The New Sudoku Players' Forum

by **mith** » Thu Nov 10, 2022 11:28 pm

Code: Select all: +-------+-------+-------+ | . . . | . . . | . . 1 | | . . . | . . . | . 2 3 | | . . . | . 4 5 | . . . | +-------+-------+-------+ | . . 4 | 6 7 . | . . 8 | | . 6 . | . 5 8 | . . . | | 8 . . | 9 . 4 | . 6 . | +-------+-------+-------+ | . 8 6 | . . 9 | 7 . . | | 4 5 . | . 6 7 | 8 . . | | 9 . 7 | . . . | . . . | +-------+-------+-------+ ........1.......23....45.....467...8.6..58...8..9.4.6..86..97..45..678..9.7......

Wasn't planning to post this here, but marek found a gorgeous way through it.

by **totuan** » Fri Nov 11, 2022 1:31 pm

Code: Select all: *-----------------------------------------------------------------------------* | 23567 23479 23589 | 2378 239 236 | 4569 45789 1 | | 1567 1479 1589 | 178 19 16 | 4569 2 3 | | 12367 12379 12389 | 12378 4 5 | 69 789 679 | |-------------------------+-------------------------+-------------------------| |*1235A #1239B 4 | 6 7 &123 | 12359 1359 8 | |#1237B 6 *1239A |&123 5 8 | 12349 13479 2479 | | 8 *1237A #1235B | 9 &123 4 | 1235 6 257 | |-------------------------+-------------------------+-------------------------| |&123 8 6 | 45 &123 9 | 7 1345 245 | | 4 5 &123 |&123 6 7 | 8 139 29 | | 9 &123 7 | 45 8 &123 | 123456 1345 2456 | *-----------------------------------------------------------------------------*

Look at tridagon (123)B4578 and box 4 then see that two tridagon, one (*) and other (#) – call A & B.
Guardians 5’s, 7’s, 9’s on tridagon A & B can not all in A or B => one tridagon A or B contains two of (5,7,9) and other contains the rest.

The present quite complex, but I see that: Look at 5’s and 7’s then have 4 cases
1. r4c1=5 & r5c1=7 => impossible
2. r5c1=7 & r6c3=5 => r5c3=9 => impossible
3. r4c1=5 & r6c2=7 => r4c2=9 => impossible
4. r6c2=7 & r6c3=5 then the puzzle collapse

I’ll study more…

Thanks for nice puzzle!
totuan

by **eleven** » Fri Nov 11, 2022 2:26 pm

Two moves, but then i'm stuck:

Code: Select all: +-------------------------+-------------------------+-------------------------+ | 23567 23479 23589 | 2378 239 236 | 4569 45789 1 | | 1567 1479 1589 | 178 19 16 | 4569 2 3 | |*123+67 *123+79 *123+89 |a123-78 4 5 |a69 a789 a679 | +-------------------------+-------------------------+-------------------------+ |A1235 B1239 4 | 6 7 *123 | 12359 1359 8 | |B1237 6 A1239 |*123 5 8 | 12349 13479 2479 | | 8 A1237 B1235 | 9 *123 4 | 1235 6 257 | +-------------------------+-------------------------+-------------------------+ |*123 8 6 | 12345 *123 9 | 7 1345 245 | | 4 5 *123 |*123 6 7 | 8 139 29 | | 9 *123 7 | 12345 8 *123 | 123456 1345 2456 | +-------------------------+-------------------------+-------------------------+

Note that this pattern (with 123 in r3c123) is impossible. Whatever extra-candidate you set in b4, you get triples 123, which set the other 2 of one of the 2 tridagons - and you are left with the other tridagon.
=> -78r3c4 (would build a quad 6789 in r3c4789)

Code: Select all: +-------------------------+-------------------------+-------------------------+ | 23567 23479 23589 | 78 239 236 |a4569 45789 1 | | 1567 1479 1589 | 78 19 16 |a4569 2 3 | | 12367 12379 12389 | 123 4 5 |a69 789 679 | +-------------------------+-------------------------+-------------------------+ |A1235 B1239 4 | 6 7 *123 |*123+59 1359 8 | |B1237 6 A1239 |*123 5 8 |*123+49 13479 2479 | | 8 A1237 B1235 | 9 *123 4 |*123+5 6 257 | +-------------------------+-------------------------+-------------------------+ |*123 8 6 | 45 *123 9 | 7 1345 245 | | 4 5 *123 |*123 6 7 | 8 139 29 | | 9 *123 7 | 45 8 *123 |a123-456 1345 2456 | +-------------------------+-------------------------+-------------------------+

Same way this pattern is impossible => -456r9c7

[Added:] After UR 45 i just found one unusual step.

Code: Select all: +----------------------+----------------------+----------------------+ | 23567 23479 23589 | 78 239 236 | 4569 45789 1 | | 1567 1479 1589 | 78 19 16 | 4569 2 3 | | 12367 12379 12389 | 123 4 5 | 69 789 79 | +----------------------+----------------------+----------------------+ | 1235 1239 4 | 6 7 *123 |*123+59 1359 8 | | 1237 6 *123+9 |*123 5 8 | 12349 13479 2479 | | 8 1237 *123+5 | 9 *123 4 |*123+5 6 257 | +----------------------+----------------------+----------------------+ |*123 8 6 | 45 *123 9 | 7 13 45 | | 4 5 *123 |*123 6 7 | 8 139 29 | | 9 *123 7 | 45 8 *123 |*123 45 6 | +----------------------+----------------------+----------------------*

One more impossible pattern 123 (*), see here, extra candidates in r56c3,r46c7.
(5|9)r56c3,r4c7 == *5r6c7 - (5=469)r123c7 - (9=7)r3c9 - (*5|7=29)r68c9 - (9=13)r78c8 - (1|3=59)r4c8 => -5&9r4c12

Two more chains:
(5|7)b6p12456 = 57r6c79 - (5=469)r123c7 - (9=7)r3c9 - r6c9 = 7r6c2 => -5r4c1 & 7r5c1 => -5&7r45c1
=> -5r4c1
(9|7)b6p129 = 97b6p45 - 9r3c7,7r13c8 = 67r3c79 - (6|7=123)r347c1 - (1|2|3=7)r5c1 => -9&7r46c2 => -9r5c2
But the last one (-7r5c1) i don't want to post.

by **totuan** » Mon Nov 14, 2022 8:43 pm

eleven wrote:One more impossible pattern 123 (*), see here.

Very nice pattern!

Code: Select all: *-----------------------------------------------------------------------------* | 23567 23479 23589 | 2378 239 236 | 4569 45789 1 | | 1567 1479 1589 | 178 19 16 | 4569 2 3 | | 12367 12379 12389 | 12378 4 5 | 69 789 679 | |-------------------------+-------------------------+-------------------------| |A1235 B1239* 4 | 6 7 &123 | 12359 1359 8 | |B1237 6 A1239 |&123 5 8 |*12349 13479 2479 | | 8 A1237* B1235 | 9 &123 4 |*1235 6 25-7 | |-------------------------+-------------------------+-------------------------| |&123 8 6 | 45 &123 9 | 7 1345 245 | | 4 5 &123 |&123 6 7 | 8 139 29 | | 9 &123 7 | 45 8 &123 |*123456 1345 2456 | *-----------------------------------------------------------------------------*

My path based on eleven’s impossible pattern, not nice

Edit: I'll check my path based on eleven's explanation.

Again, thanks for the puzzle!
totuan

by **eleven** » Tue Nov 15, 2022 12:47 am

You have to be very careful with that pattern.
To check it, look at the rectangles. In the column with the 2 cells in a mini-column, there must be a rectangle with 2 of the 3 cells and 2 cells of a column of the other 2 boxes.
Say a is part of the rectangle, b not.
In the column outside one cell must be in the line of b, the other 2 cells must form a rectangle with another column.

Same with patterns in 2 stacks (2 cells in a minirow etc).

Examples:

Code: Select all: -----------------------+-----------------+----------------| | . a123 . | . . a123 | - . . | | . . . | c123 . . | c123 . . | | . b123 . | . 123 . | b123 . . | |----------------------+-----------------+----------------| | 123 . . | . 123 . | . . . | | . . 123 | c123 . . | c123 . . | | . a123 . | . . a123 | - . . | *---------------------------------------------------------*

Code: Select all: -----------------------+-----------------+----------------| | . b123 . | . . 123 | b123 . . | | . a123 . | a123 . . | . . . | | . . . | . c123 . | c123 . . | |----------------------+-----------------+----------------| | 123 . . | . c123 . | c123 . . | | . a123 . | a123 . . | . . . | | . . 123 | . . 123 | . . . | *---------------------------------------------------------*

Code: Select all: -----------------------+-----------------+----------------| | . . . | . . c123 | c123 . . | | . b123 . | 123 . . | b123 . . | | . a123 . | . a123 . | . . . | |----------------------+-----------------+----------------| | 123 . . | 123 . . | . . . | | . a123 . | . a123 . | . . . | | . . 123 | . . c123 | c123 . . | *---------------------------------------------------------*

by **totuan** » Tue Nov 15, 2022 2:17 am

eleven wrote:You have to be very careful with that pattern.

Yes, thank you for your explanation. I'll check my path… :oops:

totuan

by **totuan** » Tue Nov 15, 2022 2:53 pm

My ugly path, used two first moves from eleven’s path. Don't know how marek solve it easy

Code: Select all: *-----------------------------------------------------------------------------* | 23567 23479 23589 | 2378 239 236 |a4569 45789 1 | | 1567 1479 1589 | 178 19 16 |a4569 2 3 | |#12367 #12379 #12389 |#123-78 4 5 |%69 b789 b679 | |-------------------------+-------------------------+-------------------------| |A1235 B1239 4 | 6 7 &123 |*12359 1359 8 | |B1237 6 A1239 |&123 5 8 |*12349 13479 2479 | | 8 A1237 B1235 | 9 &123 4 |*1235 6 25-7 | |-------------------------+-------------------------+-------------------------| |&123 8 6 | 45 &123 9 | 7 1345 245 | | 4 5 &123 |&123 6 7 | 8 139 29 | | 9 &123 7 | 45 8 &123 |*123-456 1345 2456 | *-----------------------------------------------------------------------------*

01 & 02: eleven’s path => r3c4<>78, r9c7<>456, r9c9=6

Code: Select all: *--------------------------------------------------------------------* | 23567 23479 &23589 | 78 239 236 | 4569 45789 1 | | 1567 1479 &1589 | 78 19 16 | 4569 2 3 | |%12367 @12379 %12389 |%123 4 5 |#69 #789 #79 | |----------------------+----------------------+----------------------| |A1235 %1239 4 | 6 7 123 | 12359 1359 8 | | 1237 6 A1239 | 123 5 8 | 12349 13479 2479 | | 8 A1237 *1235 | 9 123 4 |*1235 6 *25-7 | |----------------------+----------------------+----------------------| | 123 8 6 | 45 123 9 | 7 1345 245 | | 4 5 123 | 123 6 7 | 8 139 29 | | 9 123 7 | 45 8 123 | 123 45 6 | *--------------------------------------------------------------------*

03: Present as diagram: => r6c9<>7, r6c2=7
- Tridagon or TH-A (123) => (5)r4c1=(7)r6c2=(9)r5c3
- eleven’s impossible pattern (123)r3c134 & r4c12 => (59)r4c12/r3c3=(8)r3c3=(67)r3c1

Code: Select all: (5)r6c9* eleven’s impossible pattern || || || TH-A(123)B4578 -----------------------------------(59)r4c12/r3c3 || || | || || (7)r6c2* | AAALS(12356789)r3c12/r1c3 (6)r3c1-(69=7)r3c9* || || | || || (5)r6c3--(5)r4c1 | (8)r1c3----------------------(8)r3c3 || || | || | || || (59)r56c3----------(59) | (7)r3c1-r5c1=r6c2* || | || | || | (679)r3c12-(679=8)r3c789--| || | || | || | (123)r3c12/r1c3—(1)r2c3 | || | || | || | (8)r2c3--- || | || || ----------------------(59)r2c3 || (5)r6c7-(5=469)r123c7-(69=7)r3c9*

Code: Select all: *--------------------------------------------------------------------* | 23567 2349 23589 | 78 239 236 | 4569 45789 1 | | 1567 149 1589 | 78 19 16 | 4569 2 3 | | 12367 1239 12389 | 123 4 5 | 69 789 79 | |----------------------+----------------------+----------------------| | 1235 #1239 4 | 6 7 123 |*12359 1359 8 | | 123 6 *1239 | 123 5 8 | 12349 13479 2479 | | 8 7 #1235 | 9 123 4 |*1235 6 25 | |----------------------+----------------------+----------------------| | 123 8 6 | 45 123 9 | 7 1345 245 | | 4 5 123 | 123 6 7 | 8 139 29 | | 9 123 7 | 45 8 123 |*123 45 6 | *--------------------------------------------------------------------*

- Tridagon (123)B4578 => (5)r6c3=(9)r4c2
- eleven’s impossible pattern (123)r56c3 & r469c7 => (59)r46c7/r5c3=(5)r6c3

04: (5)r6c3=(9)r4c2-(9*=123)r5c134-(123=4569)r1235c7-(59)r46c7/r5c3*==(5)r6c3 => r6c3=5, some singles

Code: Select all: *--------------------------------------------------* | 7 49 23 | 8 239 236 | 46 5 1 | | 5 49 8 | 7 19 16 | 46 2 3 | | 6 123 123 | 23 4 5 | 9 8 7 | |----------------+----------------+----------------| | 123 123 4 | 6 7 123 | 5 9 8 | |b123 6 9 |a123 5 8 | 13 7 4 | | 8 7 5 | 9 13 4 | 13 6 2 | |----------------+----------------+----------------| |c123 8 6 | 4 123 9 | 7 13 5 | | 4 5 d123 | 13-2 6 7 | 8 13 9 | | 9 13 7 | 5 8 13 | 2 4 6 | *--------------------------------------------------*

05: 2’s r5c4=r5c1-r7c1=r8c3 => r8c4<>2, stte

totuan

by **marek stefanik** » Tue Nov 15, 2022 5:43 pm

Interesting solutions from both of you.

My solution, similarly to totuan's first post, was based on eliminating 57 combinations in b4.
Note that both 9s in b4 are in the rectangles of their respective THs.

Code: Select all: .---------------------.-------------------.---------------------. | 23567 23479 23589 | 2378 2389 236 | 4569 45789 1 | | 1567 1479 1589 | 178 189 16 | 4569 2 3 | | 12367 12379 12389 | 12378 4 5 | 69 789 679 | :---------------------+-------------------+---------------------: | 1235 #123+9 4 | 6 7 A#123 |#12359 #1359 8 | |#123+7 6 1239 |#123 5 8 |#12349 13479 2479 | | 8 1237 #123+5 | 9 #123 4 |#1235 6 257 | :---------------------+-------------------+---------------------: |#123 8 6 | 12345 #123 9 | 7 1345 245 | | 4 5 #123 |#123 6 7 | 8 139 29 | | 9 A#123 7 | 123458 1238 A#123 |#123456 1345 2456 | '---------------------'-------------------'---------------------'

Suppose 9r4c2 is the only guardian of its TH.
RTs 123A..., 13c7A \ r49b6 => -13r4c8
Hence 5r4c8 and 5b4 is forced onto the TH, ie. contra.
Therefore 5r6c3 = 7r5c1.

Code: Select all: .---------------------.-------------------.---------------------. | 23567 23479 #23589 | 2378 2389 236 | 4569 45789 1 | | 1567 1479 #1589 | 178 189 16 | 4569 2 3 | |#12367 #12379 #12389 |#12378 4 5 | 69 789 679 | :---------------------+-------------------+---------------------: |#1235 1239 4 | 6 7 #123 | 12359 1359 8 | | 1237 6 #1239 |A#123 5 8 | 12349 13479 2479 | | 8 #1237 1235 | 9 #123 4 | 1235 6 257 | :---------------------+-------------------+---------------------: |#123 8 6 | 12345 #123 9 | 7 1345 245 | | 4 5 A#123 |A#123 6 7 | 8 139 29 | | 9 #123 7 | 123458 1238 #123 | 123456 1345 2456 | '---------------------'-------------------'---------------------'

Suppose 9r5c3 is the only guardian of its TH.
RTs 123A..., 123r3A \ c34b1 => -123r12c3
Hence 58r12c3 and 5b4 is forced onto the TH, ie. contra.
Therefore 5r4c1 = 7r6c2.

Therefore 57r45c1 or 57r6c23.

Code: Select all: .---------------------.-------------------.---------------------. | 23567 23479 23589 | 2378 2389 236 |c4569 45789 1 | | 1567 1479 1589 | 178 189 16 |c4569 2 3 | | 12367 12379 12389 | 12378 4 5 |c69 789 c679 | :---------------------+-------------------+---------------------: | 1235 1239 4 | 6 7 123 | 12359 1359 8 | | 1237 6 1239 | 123 5 8 | 12349 13479 2479 | | 8 a1237 a1235 | 9 123 4 |b1235 6 b257 | :---------------------+-------------------+---------------------: | 123 8 6 | 12345 123 9 | 7 1345 245 | | 4 5 123 | 123 6 7 | 8 139 29 | | 9 123 7 | 123458 1238 123 | 123456 1345 2456 | '---------------------'-------------------'---------------------'

(5|7)r6c23 = 57r6c79 – (5|7=469.)b3p1479 => (5|7)r6c23 => 57r6c23
Solves with a skyscaper.

The first two steps found by eleven are really nice.
I found a few more eliminations with the 123 ...quadri? tetra? -locals in r3 and c7.

Code: Select all: 2r6c9 – 123r459c7 = (1|2|3)r6c7 5r6c9 7r6c9 – (7=4569)b1p1479 => –5r6c7

Similarly you can eliminate 123r12c1, further reducing the puzzle to 9.8 skfr.

Marek

by **eleven** » Wed Nov 16, 2022 7:35 am

Ah, very nice !
And thanks for showing me, how i could finish my path easily with the (impossible) remote triple from 9r5c3

Website · by **denis_berthier** » Wed Nov 16, 2022 9:52 am

.
I had missed this puzzle - a tough one.
Applying W12+OR5W12 leaves it in T&E(2).

The solution I'll propose here will use both eleven replacement and anti-tridagon rules.
it will rely on the following observation:

If an ORk-relation based on an anti-tridagon pattern with k guardians is found and eleven replacement is later applied in some of the 4 blocks to the 3 digits of the anti-tridagon, then the ORk-relation remains valid in the modified puzzle.

Proof: obvious. We only have to consider the 12 cells of the pattern. In any case, eleven replacement doesn't change anything to the digits not in the pattern and when applying it in some of the 4 blocks to the 3 digits of the anti-tridagon, it doesn't change anything to the anti-tridagon pattern (It may even re-add digits of the pattern that were eliminated before replacement, thus restoring a non-degenerated anti-tridagon - except of course the 3 that are fixed by replacement)

We have two OR3-anti-tridagons in the same blocks at the start:

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 23567 23479 23589 ! 2378 2389 236 ! 4569 45789 1 ! ! 1567 1479 1589 ! 178 189 16 ! 4569 2 3 ! ! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 ! +----------------------+----------------------+----------------------+ ! 1235 1239 4 ! 6 7 123 ! 12359 1359 8 ! ! 1237 6 1239 ! 123 5 8 ! 12349 13479 2479 ! ! 8 1237 1235 ! 9 123 4 ! 1235 6 257 ! +----------------------+----------------------+----------------------+ ! 123 8 6 ! 12345 123 9 ! 7 1345 245 ! ! 4 5 123 ! 123 6 7 ! 8 139 29 ! ! 9 123 7 ! 123458 1238 123 ! 123456 1345 2456 ! +----------------------+----------------------+----------------------+ 209 candidates. hidden-pairs-in-a-column: c4{n4 n5}{r7 r9} ==> r9c4≠8, r9c4≠3, r9c4≠2, r9c4≠1, r7c4≠3, r7c4≠2, r7c4≠1 hidden-single-in-a-block ==> r9c5=8 +----------------------+----------------------+----------------------+ ! 23567 23479 23589 ! 2378 239 236 ! 4569 45789 1 ! ! 1567 1479 1589 ! 178 19 16 ! 4569 2 3 ! ! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 ! +----------------------+----------------------+----------------------+ ! 1235 1239 4 ! 6 7 123 ! 12359 1359 8 ! ! 1237 6 1239 ! 123 5 8 ! 12349 13479 2479 ! ! 8 1237 1235 ! 9 123 4 ! 1235 6 257 ! +----------------------+----------------------+----------------------+ ! 123 8 6 ! 45 123 9 ! 7 1345 245 ! ! 4 5 123 ! 123 6 7 ! 8 139 29 ! ! 9 123 7 ! 45 8 123 ! 123456 1345 2456 ! +----------------------+----------------------+----------------------+ OR3-anti-tridagon[12] for digits 1, 2 and 3 in blocks: b4, with cells: r4c2, r5c1, r6c3 b5, with cells: r4c6, r5c4, r6c5 b7, with cells: r9c2, r7c1, r8c3 b8, with cells: r9c6, r7c5, r8c4 with 3 guardians: n9r4c2 n7r5c1 n5r6c3 OR3-anti-tridagon[12] for digits 1, 2 and 3 in blocks: b4, with cells: r4c1, r5c3, r6c2 b5, with cells: r4c6, r5c4, r6c5 b7, with cells: r7c1, r8c3, r9c2 b8, with cells: r7c5, r8c4, r9c6 with 3 guardians: n5r4c1 n9r5c3 n7r6c2

Based on the first anti-tridagon:
Trid-OR3-whip[7]: b3n5{r2c7 r1c8} - c8n8{r1 r3} - c8n7{r3 r5} - OR3{{n7r5c1 n5r6c3 | n9r4c2}} - c8n9{r4 r8} - r8c9{n9 n2} - r6c9{n2 .} ==> r6c7≠5
Trid-OR3-ctr-whip[8]: c7n2{r6 r9} - r9n6{c7 c9} - c9n4{r9 r7} - c9n5{r7 r6} - b6n7{r6c9 r5c8} - r5n4{c8 c7} - r5n9{c7 c3} - OR3{{n5r6c3 n7r5c1 n9r4c2 | .}} ==> r5c9≠2

Code: Select all: ***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE FOR GENERAL TRIDAGON in resolution state: ***** RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens. +----------------------+----------------------+----------------------+ ! 23567 23479 23589 ! 2378 239 236 ! 4569 45789 1 ! ! 1567 1479 1589 ! 178 19 16 ! 4569 2 3 ! ! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 ! +----------------------+----------------------+----------------------+ ! 1235 1239 4 ! 6 7 123 ! 12359 1359 8 ! ! 1237 6 1239 ! 123 5 8 ! 12349 13479 479 ! ! 8 1237 1235 ! 9 123 4 ! 123 6 257 ! +----------------------+----------------------+----------------------+ ! 123 8 6 ! 45 123 9 ! 7 1345 245 ! ! 4 5 123 ! 123 6 7 ! 8 139 29 ! ! 9 123 7 ! 45 8 123 ! 123456 1345 2456 ! +----------------------+----------------------+----------------------+ Trying in block 8 +----------------------+----------------------+----------------------+ ! 123567 123479 123589 ! 12378 1239 1236 ! 4569 45789 123 ! ! 123567 123479 123589 ! 12378 1239 1236 ! 4569 123 123 ! ! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 ! +----------------------+----------------------+----------------------+ ! 1235 1239 4 ! 6 7 123 ! 12359 12359 8 ! ! 1237 6 1239 ! 123 5 8 ! 12349 123479 479 ! ! 8 1237 1235 ! 9 123 4 ! 123 6 12357 ! +----------------------+----------------------+----------------------+ ! 123 8 6 ! 45 3 9 ! 7 12345 12345 ! ! 4 5 123 ! 2 6 7 ! 8 1239 1239 ! ! 9 123 7 ! 45 8 1 ! 123456 12345 123456 ! +----------------------+----------------------+----------------------+ whip[1]: r3n2{c1 .} ==> r2c3≠2, r2c2≠2, r2c1≠2, r1c3≠2, r1c2≠2, r1c1≠2 whip[1]: c7n1{r4 .} ==> r4c8≠1, r5c8≠1, r6c9≠1 whip[7]: r9c2{n3 n2} - r7c1{n2 n1} - r4n1{c1 c7} - r4n9{c7 c8} - b4n9{r4c2 r5c3} - r5n1{c3 c4} - b5n3{r5c4 .} ==> r4c2≠3 whip[6]: b4n5{r6c3 r4c1} - b6n5{r4c7 r6c9} - r6n7{c9 c2} - r6n3{c2 c7} - r4n3{c8 c6} - b5n2{r4c6 .} ==> r6c3≠2

This is where the fun begins:
Based on the second anti-tridagon:
Trid-OR3-ctr-whip[6]: c3n2{r3 r5} - b4n9{r5c3 r4c2} - c2n1{r4 r6} - r6n7{c2 c9} - r6n5{c9 c3} - OR3{{n5r4c1 n9r5c3 n7r6c2 | .}} ==> r3c3≠1
Trid-OR3-whip[6]: r6n7{c2 c9} - r6n5{c9 c3} - OR3{{n5r4c1 n7r6c2 | n9r5c3}} - r4c2{n9 n2} - r4c6{n2 n3} - r4c1{n3 .} ==> r6c2≠1
whip[6]: r5c4{n3 n1} - r6c5{n1 n2} - r6c2{n2 n7} - r5c1{n7 n2} - b7n2{r7c1 r9c2} - b7n3{r9c2 .} ==> r5c3≠3
whip[6]: r4c6{n2 n3} - r5c4{n3 n1} - r5c3{n1 n9} - r4c2{n9 n1} - r3n1{c2 c1} - r7c1{n1 .} ==> r4c1≠2
z-chain[5]: r6n7{c2 c9} - r6n5{c9 c3} - r4c1{n5 n1} - b7n1{r7c1 r8c3} - b7n3{r8c3 .} ==> r6c2≠3
z-chain[3]: r6n3{c9 c3} - b7n3{r8c3 r9c2} - c7n3{r9 .} ==> r5c8≠3, r4c8≠3

Based on the fist anti-tridagon:
Trid-OR3-whip[5]: c9n5{r7 r6} - r6n7{c9 c2} - OR3{{n7r5c1 n5r6c3 | n9r4c2}} - r4c8{n9 n2} - c7n2{r4 .} ==> r9c7≠5
Trid-OR3-whip[6]: r6n3{c9 c3} - b4n5{r6c3 r4c1} - r4n1{c1 c2} - OR3{{n9r4c2 n5r6c3 | n7r5c1}} - b6n7{r5c8 r6c9} - r6n5{c9 .} ==> r4c7≠3

The rest is almost trivial:

Code: Select all: whip[6]: b4n3{r4c1 r6c3} - b6n3{r6c9 r5c7} - r5c4{n3 n1} - r3n1{c4 c2} - b4n1{r4c2 r4c1} - b4n5{r4c1 .} ==> r3c1≠3 whip[5]: r6c2{n7 n2} - r6c5{n2 n1} - r5c4{n1 n3} - r3n3{c4 c3} - c3n2{r3 .} ==> r3c2≠7 whip[6]: c3n2{r5 r3} - c1n2{r3 r7} - r9c2{n2 n3} - r3n3{c2 c4} - b5n3{r5c4 r4c6} - b5n2{r4c6 .} ==> r6c2≠2 naked-single ==> r6c2=7

Note that the two anti-tridagons come again into play:

Code: Select all: At least one candidate of a previous Trid-OR3-relation has just been eliminated. There remains a Trid-OR2-relation between candidates: n5r6c3 n9r4c2 +-------------------+-------------------+-------------------+ ! 13567 13479 13589 ! 1378 129 236 ! 4569 45789 123 ! ! 13567 13479 13589 ! 1378 129 236 ! 4569 123 123 ! ! 1267 1239 2389 ! 1378 4 5 ! 69 789 679 ! +-------------------+-------------------+-------------------+ ! 135 129 4 ! 6 7 23 ! 1259 259 8 ! ! 123 6 129 ! 13 5 8 ! 12349 2479 479 ! ! 8 7 135 ! 9 12 4 ! 123 6 235 ! +-------------------+-------------------+-------------------+ ! 12 8 6 ! 45 3 9 ! 7 1245 1245 ! ! 4 5 13 ! 2 6 7 ! 8 139 139 ! ! 9 23 7 ! 45 8 1 ! 2346 2345 23456 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous Trid-OR3-relation has just been eliminated. There remains a Trid-OR2-relation between candidates: n9r4c2 n5r6c3 +-------------------+-------------------+-------------------+ ! 13567 13479 13589 ! 1378 129 236 ! 4569 45789 123 ! ! 13567 13479 13589 ! 1378 129 236 ! 4569 123 123 ! ! 1267 1239 2389 ! 1378 4 5 ! 69 789 679 ! +-------------------+-------------------+-------------------+ ! 135 129 4 ! 6 7 23 ! 1259 259 8 ! ! 123 6 129 ! 13 5 8 ! 12349 2479 479 ! ! 8 7 135 ! 9 12 4 ! 123 6 235 ! +-------------------+-------------------+-------------------+ ! 12 8 6 ! 45 3 9 ! 7 1245 1245 ! ! 4 5 13 ! 2 6 7 ! 8 139 139 ! ! 9 23 7 ! 45 8 1 ! 2346 2345 23456 ! +-------------------+-------------------+-------------------+

Code: Select all: z-chain[4]: r3n1{c1 c4} - r5c4{n1 n3} - r5c1{n3 n2} - r7c1{n2 .} ==> r1c1≠1, r2c1≠1 t-whip[4]: c2n1{r1 r4} - b4n9{r4c2 r5c3} - b4n2{r5c3 r5c1} - r7c1{n2 .} ==> r3c1≠1 z-chain[4]: r3n1{c2 c4} - r3n3{c4 c3} - c3n2{r3 r5} - c3n9{r5 .} ==> r3c2≠9 z-chain[5]: b5n2{r4c6 r6c5} - b5n1{r6c5 r5c4} - r3n1{c4 c2} - c2n2{r3 r9} - c7n2{r9 .} ==> r4c8≠2

Trid-OR2-whip[2]: OR2{{n5r6c3 | n9r4c2}} - r4c8{n9 .} ==> r6c9≠5 (possibly not necessary)

Code: Select all: hidden-single-in-a-row ==> r6c3=5 whip[1]: r6n3{c9 .} ==> r5c7≠3 whip[1]: b4n3{r4c1 .} ==> r1c1≠3, r2c1≠3 whip[1]: c9n5{r9 .} ==> r9c8≠5, r7c8≠5 naked-triplets-in-a-column: c1{r4 r5 r7}{n1 n3 n2} ==> r3c1≠2 naked-triplets-in-a-row: r3{c1 c7 c9}{n7 n6 n9} ==> r3c8≠9, r3c8≠7, r3c4≠7, r3c3≠9 naked-single ==> r3c8=8 whip[1]: r3n9{c9 .} ==> r2c7≠9, r1c8≠9, r1c7≠9 naked-pairs-in-a-column: c4{r3 r5}{n1 n3} ==> r2c4≠3, r2c4≠1, r1c4≠3, r1c4≠1 naked-triplets-in-a-column: c9{r1 r2 r6}{n2 n1 n3} ==> r9c9≠3, r9c9≠2, r8c9≠3, r8c9≠1, r7c9≠2, r7c9≠1 singles ==> r8c9=9, r3c7=9 whip[1]: b9n1{r8c8 .} ==> r2c8≠1 naked-pairs-in-a-row: r7{c4 c9}{n4 n5} ==> r7c8≠4 naked-triplets-in-a-column: c8{r2 r7 r8}{n3 n2 n1} ==> r9c8≠3, r9c8≠2, r5c8≠2 singles ==> r9c8=4, r7c9=5, r7c4=4, r9c9=6, r3c9=7, r1c8=5, r4c8=9, r5c8=7, r3c1=6, r1c1=7, r1c4=8, r2c4=7, r2c1=5, r5c9=4, r9c4=5, r2c3=8, r5c3=9, r3c3=2, r4c7=5 whip[1]: r4n1{c1 .} ==> r5c1≠1 naked-pairs-in-a-block: b1{r1c3 r3c2}{n1 n3} ==> r2c2≠3, r2c2≠1, r1c2≠3, r1c2≠1 finned-x-wing-in-columns: n3{c3 c8}{r8 r1} ==> r1c9≠3 whip[1]: b3n3{r2c9 .} ==> r2c6≠3 finned-x-wing-in-rows: n2{r5 r7}{c1 c7} ==> r9c7≠2 stte

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Triple Double (SER 10.9, te3 ID 587982)

Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)

Re: Triple Double (SER 10.9, te3 ID 587982)