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Website · by **denis_berthier** » Mon Jan 09, 2023 4:55 am

.
That's a real monster. I don't know if there is another T&E(3) pattern that'd simplify it, but if not, it requires gW15+OR4gW15 (maybe slightly less if we also allow Forcing-g-whips).
As of now, it's the hardest puzzle (in mith's 63137 collection) I've solved without uniqueness or replacement. Thanks for submitting it.

Notice that we start with an OR4 relation, which degenerates to an OR3 one and then to an OR2 one (using ORk ultra-persistency rules, see https://www.researchgate.net/publication/365186265_Augmented_User_Manual_for_CSP-Rules-V21). The 3 of them lead to ORk eliminations.
It's the first time I see all this in a puzzle.

Code: Select all: Resolution state after Singles and whips[1]: +-------------------------+-------------------------+-------------------------+ ! 15 1259 2359 ! 4 1259 6 ! 7 8 1239 ! ! 14567 12569 2345679 ! 1259 8 1279 ! 129 1239 123469 ! ! 8 1269 24679 ! 129 3 1279 ! 5 129 12469 ! +-------------------------+-------------------------+-------------------------+ ! 2 56 1 ! 35689 579 389 ! 4 3579 379 ! ! 3 7 568 ! 12569 1259 4 ! 1289 1259 129 ! ! 9 4 58 ! 1235 1257 123 ! 128 6 1237 ! +-------------------------+-------------------------+-------------------------+ ! 145 1259 2459 ! 12389 129 12389 ! 6 1279 1279 ! ! 16 8 269 ! 7 4 129 ! 3 129 5 ! ! 17 3 279 ! 129 6 5 ! 129 4 8 ! +-------------------------+-------------------------+-------------------------+ 197 candidates.

Code: Select all: hidden-pairs-in-a-row: r7{n3 n8}{c4 c6} ==> r7c6≠9, r7c6≠2, r7c6≠1, r7c4≠9, r7c4≠2, r7c4≠1 hidden-pairs-in-a-column: c9{n4 n6}{r2 r3} ==> r3c9≠9, r3c9≠2, r3c9≠1, r2c9≠9, r2c9≠3, r2c9≠2, r2c9≠1 152 g-candidates, 834 csp-glinks and 471 non-csp glinks +-------------------------+-------------------------+-------------------------+ ! 15 1259 2359 ! 4 1259 6 ! 7 8 1239 ! ! 14567 12569 2345679 ! 1259 8 1279 ! 129 1239 46 ! ! 8 1269 24679 ! 129 3 1279 ! 5 129 46 ! +-------------------------+-------------------------+-------------------------+ ! 2 56 1 ! 35689 579 389 ! 4 3579 379 ! ! 3 7 568 ! 12569 1259 4 ! 1289 1259 129 ! ! 9 4 58 ! 1235 1257 123 ! 128 6 1237 ! +-------------------------+-------------------------+-------------------------+ ! 145 1259 2459 ! 38 129 38 ! 6 1279 1279 ! ! 16 8 269 ! 7 4 129 ! 3 129 5 ! ! 17 3 279 ! 129 6 5 ! 129 4 8 ! +-------------------------+-------------------------+-------------------------+ OR4-anti-tridagon[12] for digits 1, 2 and 9 in blocks: b2, with cells: r1c5, r2c6, r3c4 b3, with cells: r1c9, r2c7, r3c8 b8, with cells: r7c5, r8c6, r9c4 b9, with cells: r7c9, r8c8, r9c7 with 4 guardians: n5r1c5 n3r1c9 n7r2c6 n7r7c9 biv-chain[3]: r5n6{c4 c3} - r4c2{n6 n5} - b6n5{r4c8 r5c8} ==> r5c4≠5 t-whip[5]: b4n5{r6c3 r4c2} - r4n6{c2 c4} - c4n8{r4 r7} - c4n3{r7 r6} - c4n5{r6 .} ==> r2c3≠5 whip[7]: r1c1{n5 n1} - c2n1{r3 r7} - c2n5{r7 r4} - r4n6{c2 c4} - c4n5{r4 r6} - c4n3{r6 r7} - c4n8{r7 .} ==> r2c1≠5 whip[8]: r1c1{n5 n1} - c2n1{r3 r7} - c2n5{r7 r4} - c8n5{r4 r5} - c5n5{r5 r6} - c5n1{r6 r5} - c9n1{r5 r6} - r6n7{c9 .} ==> r1c3≠5

Nothing unusual until now. Not even a large number of guardians.
This is where we take off (hardest steps in blue):

g-whip[9]: c2n2{r3 r7} - c2n1{r7 r123} - r1c1{n1 n5} - c2n5{r2 r4} - c8n5{r4 r5} - c5n5{r5 r6} - r6n7{c5 c9} - c9n2{r6 r5} - c5n2{r5 .} ==> r1c3≠2
whip[13]: c6n7{r3 r2} - c1n7{r2 r9} - c3n7{r9 r3} - r3n4{c3 c9} - r3n6{c9 c2} - b4n6{r4c2 r5c3} - c4n6{r5 r4} - r4n8{c4 c6} - c6n9{r4 r8} - r8c3{n9 n2} - r8c8{n2 n1} - r3n1{c8 c4} - r9n1{c4 .} ==> r3c6≠2
Trid-OR4-whip[13]: r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 n5} - r2n5{c2 c4} - r6n5{c4 c5} - c3n5{r6 r7} - c3n4{r7 r2} - b1n7{r2c3 r2c1} - c1n6{r2 r8} - r8c3{n6 n9} - r1c3{n9 n3} - OR4{{n3r1c9 n5r1c5 n7r2c6 | n7r7c9}} - r6n7{c9 .} ==> r3c3≠2
Trid-OR4-whip[13]: r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 n5} - c3n5{r6 r7} - c3n4{r7 r2} - b1n7{r2c3 r2c1} - c1n4{r2 r7} - c1n5{r7 r1} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - OR4{{n7r7c9 n5r1c5 n7r2c6 | n3r1c9}} - r1c3{n3 .} ==> r3c3≠9
g-whip[13]: b3n3{r2c8 r1c9} - r1c3{n3 n9} - b7n9{r7c3 r7c2} - c2n1{r7 r123} - r1c1{n1 n5} - c2n5{r1 r4} - b4n6{r4c2 r5c3} - r8c3{n6 n2} - r8c8{n2 n9} - r9n9{c7 c4} - r3n9{c4 c6} - r3n7{c6 c3} - r9c3{n7 .} ==> r2c8≠1
Trid-OR4-gwhip[13]: b7n1{r7c2 r789c1} - r1c1{n1 n5} - r7n5{c1 c3} - c3n2{r7 r2} - c3n4{r2 r3} - b1n7{r3c3 r2c1} - r9n7{c1 c3} - b7n9{r9c3 r8c3} - r1c3{n9 n3} - OR4{{n3r1c9 n5r1c5 n7r2c6 | n7r7c9}} - r6n7{c9 c5} - r6n5{c5 c4} - r2n5{c4 .} ==> r7c2≠2
whip[1]: b7n2{r9c3 .} ==> r2c3≠2
Trid-OR4-gwhip[13]: c2n9{r3 r7} - c2n1{r7 r123} - r1c1{n1 n5} - r7n5{c1 c3} - c3n4{r7 r3} - b1n7{r3c3 r2c1} - r2n4{c1 c9} - r2n6{c9 c2} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - OR4{{n7r7c9 n5r1c5 n7r2c6 | n3r1c9}} - r1c3{n3 .} ==> r2c3≠9

Code: Select all: +-------------------+-------------------+-------------------+ ! 15 1259 39 ! 4 1259 6 ! 7 8 1239 ! ! 1467 12569 3467 ! 1259 8 1279 ! 129 239 46 ! ! 8 1269 467 ! 129 3 179 ! 5 129 46 ! +-------------------+-------------------+-------------------+ ! 2 56 1 ! 35689 579 389 ! 4 3579 379 ! ! 3 7 568 ! 1269 1259 4 ! 1289 1259 129 ! ! 9 4 58 ! 1235 1257 123 ! 128 6 1237 ! +-------------------+-------------------+-------------------+ ! 145 159 2459 ! 38 129 38 ! 6 1279 1279 ! ! 16 8 269 ! 7 4 129 ! 3 129 5 ! ! 17 3 279 ! 129 6 5 ! 129 4 8 ! +-------------------+-------------------+-------------------+

whip[14]: r5n6{c4 c3} - c3n8{r5 r6} - b4n5{r6c3 r4c2} - r4n6{c2 c4} - c4n8{r4 r7} - c4n3{r7 r6} - r6c6{n3 n1} - r6c7{n1 n2} - r9n2{c7 c3} - r8c3{n2 n9} - r8c6{n9 n2} - c5n2{r7 r1} - c5n1{r1 r7} - r7c2{n1 .} ==> r5c4≠2

whip[15]: c6n7{r3 r2} - c1n7{r2 r9} - c3n7{r9 r3} - r3n4{c3 c9} - r3n6{c9 c2} - b4n6{r4c2 r5c3} - r8n6{c3 c1} - r8n1{c1 c8} - r9n1{c7 c4} - r5c4{n1 n9} - c6n9{r4 r8} - r8n2{c6 c3} - r9n2{c3 c7} - c7n9{r9 r2} - r3n9{c8 .} ==> r3c6≠1
whip[12]: r2n3{c8 c3} - r1c3{n3 n9} - r8n9{c3 c6} - b2n9{r2c6 r3c4} - r3c6{n9 n7} - c3n7{r3 r9} - r9c1{n7 n1} - b8n1{r9c4 r7c5} - b9n1{r7c8 r8c8} - r3c8{n1 n2} - r2c7{n2 n1} - b2n1{r2c4 .} ==> r2c8≠9
Trid-OR4-whip[14]: c4n6{r5 r4} - r4c2{n6 n5} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - r6n1{c9 c7} - r9n1{c7 c1} - c1n7{r9 r2} - OR4{{n7r2c6 n5r1c5 n7r7c9 | n3r1c9}} - r2c8{n3 n2} - r2c7{n2 n9} - r3c8{n9 n1} - r8n1{c8 c6} - r2c6{n1 .} ==> r5c4≠1
g-whip[7]: c3n9{r9 r1} - c5n9{r1 r456} - r5c4{n9 n6} - b4n6{r5c3 r4c2} - c2n5{r4 r123} - r1c1{n5 n1} - b7n1{r7c1 .} ==> r7c2≠9
whip[1]: b7n9{r9c3 .} ==> r1c3≠9
naked-single ==> r1c3=3

Code: Select all: At least one candidate of a previous Trid-OR4-relation has just been eliminated. There remains a Trid-OR3-relation between candidates: n5r1c5 n7r2c6 n7r7c9 +-------------------+-------------------+-------------------+ ! 15 1259 3 ! 4 1259 6 ! 7 8 129 ! ! 1467 12569 467 ! 1259 8 1279 ! 129 23 46 ! ! 8 1269 467 ! 129 3 79 ! 5 129 46 ! +-------------------+-------------------+-------------------+ ! 2 56 1 ! 35689 579 389 ! 4 3579 379 ! ! 3 7 568 ! 69 1259 4 ! 1289 1259 129 ! ! 9 4 58 ! 1235 1257 123 ! 128 6 1237 ! +-------------------+-------------------+-------------------+ ! 145 15 2459 ! 38 129 38 ! 6 1279 1279 ! ! 16 8 269 ! 7 4 129 ! 3 129 5 ! ! 17 3 279 ! 129 6 5 ! 129 4 8 ! +-------------------+-------------------+-------------------+

hidden-single-in-a-block ==> r2c8=3
Trid-OR3-whip[7]: r4c2{n6 n5} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - OR3{{n7r7c9 n5r1c5 | n7r2c6}} - r2n2{c6 c7} - r2n9{c7 .} ==> r2c2≠6
Trid-OR3-whip[7]: c8n5{r5 r4} - c8n7{r4 r7} - OR3{{n7r7c9 n5r1c5 | n7r2c6}} - b1n7{r2c1 r3c3} - r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 .} ==> r5c5≠5
whip[7]: c5n7{r6 r4} - c5n5{r4 r1} - r1c1{n5 n1} - b7n1{r7c1 r7c2} - c5n1{r7 r5} - c9n1{r5 r6} - r6n7{c9 .} ==> r6c5≠2
Trid-OR3-whip[7]: r7c2{n1 n5} - b1n5{r1c2 r1c1} - OR3{{n5r1c5 n7r7c9 | n7r2c6}} - b1n7{r2c1 r3c3} - r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 .} ==> r7c9≠1
Trid-OR3-whip[9]: r5n5{c8 c3} - c3n8{r5 r6} - r6c7{n8 n1} - c9n1{r6 r1} - c9n2{r1 r7} - c5n2{r7 r1} - OR3{{n5r1c5 n7r7c9 | n7r2c6}} - r3c6{n7 n9} - r3c8{n9 .} ==> r5c8≠2
Trid-OR3-whip[10]: c8n5{r5 r4} - c8n7{r4 r7} - b9n1{r7c8 r9c7} - b3n1{r2c7 r1c9} - r1c1{n1 n5} - OR3{{n5r1c5 n7r7c9 | n7r2c6}} - b1n7{r2c1 r3c3} - r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 .} ==> r5c8≠1
Trid-OR3-ctr-whip[12]: r6n8{c7 c3} - r5n8{c3 c7} - r5n1{c7 c5} - c9n1{r5 r1} - r1c1{n1 n5} - b2n5{r1c5 r2c4} - r6n5{c4 c5} - c5n7{r6 r4} - c8n7{r4 r7} - c8n1{r7 r8} - c6n1{r8 r2} - OR3{{n5r1c5 n7r2c6 n7r7c9 | .}} ==> r6c7≠1
whip[11]: r1c1{n1 n5} - r7c1{n5 n4} - b7n1{r7c1 r7c2} - r7n5{c2 c3} - r6c3{n5 n8} - r6c7{n8 n2} - r2c7{n2 n9} - r9c7{n9 n1} - b8n1{r9c4 r8c6} - c6n2{r8 r2} - r2c2{n2 .} ==> r2c1≠1
naked-triplets-in-a-block: b1{r2c1 r2c3 r3c3}{n4 n6 n7} ==> r3c2≠6
singles ==> r4c2=6, r5c4=6
whip[1]: b4n5{r6c3 .} ==> r7c3≠5
hidden-pairs-in-a-row: r3{n4 n6}{c3 c9} ==> r3c3≠7
hidden-single-in-a-row ==> r3c6=7

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: n5r1c5 n7r7c9 +----------------+----------------+----------------+ ! 15 1259 3 ! 4 1259 6 ! 7 8 129 ! ! 467 1259 467 ! 1259 8 129 ! 129 3 46 ! ! 8 129 46 ! 129 3 7 ! 5 129 46 ! +----------------+----------------+----------------+ ! 2 6 1 ! 3589 579 389 ! 4 579 379 ! ! 3 7 58 ! 6 129 4 ! 1289 59 129 ! ! 9 4 58 ! 1235 157 123 ! 28 6 1237 ! +----------------+----------------+----------------+ ! 145 15 249 ! 38 129 38 ! 6 1279 279 ! ! 16 8 269 ! 7 4 129 ! 3 129 5 ! ! 17 3 279 ! 129 6 5 ! 129 4 8 ! +----------------+----------------+----------------+

Trid-OR2-whip[2]: OR2{{n5r1c5 | n7r7c9}} - r6n7{c9 .} ==> r6c5≠5
Trid-OR2-whip[9]: OR2{{n7r7c9 | n5r1c5}} - r1c1{n5 n1} - c2n1{r3 r7} - r7c5{n1 n9} - r7c8{n9 n7} - c8n2{r7 r3} - b3n1{r3c8 r2c7} - b2n1{r2c4 r3c4} - r9n1{c4 .} ==> r7c9≠2
Trid-OR2-whip[9]: r5c3{n8 n5} - c8n5{r5 r4} - c8n7{r4 r7} - OR2{{n7r7c9 | n5r1c5}} - c1n5{r1 r7} - b7n4{r7c1 r7c3} - r7n2{c3 c5} - r5n2{c5 c9} - r6c7{n2 .} ==> r5c7≠8
singles ==> r6c7=8, r6c3=5, r5c3=8, r5c8=5
Trid-OR2-whip[3]: OR2{{n7r7c9 | n5r1c5}} - r4c5{n5 n9} - r4c8{n9 .} ==> r4c9≠7
Trid-OR2-whip[9]: b6n7{r4c8 r6c9} - r7c9{n7 n9} - OR2{{n7r7c9 | n5r1c5}} - r1c1{n5 n1} - r1c9{n1 n2} - b3n9{r1c9 r2c7} - r3c8{n9 n1} - r8n1{c8 c6} - c6n9{r8 .} ==> r4c8≠9
singles ==> r4c8=7, r6c5=7, r7c9=7
whip[8]: b2n5{r2c4 r1c5} - r1c1{n5 n1} - c2n1{r3 r7} - c5n1{r7 r5} - c5n2{r5 r7} - r9c4{n2 n1} - r3n1{c4 c8} - b9n1{r7c8 .} ==> r2c4≠9

Resolution state RS3:

Code: Select all: +----------------+----------------+----------------+ ! 15 1259 3 ! 4 1259 6 ! 7 8 129 ! ! 467 1259 467 ! 125 8 129 ! 129 3 46 ! ! 8 129 46 ! 129 3 7 ! 5 129 46 ! +----------------+----------------+----------------+ ! 2 6 1 ! 3589 59 389 ! 4 7 39 ! ! 3 7 8 ! 6 129 4 ! 129 5 129 ! ! 9 4 5 ! 123 7 123 ! 8 6 123 ! +----------------+----------------+----------------+ ! 145 15 249 ! 38 129 38 ! 6 129 7 ! ! 16 8 269 ! 7 4 129 ! 3 129 5 ! ! 17 3 279 ! 129 6 5 ! 129 4 8 ! +----------------+----------------+----------------+

Still in T&E(2)

[Edit: simplified printing for splitting rule]
[Edit 2: corrected the end]

by **totuan** » Sun Jan 15, 2023 7:10 am

Thanks for your solution!

Code: Select all: *--------------------------------------------------------------------------------------* | 15 1259 2359 | 4 1259 6 | 7 8 1239 | | 14567 12569 234679-5 |%1259 8 1279 | 129 1239 46 | | 8 1269 24679 | 129 3 1279 | 5 129 46 | |----------------------------+----------------------------+----------------------------| | 2 &56 1 |*35689 579 *389 | 4 &3579 379 | | 3 7 %568 |&1269-5 1259 4 | 1289 &1259 129 | | 9 4 %58 |%1235 1257 123 | 128 6 127-3 | |----------------------------+----------------------------+----------------------------| | 145 1259 2459 |*38 129 *38 | 6 1279 1279 | | 16 8 269 | 7 4 129 | 3 129 5 | | 17 3 279 | 129 6 5 | 129 4 8 | *--------------------------------------------------------------------------------------*

01: UR (38)r47c46 => r6c9<>3
02: (6)r5c4=r4c4-(6=5)r4c2-r4c8=r5c8 => r5c4<>5
03: (5)r56c3=(5-6)r4c2=(368-5)r467c4=r2c4 => r2c3<>5

Code: Select all: *-----------------------------------------------------------------------------* |#15 %1259 239-5 | 4 *1259 6 | 7 8 *1239 | | 14567 %12569 234679 |&1259 8 1279 | 129 1239 46 | | 8 %1269 24679 | 129 3 1279 | 5 129 46 | |-------------------------+-------------------------+-------------------------| | 2 56 1 | 5689 579 89 | 4 3579 379 | | 3 7 568 | 1269 *1259 4 | 1289 1259 *129 | | 9 4 58 |&1235 &1257 123 | 128 6 &127 | |-------------------------+-------------------------+-------------------------| | 145 %1259 2459 | 38 %129 38 | 6 1279 &1279 | | 16 8 269 | 7 4 129 | 3 129 5 | | 17 3 279 | 129 6 5 | 129 4 8 | *-----------------------------------------------------------------------------*

04: Present as diagram: => r1c3<>5

Code: Select all: (X-wing: 1’s)r15c59-(1=5)r1c1* || (17)r6c59 – (5)r6c5 || || || (5)r6c3* || || || (5)r6c4-r2c4=r1c5* || (1)r7c59-r7c2=r123c2-(1=5)r1c1*

Code: Select all: *-----------------------------------------------------------------------------* |&15 c1259 #239 | 4 c1259 6 | 7 8 129-3 | |&14567 12569 ^234679 | 1259 8 a1279 | 129 a1239 46 | | 8 b1269 @24679 |*129 3 @1279 | 5 b129 @46 | |-------------------------+-------------------------+-------------------------| | 2 ^56 1 | 5689 579 89 | 4 3579 379 | | 3 7 ^568 | 1269 1259 4 | 1289 1259 129 | | 9 4 ^58 | 1235 1257 123 | 128 6 127 | |-------------------------+-------------------------+-------------------------| |^145 c1259 ^2459 | 38 c129 38 | 6 1279 *1279 | |&16 8 %269 | 7 4 a129 | 3 a129 5 | |&17 3 %279 |*129 6 5 |*129 4 8 | *-----------------------------------------------------------------------------*

eleven’s impossible pattern (129) => (5)r1c25=(5)r7c2=(7)r2c6=(7)r7c9=(6)r3c2=(3)r2c8

Hidden Text: Show

05: Present as diagram: => r1c9<>3, some singles

Code: Select all: (56)r47c2-(56=129)r123c2-(29=3)r1c3* || (7)r7c9-r46c9=(7-3)r4c8=r4c9* || (7)r2c6-r3c6=(467)r3c239-(67)r2c1=(67)r89c1-(67=29)r89c3-(29=3)r1c3* || (6)r3c2---(6=5)r4c2-r56c3=(5-4)r7c3=r7c1--(4)2c1 || | || || -(6=4)r3c9-----------------------(4)r3c3 || || (3)r2c8* (4-3)r2c3=r1c3* || (5)r1c25-(5=1)r1c1-(1=67)r89c1-(67=29)r89c3-(29=3)r1c3*

Code: Select all: *--------------------------------------------------------------------* | 15 1259 3 | 4 A1259 6 | 7 8 *129 | | 14567 12569 24679 |#1259 8 1279 |*129 3 46 | | 8 ^1269 ^24679 |*129 3 *1279 | 5 *129 ^46 | |----------------------+----------------------+----------------------| | 2 #56 1 | 5689 579 89 | 4 %579 3 | | 3 7 #568 |*1269 *1259 4 | 1289 %1259 *129 | | 9 4 &58 |&1235 &1257 123 | 128 6 &127 | |----------------------+----------------------+----------------------| | 145 1259 2459 | 38 *129 38 | 6 1279 *1279 | | 16 8 269 | 7 4 *129 | 3 *129 5 | | 17 3 279 |*129 6 5 |*129 4 8 | *--------------------------------------------------------------------*

Impossible pattern (129) => (5)r1c5=(5)r5c5=(7)r7c9=(6)r5c4=(7)r3c6

Hidden Text: Show

Code: Select all: *-----------------------------------------------------------* | . . . | . A129 . | . . 129 | | . . . | . . . | 129 . . | | . . . | 129 . 129 | . 129 . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | 129 129 . | . . 129 | | . . . | . . . | . . . | |-------------------+-------------------+-------------------| | . . . | . 129 . | . . 129 | | . . . | . . 129 | . 129 . | | . . . | 129 . . | 129 . . | *-----------------------------------------------------------* A=(1|2|9) => impossible Prove: A=1 => r3c8=1 => *-----------------------------------------------------------* | . . . | . 1 . | . . 29 | | . . . | . . . | 29 . . | | . . . | 29 . 29 | . 1 . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . |c129 29 . | . . b129 | | . . . | . . . | . . . | |-------------------+-------------------+-------------------| | . . . | . 29 . | . . a129 | | . . . | . . 129 | . 29 . | | . . . |d129 . . | 29-1 . . | *-----------------------------------------------------------* abcd => r9c7<>1 => a=1 => c=1 => r8c6=1 => oddagon (29) * marked cells *-----------------------------------------------------------* | . . . | . 1 . | . . *29 | | . . . | . . . |*29 . . | | . . . | 29 . 29 | . 1 . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | 1 *29 . | . . *29 | | . . . | . . . | . . . | |-------------------+-------------------+-------------------| | . . . | . *29 . | . . 1 | | . . . | . . 1 | . 29 . | | . . . |*29 . . |*29 . . | *-----------------------------------------------------------*

06: Present as diagram: => r2c6<>7, r3c6=7

Code: Select all: (5)r1c5-(5=129)r239c3-(129=6)r5c4-r5c3=r4c2-- || | (5)r5c5-r5c8=r4c8-(5=6)r4c2--------- | || | | (6)r5c4-r5c3=r4c2------------------->-------->--r3c2=(46-7)r3c39=r3c6* || | | (7)r7c9-r6c9=r6c5—(5)r6c5 | | || || | | || (5)r6c3-(5=6)r4c2- | || || | (7)r3c6* (5-3)r6c4=(38-6)r47c4=r4c2-

Code: Select all: *--------------------------------------------------------------------* | 15 1259 3 | 4 *1259 6 | 7 8 *129 | | 14567 12569 24679 | 1259 8 *129 |*129 3 46 | | 8 1269 2469 |*129 3 7 | 5 *129 46 | |----------------------+----------------------+----------------------| | 2 6-5 1 | 5689 579 89 | 4 579 3 | | 3 7 568 | 1269 1259 4 | 1289 1259 129 | | 9 4 58 | 1235 127-5 123 | 128 6 127 | |----------------------+----------------------+----------------------| | 145 1259 2459 | 38 *129 38 | 6 1279 *1279 | | 16 8 269 | 7 4 *129 | 3 *129 5 | | 17 3 279 |*129 6 5 |*129 4 8 | *--------------------------------------------------------------------*

Tridagon (129) => (5)r1c5=(7)r7c9
06: (5)r1c5==(7)r7c9-r6c9=r6c5 => r6c5<>5
07: (5)r6c2=(5-3)r6c4=(38-6)r47c4=r4c2 => r4c2<>5, some singles

From here, the puzzle is solved the same #27717 in mith’s 63137 list that was posted by Denis.

Hidden Text: Show

Code: Select all: *-----------------------------------------------------------* |#15 c1259 3 | 4 *1259 6 | 7 8 c129 | | 467 #1259 467 |a1259 8 *129 |a129 3 46 | | 8 b129 46 |b129 3 7 | 5 *129 46 | |-------------------+-------------------+-------------------| | 2 6 1 | 589 579 89 | 4 579 3 | | 3 7 58 | 6 1259 4 | 1289 1259 129 | | 9 4 58 | 1235 127 123 | 128 6 127 | |-------------------+-------------------+-------------------| | 145 c1259 249 | 38 *129 38 | 6 1279 c129+7 | | 16 8 269 | 7 4 *129 | 3 *129 5 | | 17 3 279 |a129 6 5 |a129 4 8 | *-----------------------------------------------------------* eleven’s impossible pattern (129) => (7)r7c9=(5)r17c2/r2c4 Tridagon (129) => (5)r1c5=(7)r7c9 *-----------------------------------------------------------* | . c129 . | . . . | . . c129 | | . . . |a129 . . |a129 . . | | . b129 . |b129 . . | . 129 . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | |-------------------+-------------------+-------------------| | . c129 . | . 129 . | . . c129 | | . . . | . . 129 | . 129 . | | . . . |a129 . . |a129 . . | *-----------------------------------------------------------* 08: (7)r7c9==(5)r17c2/r2c4-r2c2=r1c12-(5)r1c5==(7)r7c9 => r7c9=7, some singles *-----------------------------------------------------------* | 15 1259 3 | 4 A129+5 6 | 7 8 A129 | | 467 1259 467 | 1259 8 *129 |*129 3 46 | | 8 129 46 |*129 3 7 | 5 *129 46 | |-------------------+-------------------+-------------------| | 2 6 1 | 589 59 89 | 4 7 3 | | 3 7 8 | 6 #129 4 |#129 5 #129 | | 9 4 5 | 123 7 123 | 8 6 #12 | |-------------------+-------------------+-------------------| | 145 1259 249 | 38 A129 38 | 6 129 7 | | 16 8 269 | 7 4 *129 | 3 *129 5 | | 17 3 279 |*129 6 5 |*129 4 8 | *-----------------------------------------------------------* 09: A is not RT(129) by (129)r5c5 & B6 => TH[11] on * marked cells => r1c5=5, some singles (the other way is my path on #27717 posted by Denis. – used other impossible pattern :D ) *--------------------------------------------------* | 1 29 3 | 4 5 6 | 7 8 29 | | 4 5 7 | 129 8 19 |c129 3 6 | | 8 29 6 | 29-1 3 7 | 5 d12 4 | |----------------+----------------+----------------| | 2 6 1 | 5 9 8 | 4 7 3 | | 3 7 8 | 6 1 4 | 29 5 29 | | 9 4 5 | 3 7 2 | 8 6 1 | |----------------+----------------+----------------| | 5 1 4 | 8 2 3 | 6 9 7 | | 6 8 29 | 7 4 19 | 3 12 5 | | 7 3 29 |a19 6 5 |b12 4 8 | *--------------------------------------------------* 10: 1’s a=b-c=d => r3c4<>1, stte

totuan

by **totuan** » Sun Jan 15, 2023 7:17 am

for Bonus puzzle.

Code: Select all: *-----------------------------------------------------------------------------* | 1 238 34 |*248 5 6 | 7 *248 9 | |#2458 2578 47 | 1 *248 9 |*248 3 6 | | 6 289 49 | 3 7 *248 | 1 5 *248 | |-------------------------+-------------------------+-------------------------| | 23489 23789 3479 | 6 *2348 23478 |*2489 1 5 | | 23489 23789 5 |*248 12348 123478 |#24689 #24689 #248 | |A248 1 6 | 5 9 *248 | 3 #248 7 | |-------------------------+-------------------------+-------------------------| | 359 4 2 | 7 1368 1358 | 5689 689 18 | | 7 3569 8 | 249 12346 12345 | 24569 2469 124 | | 59 569 1 | 2489 2468 2458 | 245689 7 3 | *-----------------------------------------------------------------------------*

Tridagon (248) => (3)r4c5=(9)r4c7
Impossible pattern (248) => (5)r2c1=(3)r4c5=(9)r5c78

Hidden Text: Show

01: Present as diagram: => r4c5=3, r1c3=3

Code: Select all: (3)r4c5* || (5)r2c1-(5=39)r79c1-(39=248)r5c149-(248=69)r5c78-(9)r4c7==(3)r4c5* || (9)r5c78-r4c7==(3)r4c5*

Code: Select all: *-----------------------------------------------------------------------------* | 1 28 3 |*248 5 6 | 7 *248 9 | |*2458 2578 47 | 1 *248 9 |*248 3 6 | | 6 289 49 | 3 7 *248 | 1 5 *248 | |-------------------------+-------------------------+-------------------------| | 2489 2789 479 | 6 3 2478 |*2489 1 5 | | 23489 23789 5 |*248 *1248 12478 | 24689 24689 *248 | |A248 1 6 | 5 9 *248 | 3 *248 7 | |-------------------------+-------------------------+-------------------------| | 359 4 2 | 7 168 1358 | 5689 689 18 | | 7 3569 8 | 249 1246 12345 | 24569 2469 124 | | 59 569 1 | 2489 2468 2458 | 245689 7 3 | *-----------------------------------------------------------------------------*

Impossible pattern (248) => (5)r2c1=(9)r4c7=(1)r5c5

Hidden Text: Show

02: Present as diagram: => r5c5=1, some singles

Code: Select all: (1)r5c5* || ------------------------(9)r4c2 || | || (9)r4c7---r4c3=r3c3-(9=28)r13c2--(28)r4c2 || || || (7)r4c2-r4c6=(7-1)r5c6=r5c5* || (5)r2c1-(5=39)r79c1-(39=248)r5c149-(248=1)r5c5*

Code: Select all: *--------------------------------------------------------------------* | 1 28 3 |*248 5 6 | 7 *248 9 | |*2458 2578 47 | 1 *248 9 |*248 3 6 | | 6 289 49 | 3 7 *248 | 1 5 *248 | |----------------------+----------------------+----------------------| | 248 2789 479 | 6 3 2478 |*2489 1 5 | | 2348 23789 5 |*248 1 *2478 | 24689 24689 *248 | |A248 1 6 | 5 9 *248 | 3 *248 7 | |----------------------+----------------------+----------------------| | 35 4 2 | 7 68 13 | 5689 689 18 | | 7 35 8 | 9 246 13 | 2456 246 124 | | 9 6 1 | 248 248 5 | 248 7 3 | *--------------------------------------------------------------------*

The same for 7r5c6, Impossible pattern (248) => (5)r2c1=(9)r4c7=(7)r5c6

Hidden Text: Show

03: Present as diagram: => r5c6=7

Code: Select all: (7)r5c6* || ------------------------(9)r4c2 || | || (9)r4c7---r4c3=r3c3-(9=28)r13c2--(28)r4c2 || || || (7)r4c2-r4c6=r5c6* || (5)r2c1-(5=39)r79c1-(39=248)r5c149-(248=7)r5c6*

Code: Select all: *--------------------------------------------------------------------* | 1 28 3 |*248 5 6 | 7 *248 9 | | 2458 2578 47 | 1 *248 9 |*248 3 6 | | 6 289 49 | 3 7 *248 | 1 5 *248 | |----------------------+----------------------+----------------------| | 248 2789 479 | 6 3 *248 |*2489 1 5 | | 2348 2389 5 |*248 1 7 | 24689 24689 *248 | | 248 1 6 | 5 9 *248 | 3 *248 7 | |----------------------+----------------------+----------------------| | 35 4 2 | 7 68 13 | 5689 689 18 | | 7 35 8 | 9 246 13 | 2456 246 124 | | 9 6 1 |A248 248 5 |*248 7 3 | *--------------------------------------------------------------------*

Impossible pattern (248)

Hidden Text: Show

04: Impossible pattern (248) => r4c7=9, some singles

Code: Select all: *--------------------------------------------------* | 1 28 3 | 24 5 6 | 7 48 9 | | 4 5 7 | 1 8 9 | 2 3 6 | | 6 28 9 | 3 7 24 | 1 5 48 | |----------------+----------------+----------------| | 28 7 4 | 6 3 28 | 9 1 5 | | 3 9 5 | 24 1 7 | 6 248 48 | | 28 1 6 | 5 9 248 | 3 24 7 | |----------------+----------------+----------------| | 5 4 2 | 7 6 3 | 8 9 1 | | 7 3 8 | 9 4 1 | 5 6 2 | | 9 6 1 | 8 2 5 | 4 7 3 | *--------------------------------------------------*

05: UR (28)r46c16 => r6c6=4, stte

totuan

The New Sudoku Players' Forum

Other hard one from mith’s “Trivalue Oddagon” list

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Re: Other hard one from mith’s “Trivalue Oddagon” list

Re: Other hard one from mith’s “Trivalue Oddagon” list