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Website · by **denis_berthier** » Sat Sep 10, 2022 8:04 am

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#25591 in mith's list of 63137 min-expands
Goal: find many tridagon-related direct eliminations.

Code: Select all: +-------+-------+-------+ ! 1 . 3 ! . . . ! . . . ! ! . 5 . ! 1 . 9 ! . . 6 ! ! 6 9 . ! 2 3 . ! . . . ! +-------+-------+-------+ ! 2 . . ! . . . ! 4 . 8 ! ! 3 6 . ! . . . ! 1 2 7 ! ! . . . ! . . 2 ! . 9 5 ! +-------+-------+-------+ ! 5 . 6 ! . 2 3 ! . . . ! ! . . 1 ! 6 9 . ! . 5 . ! ! 9 . . ! 5 . 1 ! . . . ! +-------+-------+-------+ 1.3.......5.1.9..669.23....2.....4.836....127.....2.955.6.23.....169..5.9..5.1...;5250;73296 SER = 11.0

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 1 2478 3 ! 478 45678 45678 ! 25789 478 249 ! ! 478 5 2478 ! 1 478 9 ! 2378 3478 6 ! ! 6 9 478 ! 2 3 4578 ! 578 1478 14 ! +-------------------+-------------------+-------------------+ ! 2 17 579 ! 379 1567 567 ! 4 36 8 ! ! 3 6 4589 ! 489 458 458 ! 1 2 7 ! ! 478 1478 478 ! 3478 14678 2 ! 36 9 5 ! +-------------------+-------------------+-------------------+ ! 5 478 6 ! 478 2 3 ! 789 1478 149 ! ! 478 23478 1 ! 6 9 478 ! 278 5 234 ! ! 9 23478 2478 ! 5 478 1 ! 2678 4678 234 ! +-------------------+-------------------+-------------------+ 169 candidates

by **shye** » Sat Sep 10, 2022 11:04 am

Code: Select all: .------------------.---------------------.-------------------. | 1 #2-478 3 |#*478 *45678 *45678 | 25789 *478 249 | |#478 5 2478 | 1 #478 9 | 2378 3478 6 | | 6 9 #478 | 2 3 #4578 | 578 1478 14 | :------------------+---------------------+-------------------: | 2 17 579 | 379 1567 567 | 4 36 8 | | 3 6 4589 | 489 458 458 | 1 2 7 | | 478 1478 478 | 3478 14678 2 | 36 9 5 | :------------------+---------------------+-------------------: | 5 #478 6 | #478 2 3 | 789 1478 149 | |#478 *23478 1 | 6 9 #478 | 2378 5 234 | | 9 *23478 #2478 | 5 #478 1 | 23678 34678 234 | '------------------'---------------------'-------------------'

kraken trivalue oddagon
||2r1c2
||2r9c3 - 2r89c2 = 2r1c2
||5r3c6 - (5=4678)r1c4568
=> -478r1c2 stte

by **totuan** » Sat Sep 10, 2022 1:21 pm

My path for this one - like shye’s with a bit change on presenting

Tridagon (478)B1278 => (2)r1c2/r9c3=(5)r3c6
(2)r1c2/r9c3==(5)r3c6-r3c7=(5-29)r1c79=(2)r2c7 => r2c3<>2, stte

Thanks for the puzzle!
totuan

by **Cenoman** » Sat Sep 10, 2022 9:06 pm

After lcls,

Code: Select all: +-----------------------+------------------------+-----------------------+ | 1 2478 3 | 478 45678 45678 | 25789 478 249 | | 478 5 2478 | 1 478 9 | 2378 3478 6 | | 6 9 478 | 2 3 4578 | 578 1478 14 | +-----------------------+------------------------+-----------------------+ | 2 17 59 | 379 1567 567 | 4 36 8 | | 3 6 59 | 489 458 458 | 1 2 7 | | 478 1478 478 | 37 167 2 | 36 9 5 | +-----------------------+------------------------+-----------------------+ | 5 478 6 | 478 2 3 | 789 1478 149 | | 478 23478 1 | 6 9 478 | 278 5 234 | | 9 23478 2478 | 5 478 1 | 2678 4678 234 | +-----------------------+------------------------+-----------------------+

(162 candidates)
I had found the same one-step as totuan's, i.e. one direct elimination, out of a total of 113 eliminations.
Not all are direct eliminations. According to my solver, 66 candidates (no guarantee about the accuracy of this count) are in a derived weak link to the three guardians of TH(478)b1278: 2r1c2, 2r9c3, 5r3c6 (no guarantee about the accuracy of this count).
So if they are eliminated, one by one, without clicking the "Find singles" button, the corresponding elimination state could be seen. So what ???
Oh! yes, the puzzle is solved (miraculous, isn't it ?)

Please, don't ask me the 66x3 triplets of chains, showing each of these eliminations (there could be less than 198, of course, groups of eliminations have the same chains, and two guardians are conjugates, but definitely I would prefer not to !)

Website · by **denis_berthier** » Sun Sep 11, 2022 3:33 am

Cenoman wrote:I had found the same one-step as totuan's, i.e. one direct elimination, out of a total of 113 eliminations.
Not all are direct eliminations. According to my solver, 66 candidates (no guarantee about the accuracy of this count) are in a derived weak link to the three guardians of TH(478)b1278: 2r1c2, 2r9c3, 5r3c6 (no guarantee about the accuracy of this count).

I guess that using Forcing3-T&E based on the 3 guardians would lead to a big number of direct eliminations and then you can choose the 'best" one (leading to the solution after a few more steps). The problem with Forcing-T&E is, it doesn't give any information on how hard each elimination would be in a pattern-based solution.

Website · by **denis_berthier** » Sun Sep 11, 2022 4:14 am

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I have chosen this puzzle as an illustration of solving with two different sets of tridagon-based rules: ORk-Forcing-Whips vs ORk-Contrad-Whips (see http://forum.enjoysudoku.com/ork-forcing-whips-and-ork-contrad-whips-t40189.html for definitions).

Both resolutions paths start with (from the RS after Whips[1]):

Code: Select all: hidden-pairs-in-a-column: c3{n5 n9}{r4 r5} ==> r5c3≠8, r5c3≠4, r4c3≠7 whip[1]: r5n4{c6 .} ==> r6c4≠4, r6c5≠4 whip[1]: r5n8{c6 .} ==> r6c4≠8, r6c5≠8 Resolution state RS2: +-------------------+-------------------+-------------------+ ! 1 2478 3 ! 478 45678 45678 ! 25789 478 249 ! ! 478 5 2478 ! 1 478 9 ! 2378 3478 6 ! ! 6 9 478 ! 2 3 4578 ! 578 1478 14 ! +-------------------+-------------------+-------------------+ ! 2 17 59 ! 379 1567 567 ! 4 36 8 ! ! 3 6 59 ! 489 458 458 ! 1 2 7 ! ! 478 1478 478 ! 37 167 2 ! 36 9 5 ! +-------------------+-------------------+-------------------+ ! 5 478 6 ! 478 2 3 ! 789 1478 149 ! ! 478 23478 1 ! 6 9 478 ! 278 5 234 ! ! 9 23478 2478 ! 5 478 1 ! 2678 4678 234 ! +-------------------+-------------------+-------------------+ OR3-anti-tridagon[12] (type diag) for digits 4, 7 and 8 in blocks: b1, with cells: r1c2, r2c1, r3c3 b2, with cells: r1c4, r2c5, r3c6 b7, with cells: r7c2, r8c1, r9c3 b8, with cells: r7c4, r8c6, r9c5 with 3 guardians: n2r1c2 n5r3c6 n2r9c3

From resolution state RS2, the path with Forcing-Whips continues:

Code: Select all: OR3-forcing-whip-elim[3] based on OR3-anti-tridagon[12] for n2r1c2, n2r9c3 and n5r3c6: || n2r1c2 - || n2r9c3 - partial-whip[1]: r2n2{c3 c7} - || n5r3c6 - partial-whip[1]: r1n5{c6 c7} - ==> r1c7≠2 OR3-forcing-whip-elim[4] based on OR3-anti-tridagon[12] for n2r1c2, n2r9c3 and n5r3c6: || n2r1c2 - || n2r9c3 - || n5r3c6 - partial-whip[3]: r1n5{c6 c7} - r1n9{c7 c9} - b3n2{r1c9 r2c7} - ==> r2c3≠2 stte

From resolution state RS2, the path with Contradiction-Whips continues:

Code: Select all: OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[3]: c3n2{r9 r2} - r1n2{c2 c7} - c7n5{r1 r3} - ==> r9c9≠2 z-chain[4]: c2n3{r8 r9} - c2n2{r9 r1} - c9n2{r1 r8} - r8n3{c9 .} ==> r8c2≠8, r8c2≠7, r8c2≠4 whip[5]: c8n6{r9 r4} - r4n3{c8 c4} - r6c4{n3 n7} - r7n7{c4 c2} - r4n7{c2 .} ==> r9c8≠7 OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[4]: r9n2{c3 c7} - r1n2{c7 c9} - r1n9{c9 c7} - c7n5{r1 r3} - ==> r8c2≠2 singles ==> r8c2=3, r9c9=3 whip[1]: r8n2{c9 .} ==> r9c7≠2 t-whip[5]: c2n2{r9 r1} - r2n2{c3 c7} - c7n3{r2 r6} - r6c4{n3 n7} - r4n7{c6 .} ==> r9c2≠7 OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[4]: r2n2{c7 c3} - r1n2{c2 c9} - r1n9{c9 c7} - c7n5{r1 r3} - ==> r8c7≠2 hidden-single-in-a-block ==> r8c9=2 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠4 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠7 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: r2n2{c7 c3} - r1c2{n2 n8} - b2n8{r1c4 r3c6} - ==> r2c7≠8 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠8 stte

Notice that, although there are more direct tridagon eliminations (7 instead of 2), a whip[5] is still necessary.
Notice that these paths are obtained by using function "solve-with-preferences", with preferences set respectively to TRIDAGON-ORk-FW and TRIDAGON-ORk-CW. This allows to have more ORk-Forcing-Whips or ORk-Contrad-Whips before any other rule is applied.

Website · by **denis_berthier** » Sun Sep 11, 2022 4:30 am

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It is also interesting to compare the two paths if no preferences are applied.

Starting from RS1, both paths have a whip[5] before the ant-tridagon pattern is detected:
(This is a mere consequence of the default priorities assigned to rules; if you don't like it, use function "solve-w-prefs" as in my previous post.)

Code: Select all: hidden-pairs-in-a-column: c3{n5 n9}{r4 r5} ==> r5c3≠8, r5c3≠4, r4c3≠7 whip[1]: r5n4{c6 .} ==> r6c4≠4, r6c5≠4 whip[1]: r5n8{c6 .} ==> r6c4≠8, r6c5≠8 z-chain[5]: c2n3{r8 r9} - b7n2{r9c2 r9c3} - r2n2{c3 c7} - r8n2{c7 c9} - r8n3{c9 .} ==> r8c2≠4, r8c2≠8, r8c2≠7 whip[5]: c8n6{r9 r4} - r4n3{c8 c4} - r6c4{n3 n7} - r7n7{c4 c2} - r4n7{c2 .} ==> r9c8≠7 Resolution state RS3: +-------------------+-------------------+-------------------+ ! 1 2478 3 ! 478 45678 45678 ! 25789 478 249 ! ! 478 5 2478 ! 1 478 9 ! 2378 3478 6 ! ! 6 9 478 ! 2 3 4578 ! 578 1478 14 ! +-------------------+-------------------+-------------------+ ! 2 17 59 ! 379 1567 567 ! 4 36 8 ! ! 3 6 59 ! 489 458 458 ! 1 2 7 ! ! 478 1478 478 ! 37 167 2 ! 36 9 5 ! +-------------------+-------------------+-------------------+ ! 5 478 6 ! 478 2 3 ! 789 1478 149 ! ! 478 23 1 ! 6 9 478 ! 278 5 234 ! ! 9 23478 2478 ! 5 478 1 ! 2678 468 234 ! +-------------------+-------------------+-------------------+ OR3-anti-tridagon[12] (type diag) for digits 4, 7 and 8 in blocks: b1, with cells: r1c2, r2c1, r3c3 b2, with cells: r1c4, r2c5, r3c6 b7, with cells: r7c2, r8c1, r9c3 b8, with cells: r7c4, r8c6, r9c5 with 3 guardians: n2r1c2 n5r3c6 n2r9c3

From resolution state RS3, the path with Forcing-Whips continues:

Code: Select all: OR3-forcing-whip-elim[3] based on OR3-anti-tridagon[12] for n2r1c2, n2r9c3 and n5r3c6: || n2r1c2 - || n2r9c3 - partial-whip[1]: r2n2{c3 c7} - || n5r3c6 - partial-whip[1]: r1n5{c6 c7} - ==> r1c7≠2 finned-x-wing-in-columns: n2{c3 c7}{r2 r9} ==> r9c9≠2 z-chain[4]: b3n2{r2c7 r1c9} - c9n9{r1 r7} - r7n1{c9 c8} - c8n7{r7 .} ==> r2c7≠7 OR3-forcing-whip-elim[4] based on OR3-anti-tridagon[12] for n2r1c2, n2r9c3 and n5r3c6: || n2r1c2 - || n2r9c3 - || n5r3c6 - partial-whip[3]: r1n5{c6 c7} - r1n9{c7 c9} - c9n2{r1 r8} - ==> r8c2≠2 singles ==> r8c2=3, r9c9=3 whip[1]: r8n2{c9 .} ==> r9c7≠2 OR3-forcing-whip-elim[4] based on OR3-anti-tridagon[12] for n2r1c2, n2r9c3 and n5r3c6: || n2r1c2 - || n2r9c3 - || n5r3c6 - partial-whip[3]: r1n5{c6 c7} - r1n9{c7 c9} - b3n2{r1c9 r2c7} - ==> r2c3≠2 stte

From resolution state RS3, the path with Contrad-Whips continues:

Code: Select all: OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[3]: c3n2{r9 r2} - r1n2{c2 c7} - c7n5{r1 r3} - ==> r9c9≠2 OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[4]: r8c2{n3 n2} - c9n2{r8 r1} - r1n9{c9 c7} - c7n5{r1 r3} - ==> r9c2≠3 singles ==> r8c2=3, r9c9=3 whip[1]: r8n2{c9 .} ==> r9c7≠2 t-whip[5]: c2n2{r9 r1} - r2n2{c3 c7} - c7n3{r2 r6} - r6c4{n3 n7} - r4n7{c6 .} ==> r9c2≠7 OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[4]: r2n2{c7 c3} - r1n2{c2 c9} - r1n9{c9 c7} - c7n5{r1 r3} - ==> r8c7≠2 hidden-single-in-a-block ==> r8c9=2 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠4 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠7 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: r2n2{c7 c3} - r1c2{n2 n8} - b2n8{r1c4 r3c6} - ==> r2c7≠8 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠8 stte

Here again, there are more Contrad-Whips than Forcing-Whips. This is not very meaningful, as the number of steps in a resolution path is not an intrinsic property of a puzzle.
What's more relevant is, here again, we need longer Contrad-Whips (length 5) than Forcing-Whips (length 4).

by **pjb** » Mon Sep 12, 2022 1:51 am

type 3 TH with extra 2 at r1c2, 5 at r3c6 and 2 at r9c3. This is a two step solution, but with simple short chains:

TH(1)
chain1: (2)r1c2 - r1c7
chain2: (5)r3c6 - r1c5 = (5-2)r1c7
chain3: (2)r9c3 - r2c3 = r1c2 - r1c7 => -2 r1c7

TH(2)
chain1: (2)r1c2 - r1c9
chain2: (5)r3c6 - r1c5 = (5-9)r1c7 = (9-2)r1c9
chain3: (2)r9c3 - r2c3 = r1c2 - r1c9 => -2 r1c9; stte

Phil

Website · by **denis_berthier** » Mon Sep 12, 2022 4:10 am

pjb wrote:type 3 TH with extra 2 at r1c2, 5 at r3c6 and 2 at r9c3. This is a two step solution, but with simple short chains:

Good.
Same eliminations as my first solution, with different chains (of same lengths as mine).

by **DEFISE** » Mon Sep 12, 2022 11:45 am

denis_berthier wrote:.
....
From resolution state RS3, the path with Contrad-Whips continues:
OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2:
partial-whip[3]: c3n2{r9 r2} - r1n2{c2 c7} - c7n5{r1 r3} -
==> r9c9≠2

OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3:
partial-whip[4]: r8c2{n3 n2} - c9n2{r8 r1} - r1n9{c9 c7} - c7n5{r1 r3} -
==> r9c2≠3

singles ==> r8c2=3, r9c9=3
whip[1]: r8n2{c9 .} ==> r9c7≠2
t-whip[5]: c2n2{r9 r1} - r2n2{c3 c7} - c7n3{r2 r6} - r6c4{n3 n7} - r4n7{c6 .} ==> r9c2≠7

OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2:
partial-whip[4]: r2n2{c7 c3} - r1n2{c2 c9} - r1n9{c9 c7} - c7n5{r1 r3} -
==> r8c7≠2
hidden-single-in-a-block ==> r8c9=2

OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3:
partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} -
==> r1c2≠4

OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3:
partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} -
==> r1c2≠7

OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3:
partial-whip[3]: r2n2{c7 c3} - r1c2{n2 n8} - b2n8{r1c4 r3c6} -
==> r2c7≠8

OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3:
partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} -
==> r1c2≠8
stte

Hi Denis,
4r1c2 is part of the tridagon pattern. If you delete it, I don't understand how you can use this pattern afterwards.

Website · by **denis_berthier** » Mon Sep 12, 2022 1:55 pm

DEFISE wrote:4r1c2 is part of the tridagon pattern. If you delete it, I don't understand how you can use this pattern afterwards.

Hi François,
Excellent remark. I hadn't noticed this.

It is obvious that the contradictory trivalue oddagon pattern remains contradictory even if one (or more) of its 3 defining candidates is absent from one of its 12 cells. I had planned to analyse such situations in the tridagon thread, but I've been busy with other things.
It is not clear if this situation is possible at the start of a puzzle, but you've just found a very interesting case where it's possible after some eliminations.

This extended form of the pattern (allowing missing candidates) being contradictory, the only questions about it are practical:
- whether proving the contradiction still requires T&E(3) - i.e. in which cases does it?;
- how to efficiently identify it in practice in a puzzle (mith's pre-filter no longer works).

As for how to use the extended form in ORk-Forcing-Whips or ORk-Contrad-Whips, it doesn't change anything in the logic.

Technically, the ORk-Forcing-Whips or ORk-Contrad-Whips rules continue to work in SudoRules because they rely only on a proven Tridagon-ORk-relation and I've made such relations persistent (once proven, they remain True, even if some candidate used to prove them disappears). This is consistent with my choice of considering these ORk-relations as independent predicates, whose length is counted only as 1 (instead of 12) in the lengths of the ORk-chains.

by **yzfwsf** » Mon Sep 12, 2022 3:24 pm

Asserting or deleting a candidate contained in a cell in the structure does not affect the validity of the structure, as long as the candidate being asserted is not a guardian.

by **Leren** » Mon Sep 12, 2022 8:44 pm

I said the same thing here. As I said, it seems possible to me that part of a trigadon could even be puzzle clues. To illustrate my point, which puzzle is easier to solve ?

Code: Select all: ..3.49......7..1.....1385.45.......9.69....5.3.4.9..16...3.16.5.359.6.416.1...9.. ..3.49......7..1.....1385.45......79.69....5.3.4.9..16...3.16.5.359.6.416.1...9..

Leren

Website · by **denis_berthier** » Tue Sep 13, 2022 3:18 am

Leren wrote:I said the same thing here. As I said, it seems possible to me that part of a trigadon could even be puzzle clues. To illustrate my point, which puzzle is easier to solve ?
Code: Select all
..3.49......7..1.....1385.45.......9.69....5.3.4.9..16...3.16.5.359.6.416.1...9.. ..3.49......7..1.....1385.45......79.69....5.3.4.9..16...3.16.5.359.6.416.1...9..

Leren

The first puzzle has an easy to find anti-tridagon wiith 2 guardians and a ORk-FW solution:

Hidden Text: Show

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 1278 12578 3 ! 256 4 9 ! 278 2678 278 ! ! 2489 24589 268 ! 7 256 25 ! 1 23689 238 ! ! 279 279 267 ! 1 3 8 ! 5 2679 4 ! +-------------------+-------------------+-------------------+ ! 5 1278 278 ! 2468 12678 2347 ! 23478 278 9 ! ! 1278 6 9 ! 248 1278 2347 ! 23478 5 278 ! ! 3 278 4 ! 258 9 257 ! 278 1 6 ! +-------------------+-------------------+-------------------+ ! 24789 24789 278 ! 3 278 1 ! 6 278 5 ! ! 278 3 5 ! 9 278 6 ! 278 4 1 ! ! 6 278 1 ! 2458 2578 2457 ! 9 2378 2378 ! +-------------------+-------------------+-------------------+ 174 candidates. hidden-pairs-in-a-row: r7{n4 n9}{c1 c2} ==> r7c2≠8, r7c2≠7, r7c2≠2, r7c1≠8, r7c1≠7, r7c1≠2 hidden-pairs-in-a-column: c7{n3 n4}{r4 r5} ==> r5c7≠8, r5c7≠7, r5c7≠2, r4c7≠8, r4c7≠7, r4c7≠2 +-------------------+-------------------+-------------------+ ! 1278 12578 3 ! 256 4 9 ! 278 2678 278 ! ! 2489 24589 268 ! 7 256 25 ! 1 23689 238 ! ! 279 279 267 ! 1 3 8 ! 5 2679 4 ! +-------------------+-------------------+-------------------+ ! 5 1278 278 ! 2468 12678 2347 ! 34 278 9 ! ! 1278 6 9 ! 248 1278 2347 ! 34 5 278 ! ! 3 278 4 ! 258 9 257 ! 278 1 6 ! +-------------------+-------------------+-------------------+ ! 49 49 278 ! 3 278 1 ! 6 278 5 ! ! 278 3 5 ! 9 278 6 ! 278 4 1 ! ! 6 278 1 ! 2458 2578 2457 ! 9 2378 2378 ! +-------------------+-------------------+-------------------+ OR2-anti-tridagon[12] (type diag) for digits 7, 8 and 2 in blocks: b4, with cells: r4c3, r5c1, r6c2 b6, with cells: r4c8, r5c9, r6c7 b7, with cells: r7c3, r8c1, r9c2 b9, with cells: r7c8, r8c7, r9c9 with 2 guardians: n1r5c1 n3r9c9 OR2-forcing-whip-elim[4] based on OR2-anti-tridagon[12] for n3r9c9 and n1r5c1: || n3r9c9 - partial-whip[1]: c8n3{r9 r2} - || n1r5c1 - partial-whip[2]: r4n1{c2 c5} - c5n6{r4 r2} - ==> r2c8≠6 OR2-forcing-whip-elim[5] based on OR2-anti-tridagon[12] for n3r9c9 and n1r5c1: || n3r9c9 - partial-whip[1]: c8n3{r9 r2} - || n1r5c1 - partial-whip[3]: c2n1{r4 r1} - c2n5{r1 r2} - r2c6{n5 n2} - ==> r2c8≠2 +-------------------+-------------------+-------------------+ ! 1278 12578 3 ! 256 4 9 ! 278 2678 278 ! ! 2489 24589 268 ! 7 256 25 ! 1 389 238 ! ! 279 279 267 ! 1 3 8 ! 5 2679 4 ! +-------------------+-------------------+-------------------+ ! 5 1278 278 ! 2468 12678 2347 ! 34 278 9 ! ! 1278 6 9 ! 248 1278 2347 ! 34 5 278 ! ! 3 278 4 ! 258 9 257 ! 278 1 6 ! +-------------------+-------------------+-------------------+ ! 49 49 278 ! 3 278 1 ! 6 278 5 ! ! 278 3 5 ! 9 278 6 ! 278 4 1 ! ! 6 278 1 ! 2458 2578 2457 ! 9 2378 2378 ! +-------------------+-------------------+-------------------+ t-whip[4]: r1n5{c2 c4} - b2n6{r1c4 r2c5} - b2n2{r2c5 r2c6} - r2c3{n2 .} ==> r1c2≠8 whip[4]: c8n9{r2 r3} - r3n6{c8 c3} - r2c3{n6 n2} - r3n2{c1 .} ==> r2c8≠8 t-whip[5]: r1n1{c1 c2} - c2n5{r1 r2} - r2c6{n5 n2} - r2c5{n2 n6} - r2c3{n6 .} ==> r1c1≠8 whip[1]: r1n8{c9 .} ==> r2c9≠8 OR2-forcing-whip-elim[4] based on OR2-anti-tridagon[12] for n1r5c1 and n3r9c9: || n1r5c1 - partial-whip[1]: c2n1{r4 r1} - || n3r9c9 - partial-whip[2]: r2c9{n3 n2} - b2n2{r2c5 r1c4} - ==> r1c2≠2 OR2-forcing-whip-elim[4] based on OR2-anti-tridagon[12] for n3r9c9 and n1r5c1: || n3r9c9 - partial-whip[1]: r2c9{n3 n2} - || n1r5c1 - partial-whip[2]: c2n1{r4 r1} - c2n5{r1 r2} - ==> r2c2≠2 OR2-forcing-whip-elim[4] based on OR2-anti-tridagon[12] for n3r9c9 and n1r5c1: || n3r9c9 - partial-whip[1]: r2c9{n3 n2} - || n1r5c1 - partial-whip[2]: r4n1{c2 c5} - c5n6{r4 r2} - ==> r2c5≠2 z-chain[4]: b2n2{r2c6 r1c4} - c4n6{r1 r4} - r4n4{c4 c7} - r4n3{c7 .} ==> r4c6≠2 OR2-forcing-whip-elim[5] based on OR2-anti-tridagon[12] for n1r5c1 and n3r9c9: || n1r5c1 - partial-whip[1]: r4n1{c2 c5} - || n3r9c9 - partial-whip[3]: r2c9{n3 n2} - b2n2{r2c6 r1c4} - b2n6{r1c4 r2c5} - ==> r4c5≠6 singles ==> r4c4=6, r2c5=6, r3c3=6, r1c8=6, r9c5=5 hidden-pairs-in-a-row: r4{n3 n4}{c6 c7} ==> r4c6≠7 OR2-forcing-whip-elim[5] based on OR2-anti-tridagon[12] for n1r5c1 and n3r9c9: || n1r5c1 - partial-whip[2]: c2n1{r4 r1} - c2n5{r1 r2} - || n3r9c9 - partial-whip[2]: r2c9{n3 n2} - r2c3{n2 n8} - ==> r2c2≠8 OR2-forcing-whip-elim[5] based on OR2-anti-tridagon[12] for n1r5c1 and n3r9c9: || n1r5c1 - partial-whip[1]: c2n1{r4 r1} - || n3r9c9 - partial-whip[3]: r2c9{n3 n2} - b2n2{r2c6 r1c4} - r1n5{c4 c2} - ==> r1c2≠7 z-chain[5]: c2n8{r6 r9} - c4n8{r9 r6} - c4n5{r6 r1} - r1c2{n5 n1} - b4n1{r4c2 .} ==> r5c1≠8 OR2-forcing-whip-elim[5] based on OR2-anti-tridagon[12] for n3r9c9 and n1r5c1: || n3r9c9 - partial-whip[1]: r2c9{n3 n2} - || n1r5c1 - partial-whip[3]: c2n1{r4 r1} - c2n5{r1 r2} - r2n4{c2 c1} - ==> r2c1≠2 OR2-forcing-whip-elim[5] based on OR2-anti-tridagon[12] for n3r9c9 and n1r5c1: || n3r9c9 - partial-whip[1]: r2c9{n3 n2} - || n1r5c1 - partial-whip[3]: c2n1{r4 r1} - c2n5{r1 r2} - r2c6{n5 n2} - ==> r2c3≠2 singles ==> r2c3=8, r8c1=8 naked-pairs-in-a-column: c1{r2 r7}{n4 n9} ==> r3c1≠9 z-chain[4]: r8n7{c5 c7} - r6n7{c7 c2} - c2n8{r6 r4} - r4n1{c2 .} ==> r4c5≠7 z-chain[4]: r8n2{c5 c7} - r6n2{c7 c2} - c2n8{r6 r4} - r4n1{c2 .} ==> r4c5≠2 z-chain[5]: r9n3{c8 c9} - b9n8{r9c9 r7c8} - r4c8{n8 n2} - c3n2{r4 r7} - b7n7{r7c3 .} ==> r9c8≠7 OR2-contrad-whip[4] based on OR2-anti-tridagon[12] for n1r5c1 and n3r9c9: partial-whip[3]: c1n7{r3 r5} - b5n7{r5c6 r6c6} - r9n7{c6 c9} - ==> r3c2≠7 whip[1]: b1n7{r3c1 .} ==> r5c1≠7 biv-chain[4]: r5c1{n2 n1} - b1n1{r1c1 r1c2} - r1n5{c2 c4} - b2n2{r1c4 r2c6} ==> r5c6≠2 whip[4]: r9n3{c8 c9} - r2c9{n3 n2} - c6n2{r2 r6} - c7n2{r6 .} ==> r9c8≠2 OR2-forcing-whip-elim[4] based on OR2-anti-tridagon[12] for n1r5c1 and n3r9c9: || n1r5c1 - || n3r9c9 - partial-whip[3]: r9c8{n3 n8} - b8n8{r9c4 r7c5} - r4n8{c5 c2} - ==> r4c2≠1 singles ==> r5c1=1, r1c2=1, r2c2=5,r2c6=2, r1c4=5, r2c9=3, r2c8=9, r2c1=4, r7c1=9, r7c2=4, r3c2=9, r9c8=3, r6c6=5, r4c5=1 whip[1]: b5n7{r5c6 .} ==> r5c9≠7 finned-x-wing-in-columns: n8{c5 c8}{r7 r5} ==> r5c9≠8 singles ==> r5c9=2, r6c4=2, r9c2=2, r7c3=7, r4c3=2 biv-chain[3]: b3n8{r1c7 r1c9} - c9n7{r1 r9} - r8c7{n7 n2} ==> r1c7≠2 stte

The second puzzle doesn't. But does it mean it's more difficult? With respect to what?
It only means that, like all the known exotic patterns, the anti-tridagon pattern is not stable under confluence. In order to define a stable pattern, we would have to consider all its possible degenerated forms. The ORk conclusion remains valid in all cases.
But, as I said before, the problem is, it leads to much higher computational complexity and to the impossibility for a human player to detect the pattern (both usually go together).

It is likely that the special case where a given is one of the tridagon candidates in a tridagon cell could be dealt with. But it is no longer in T&E(3).

[Edit]: see here http://forum.enjoysudoku.com/the-tridagon-rule-t39859-106.html / http://forum.enjoysudoku.com/the-tridagon-rule-t39859-106.html for proofs that contradiction can be proven in T&E(2) in case one of the candidates is decided / in case one of the candidates is missing.

by **DEFISE** » Tue Sep 13, 2022 12:45 pm

Thank you Denis for your study of the problem.

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#25591, different solutions

#25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions

Re: #25591, different solutions