hidden Text: Show

Code: Select all: +-------------------+--------------------+--------------------+ | 1 2 348 | 348 5 6 | 7 389 34+9 | | 348 5 7 | 1 348 9 | 2 38 6 | | 6 9 348 | 7 2 348 | 48 1 5 | +-------------------+--------------------+--------------------+ | 2 38 9 | 348 6 5 | 1 7 34 | | 5 7 348 | 29* 348 1 | 48 6 29* | | 348 1 6 | 29* 7 348 | 5 389 2349* | +-------------------+--------------------+--------------------+ | 38 38 1 | 5 9 2 | 6 4 7 | | 7 6 5 | 348 348 348 | 9 2 1 | | 9 4 2 | 6 1 7 | 3 5 8 | +-------------------+--------------------+--------------------+

4. UR(29)r56c49 using single external => +9 r1c9; lcls, 7 placements

Code: Select all: +-------------------+--------------------+-----------------+ | 1 2 c348 | 348 5 6 | 7 38 9 | | d34 5 7 | 1 3-48 9 | 2 38 6 | | 6 9 c38 | 7 2 38 | 4 1 5 | +-------------------+--------------------+-----------------+ | 2 38 9 | 348 6 5 | 1 7 34 | | 5 7 b34 | 9 a34 1 | 8 6 2 | | 348 1 6 | 2 7 348 | 5 9 34 | +-------------------+--------------------+-----------------+ | 38 38 1 | 5 9 2 | 6 4 7 | | 7 6 5 | 348 348 348 | 9 2 1 | | 9 4 2 | 6 1 7 | 3 5 8 | +-------------------+--------------------+-----------------+

5. remote pair (34): r5c5 = r5c3 - r13c3 = r2c1 => -34 r2c5; ste

Website · by **denis_berthier** » Thu Mar 09, 2023 4:47 pm

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hidden-pairs-in-a-column: c5{n1 n9}{r7 r9} ==> r9c5≠4, r9c5≠3, r7c5≠8, r7c5≠3
whip[1]: b8n3{r8c6 .} ==> r8c1≠3, r8c2≠3, r8c3≠3, r8c8≠3
whip[1]: b8n8{r8c6 .} ==> r8c1≠8, r8c2≠8, r8c3≠8
whip[1]: r9n4{c3 .} ==> r8c1≠4, r8c2≠4, r8c3≠4
hidden-pairs-in-a-column: c4{n2 n9}{r5 r6} ==> r6c4≠8, r6c4≠4, r6c4≠3, r5c4≠8, r5c4≠4, r5c4≠3, r5c4≠1
singles ==> r5c6=1, r2c4=1, r3c8=1
whip[1]: r3n9{c2 .} ==> r1c2≠9
hidden-pairs-in-a-block: b1{n6 n9}{r3c1 r3c2} ==> r3c2≠8, r3c2≠4, r3c2≠3, r3c1≠8, r3c1≠4, r3c1≠3

Code: Select all: OR2-anti-tridagon[12] for digits 4, 8 and 3 in blocks: b1, with cells (marked #): r1c2, r2c1, r3c3 b2, with cells (marked #): r1c4, r2c5, r3c6 b4, with cells (marked #): r4c2, r6c1, r5c3 b5, with cells (marked #): r4c4, r6c6, r5c5 with 2 guardians (in cells marked @): n2r1c2 n5r5c3 +----------------------+----------------------+----------------------+ ! 1 2348#@ 2348 ! 348# 5 6 ! 3478 3789 3479 ! ! 348# 5 7 ! 1 348# 9 ! 2348 238 6 ! ! 69 69 348# ! 7 2 348# ! 348 1 5 ! +----------------------+----------------------+----------------------+ ! 2 348# 9 ! 348# 6 5 ! 1 378 347 ! ! 3458 7 3458#@ ! 29 348# 1 ! 3468 3689 2349 ! ! 348# 1 6 ! 29 7 348# ! 5 389 2349 ! +----------------------+----------------------+----------------------+ ! 36789 3689 138 ! 5 19 2 ! 367 4 37 ! ! 567 26 25 ! 348 348 348 ! 9 267 1 ! ! 349 2349 1234 ! 6 19 7 ! 23 5 8 ! +----------------------+----------------------+----------------------+

Trid-OR2-whip[2]: OR2{{n2r1c2 | n5r5c3}} - r8c3{n5 .} ==> r9c2≠2, r8c2≠2
singles ==> r8c2=6, r3c2=9, r3c1=6, r7c7=6, r5c8=6, r1c7=7, r1c2=2
hidden-pairs-in-a-row: r5{n2 n9}{c4 c9} ==> r5c9≠4, r5c9≠3

We now have the same impossible pattern as in Cenoman and Marek's solutions: EL13c176(*) in eleven's list

Code: Select all: EL13c176-OR2-relation for digits: 3, 4 and 8 in cells (marked #): (r2c7 r2c1 r2c5 r3c7 r3c3 r3c6 r6c1 r6c6 r4c2 r4c4 r5c7 r5c1 r5c5) with 2 guardians (in cells marked @) : n2r2c7 n5r5c1 +----------------------+----------------------+----------------------+ ! 1 2 348 ! 348 5 6 ! 7 389 349 ! ! 348# 5 7 ! 1 348# 9 ! 2348#@ 238 6 ! ! 6 9 348# ! 7 2 348# ! 348# 1 5 ! +----------------------+----------------------+----------------------+ ! 2 348# 9 ! 348# 6 5 ! 1 378 347 ! ! 3458#@ 7 3458 ! 29 348# 1 ! 348# 6 29 ! ! 348# 1 6 ! 29 7 348# ! 5 389 2349 ! +----------------------+----------------------+----------------------+ ! 3789 38 138 ! 5 19 2 ! 6 4 37 ! ! 57 6 25 ! 348 348 348 ! 9 27 1 ! ! 349 34 1234 ! 6 19 7 ! 23 5 8 ! +----------------------+----------------------+----------------------+

biv-chain[3]: b7n1{r7c3 r9c3} - r9n2{c3 c7} - b9n3{r9c7 r7c9} ==> r7c3≠3
biv-chain[3]: b7n9{r9c1 r7c1} - r7n7{c1 c9} - b9n3{r7c9 r9c7} ==> r9c1≠3
EL13c176-OR2-whip[3]: OR2{{n2r2c7 | n5r5c1}} - r8c1{n5 n7} - r8c8{n7 .} ==> r2c8≠2

The end is trivial:

Code: Select all: singles ==> r2c7=2, r9c7=3, r7c9=7, r8c8=2, r8c3=5, r8c1=7, r9c2=4, r9c1=9, r9c5=1, r7c5=9, r9c3=2, r7c3=1, r5c1=5, r4c8=7 finned-x-wing-in-rows: n3{r5 r3}{c3 c5} ==> r2c5≠3 finned-x-wing-in-rows: n3{r3 r5}{c3 c6} ==> r6c6≠3 biv-chain[2]: r2n4{c5 c1} - b4n4{r6c1 r5c3} ==> r5c5≠4 biv-chain[2]: b3n4{r1c9 r3c7} - r5n4{c7 c3} ==> r1c3≠4 x-wing-in-columns: n4{c3 c7}{r3 r5} ==> r3c6≠4 x-wing-in-rows: n4{r1 r4}{c4 c9} ==> r8c4≠4, r6c9≠4 biv-chain[3]: r5n3{c5 c3} - b4n4{r5c3 r6c1} - b5n4{r6c6 r4c4} ==> r4c4≠3 hidden-single-in-a-block ==> r5c5=3 whip[1]: c3n3{r3 .} ==> r2c1≠3 hidden-single-in-a-row ==> r2c8=3 biv-chain[2]: c8n8{r1 r6} - b5n8{r6c6 r4c4} ==> r1c4≠8 finned-x-wing-in-rows: n8{r1 r5}{c3 c8} ==> r6c8≠8 stte

*EL13c176: 4 cells missing, 5 added (wrt tridagon)

Code: Select all: ...........X..X..X..X.X..X......X.X....X..X....X..X..X........................... isomorphic to: +-------+-------+-------+ ! . . Z ! . . X ! . . X ! ! . . . ! . - Z ! . X . ! ! . . . ! X . . ! X . . ! +-------+-------+-------+ ! . . Z ! - . Z ! . . X ! ! . . Z ! . X . ! . X . ! ! . . . ! . . - ! - . . ! +-------+-------+-------+ ! o o . ! . . . ! . . . ! ! o o . ! . . . ! . . . ! ! o o . ! . . . ! . . . ! +-------+-------+-------+

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