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Website · by **denis_berthier** » Sun May 18, 2025 5:48 pm

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This is the first puzzle I found in B24 by searching for minimals of 1-expands of min-expands in mith's T&E(3) collection .

Code: Select all: +-------+-------+-------+ ! 1 2 . ! . . . ! 7 . 9 ! ! 4 . . ! . . . ! . . . ! ! . 8 . ! . 2 . ! . 4 . ! +-------+-------+-------+ ! 2 1 . ! 3 . . ! . . . ! ! . . . ! 6 . . ! . . . ! ! 9 . . ! . 4 2 ! . 1 7 ! +-------+-------+-------+ ! . 9 . ! . . 8 ! 4 . . ! ! 7 . . ! . . . ! . 9 8 ! ! 8 4 . ! . . 7 ! 1 . 2 ! +-------+-------+-------+ 12....7.94.........8..2..4.21.3........6.....9...42.17.9...84..7......9884...71.2

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1 2 356 ! 458 3568 3456 ! 7 3568 9 ! ! 4 3567 35679 ! 15789 135689 13569 ! 23568 23568 1356 ! ! 356 8 35679 ! 1579 2 13569 ! 356 4 1356 ! +----------------------+----------------------+----------------------+ ! 2 1 45678 ! 3 5789 59 ! 5689 568 456 ! ! 35 357 34578 ! 6 15789 159 ! 23589 2358 345 ! ! 9 356 3568 ! 58 4 2 ! 3568 1 7 ! +----------------------+----------------------+----------------------+ ! 356 9 12356 ! 125 1356 8 ! 4 7 356 ! ! 7 356 12356 ! 1245 1356 13456 ! 356 9 8 ! ! 8 4 356 ! 59 3569 7 ! 1 356 2 ! +----------------------+----------------------+----------------------+ 199 candidates

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by **Cenoman** » Tue May 20, 2025 7:23 pm

Code: Select all: +-----------------------+---------------------------+-------------------------+ | 1 2 356 | 458 3568 3456 | 7 3568 9 | | 4 3567 35679 | 15789 135689 13569 | 23568 23568 1356 | | 356 8 35679 | 1579 2 13569 | 356 4 1356 | +-----------------------+---------------------------+-------------------------+ | 2 1 45678 | 3 5789 59 | 5689 568 456 | | 35 357 34578 | 6 15789 159 | 23589 2358 345 | | 9 356 3568 | 58 4 2 | 3568 1 7 | +-----------------------+---------------------------+-------------------------+ | 356 9 12 | 125 1356 8 | 4 7 356 | | 7 356 12 | 1245 1356 13456 | 356 9 8 | | 8 4 356 | 59 3569 7 | 1 356 2 | +-----------------------+---------------------------+-------------------------+

1. (4)r4c3 = r5c3 - (4=357)r5c129 => -7 r4c3; 1 placement
2. UR(12)r78c34 => +1 r23c4; lcls, 1 placement
3. (2)r2c7 = (2-*9)r5c7 = r4c7 - (9=5)r4c6 - (5=8)r6c4 - r6c7 = (829)r245c7 => -356 r2c7, -35 r5c7*
4. (356=8)r169c3 - (8=5)r6c4 - (5=9)r4c6 - r3c6 = (97)r3c34 => -356 r3c3

Code: Select all: +-----------------------+------------------------+-------------------------+ | 1 2 356* | 458 3568 3456 | 7 3568* 9 | | 4 3567* 35679 | 17 35689 3569 | 23568 23568 1356* | | 356* 8 35679 | 17 2 3569 | 356* 4 1356 | +-----------------------+------------------------+-------------------------+ | 2 1 45678 | 3 5789 59 | 5689 568 456 | | 35 357 34578 | 6 15789 159 | 23589 2358 345 | | 9 356 3568 | 58 4 2 | 3568 1 7 | +-----------------------+------------------------+-------------------------+ | 356* 9 12 | 25 1356 8 | 4 7 356* | | 7 356* 12 | 245 1356 13456 | 356* 9 8 | | 8 4 356* | 9 356 7 | 1 356* 2 | +-----------------------+------------------------+-------------------------+

Tridagon (356)b1379 having three guardians: 8r1c8, 7r2c2, 1r2c9
5.
(1)r2c9 - (1=3569)r3c1679 - r45c6 = (9)r5c5
(8)r1c8 - r1c45 = (8-9)r2c5 = (9)r5c5
(7)r2c2 - (7=3568)b4p4589 - r6c4 = (8)r5c5
=> -15 r5c5; 1 placement
6.
(1)r2c9 - (1=3569)r3c1679 - (9=5)r4c6 - (5=8)r6c4 - r6c3 = (84)r45c3
(8)r1c8 - r1c4 = r6c4 - r6c3 = (84)r45c3
(7)r2c2 - r5c2 = (74)r45c3
=> -5r45c3, -3r5c3, -6r4c3
7.
(1)r2c9 - (1=3569)r3c1679 - (9=5)r4c6
(8)r1c8 - r1c45 = (8-9)r2c5 = r5c5 - (9=5)r4c6
(7)r2c2 - r5c2 = (74-8)r45c3 = r6c3 - r6c4 = (84)r1c46
=>-5r1c6
8.
(1)r2c9 - (1=3569)r3c1679 - r4c6 = (9)r4c7
(8)r1c8 - r1c45 = (8-9)r2c5 = r5c5 - r4c6 = (9)r4c7
(7)r2c2 - r5c2 = (74-8)r45c3 = r6c3 - (8=356)r368c7
=> -56 r4c7
...yielding the following resolution state:

Code: Select all: +-----------------------+-----------------------+-----------------------+ | 1 2 356 | 458 3568 346 | 7 3568 9 | | 4 3567 35679 | 17 35689 3569 | 28 23568 1356 | | 356 8 79 | 17 2 3569 | 356 4 1356 | +-----------------------+-----------------------+-----------------------+ | 2 1 48 | 3 7 59 | 89 568 456 | | 35 357 478 | 6 89 1 | 289 2358 345 | | 9 356 3568 | 58 4 2 | 35 1 7 | +-----------------------+-----------------------+-----------------------+ | 356 9 12 | 25 1356 8 | 4 7 356 | | 7 356 12 | 245 1356 3456 | 356 9 8 | | 8 4 356 | 9 356 7 | 1 356 2 | +-----------------------+-----------------------+-----------------------+

Rating still high, solvable with a sequence of classical steps (AICs, Krakens)

Website · by **denis_berthier** » Mon May 26, 2025 7:14 am

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I had forgotten this puzzle.
Thanks for the solution. Here's mine, using 3 impossible patterns in addition to the tridagon, with chains of max length 6.
Using only the tridagon, I need Trid-ORk-whips of length 11.

hidden-pairs-in-a-column: c3{n1 n2}{r7 r8} ==> r8c3≠6, r8c3≠5, r8c3≠3, r7c3≠6, r7c3≠5, r7c3≠3

The impossible patterns used:

Code: Select all: Trid-OR3-relation for digits 3, 6 and 5 in blocks: b1, with cells (marked #): r1c3, r2c2, r3c1 b3, with cells (marked #): r1c8, r2c9, r3c7 b7, with cells (marked #): r9c3, r8c2, r7c1 b9, with cells (marked #): r9c8, r8c7, r7c9 with 3 guardians (in cells marked @): n8r1c8 n7r2c2 n1r2c9 +----------------------+----------------------+----------------------+ ! 1 2 356# ! 458 3568 3456 ! 7 3568#@ 9 ! ! 4 3567#@ 35679 ! 15789 135689 13569 ! 23568 23568 1356#@ ! ! 356# 8 35679 ! 1579 2 13569 ! 356# 4 1356 ! +----------------------+----------------------+----------------------+ ! 2 1 45678 ! 3 5789 59 ! 5689 568 456 ! ! 35 357 34578 ! 6 15789 159 ! 23589 2358 345 ! ! 9 356 3568 ! 58 4 2 ! 3568 1 7 ! +----------------------+----------------------+----------------------+ ! 356# 9 12 ! 125 1356 8 ! 4 7 356# ! ! 7 356# 12 ! 1245 1356 13456 ! 356# 9 8 ! ! 8 4 356# ! 59 3569 7 ! 1 356# 2 ! +----------------------+----------------------+----------------------+ EL13c290-OR5-relation for digits: 3, 5 and 6 in cells (marked #): (r9c3 r8c5 r8c7 r8c2 r7c5 r7c9 r7c1 r3c7 r3c1 r2c9 r2c2 r1c5 r1c3) with 5 guardians (in cells marked @) : n1r8c5 n1r7c5 n1r2c9 n7r2c2 n8r1c5 +----------------------+----------------------+----------------------+ ! 1 2 356# ! 458 3568#@ 3456 ! 7 3568 9 ! ! 4 3567#@ 35679 ! 15789 135689 13569 ! 23568 23568 1356#@ ! ! 356# 8 35679 ! 1579 2 13569 ! 356# 4 1356 ! +----------------------+----------------------+----------------------+ ! 2 1 45678 ! 3 5789 59 ! 5689 568 456 ! ! 35 357 34578 ! 6 15789 159 ! 23589 2358 345 ! ! 9 356 3568 ! 58 4 2 ! 3568 1 7 ! +----------------------+----------------------+----------------------+ ! 356# 9 12 ! 125 1356#@ 8 ! 4 7 356# ! ! 7 356# 12 ! 1245 1356#@ 13456 ! 356# 9 8 ! ! 8 4 356# ! 59 3569 7 ! 1 356 2 ! +----------------------+----------------------+----------------------+ EL13c30s-OR4-relation for digits: 3, 5 and 6 in cells (marked #): (r6c7 r6c2 r6c3 r1c3 r2c2 r3c9 r3c7 r3c1 r9c3 r8c7 r8c2 r7c9 r7c1) with 4 guardians (in cells marked @) : n8r6c7 n8r6c3 n7r2c2 n1r3c9 +----------------------+----------------------+----------------------+ ! 1 2 356# ! 458 3568 3456 ! 7 3568 9 ! ! 4 3567#@ 35679 ! 15789 135689 13569 ! 23568 23568 1356 ! ! 356# 8 35679 ! 1579 2 13569 ! 356# 4 1356#@ ! +----------------------+----------------------+----------------------+ ! 2 1 45678 ! 3 5789 59 ! 5689 568 456 ! ! 35 357 34578 ! 6 15789 159 ! 23589 2358 345 ! ! 9 356# 3568#@ ! 58 4 2 ! 3568#@ 1 7 ! +----------------------+----------------------+----------------------+ ! 356# 9 12 ! 125 1356 8 ! 4 7 356# ! ! 7 356# 12 ! 1245 1356 13456 ! 356# 9 8 ! ! 8 4 356# ! 59 3569 7 ! 1 356 2 ! +----------------------+----------------------+----------------------+ EL15c64-OR8-relation for digits: 3, 5 and 6 in cells (marked #): (r2c9 r1c5 r1c3 r1c8 r3c6 r3c1 r3c7 r9c3 r9c8 r7c5 r7c1 r7c9 r8c6 r8c2 r8c7) with 8 guardians (in cells marked @) : n1r2c9 n8r1c5 n8r1c8 n1r3c6 n9r3c6 n1r7c5 n1r8c6 n4r8c6 +-------------------------+-------------------------+-------------------------+ ! 1 2 356# ! 458 3568#@ 3456 ! 7 3568#@ 9 ! ! 4 3567 35679 ! 15789 135689 13569 ! 23568 23568 1356#@ ! ! 356# 8 35679 ! 1579 2 13569#@ ! 356# 4 1356 ! +-------------------------+-------------------------+-------------------------+ ! 2 1 45678 ! 3 5789 59 ! 5689 568 456 ! ! 35 357 34578 ! 6 15789 159 ! 23589 2358 345 ! ! 9 356 3568 ! 58 4 2 ! 3568 1 7 ! +-------------------------+-------------------------+-------------------------+ ! 356# 9 12 ! 125 1356#@ 8 ! 4 7 356# ! ! 7 356# 12 ! 1245 1356 13456#@ ! 356# 9 8 ! ! 8 4 356# ! 59 3569 7 ! 1 356# 2 ! +-------------------------+-------------------------+-------------------------+

t-whip[4]: r4n7{c5 c3} - c3n4{r4 r5} - c3n8{r5 r6} - r6c4{n8 .} ==> r4c5≠5
whip[4]: r5c1{n5 n3} - r6n3{c3 c7} - r3c7{n3 n6} - r8c7{n6 .} ==> r5c7≠5
t-whip[5]: c7n2{r2 r5} - c7n9{r5 r4} - r4c6{n9 n5} - r6c4{n5 n8} - c7n8{r6 .} ==> r2c7≠3, r2c7≠5, r2c7≠6
t-whip[5]: r4n7{c5 c3} - c3n4{r4 r5} - c3n8{r5 r6} - r6c4{n8 n5} - r4c6{n5 .} ==> r4c5≠9
whip[5]: c7n9{r5 r4} - r4c6{n9 n5} - r6c4{n5 n8} - c7n8{r6 r2} - c7n2{r2 .} ==> r5c7≠3
t-whip[6]: r5n7{c3 c5} - r5n1{c5 c6} - r5n9{c6 c7} - r5n2{c7 c8} - r5n8{c8 c3} - c3n4{r5 .} ==> r4c3≠7
hidden-single-in-a-row ==> r4c5=7

EL13c30s-OR4-whip[6]: r5n1{c6 c5} - b5n8{r5c5 r6c4} - OR4{{n8r6c3 n1r3c9 n8r6c7 | n7r2c2}} - c3n7{r3 r5} - c3n4{r5 r4} - c3n8{r4 .} ==> r3c6≠1

At least one candidate of a previous EL15c64-OR8-relation between candidates n1r2c9 n8r1c5 n8r1c8 n1r3c6 n9r3c6 n1r7c5 n1r8c6 n4r8c6 has just been eliminated.
There remains an EL15c64-OR7-relation between candidates: n1r2c9 n8r1c5 n8r1c8 n9r3c6 n1r7c5 n1r8c6 n4r8c6

EL13c30s-OR4-ctr-whip[6]: r3n1{c9 c4} - r3n7{c4 c3} - r3n9{c3 c6} - r4c6{n9 n5} - r6c4{n5 n8} - OR4{{n8r6c7 n8r6c3 n7r2c2 n1r3c9 | .}} ==> r2c9≠1

hidden-single-in-a-block ==> r3c9=1

At least one candidate of a previous Trid-OR3-relation between candidates n8r1c8 n7r2c2 n1r2c9 has just been eliminated.
There remains a Trid-OR2-relation between candidates: n8r1c8 n7r2c2

At least one candidate of a previous EL13c290-OR5-relation between candidates n1r8c5 n1r7c5 n1r2c9 n7r2c2 n8r1c5 has just been eliminated.
There remains an EL13c290-OR4-relation between candidates: n1r8c5 n1r7c5 n7r2c2 n8r1c5

At least one candidate of a previous EL15c64-OR7-relation between candidates n1r2c9 n8r1c5 n8r1c8 n9r3c6 n1r7c5 n1r8c6 n4r8c6 has just been eliminated.
There remains an EL15c64-OR6-relation between candidates: n8r1c5 n8r1c8 n9r3c6 n1r7c5 n1r8c6 n4r8c6

Trid-OR2-whip[4]: OR2{{n8r1c8 | n7r2c2}} - c3n7{r3 r5} - c3n4{r5 r4} - r4n8{c3 .} ==> r5c8≠8
Trid-OR2-whip[5]: r5n2{c8 c7} - r2c7{n2 n8} - OR2{{n8r1c8 | n7r2c2}} - r5c2{n7 n3} - r5c1{n3 .} ==> r5c8≠5
Trid-OR2-whip[5]: r5n2{c8 c7} - r2c7{n2 n8} - OR2{{n8r1c8 | n7r2c2}} - r5c2{n7 n5} - r5c1{n5 .} ==> r5c8≠3
singles ==> r5c8=2, r2c7=2
whip[1]: c7n8{r6 .} ==> r4c8≠8
Trid-OR2-whip[5]: OR2{{n8r1c8 | n7r2c2}} - c3n7{r3 r5} - c3n4{r5 r4} - r4n8{c3 c7} - r5n8{c7 .} ==> r1c5≠8

At least one candidate of a previous EL13c290-OR4-relation between candidates n1r8c5 n1r7c5 n7r2c2 n8r1c5 has just been eliminated.
There remains an EL13c290-OR3-relation between candidates: n1r8c5 n1r7c5 n7r2c2

At least one candidate of a previous EL15c64-OR6-relation between candidates n8r1c5 n8r1c8 n9r3c6 n1r7c5 n1r8c6 n4r8c6 has just been eliminated.
There remains an EL15c64-OR5-relation between candidates: n8r1c8 n9r3c6 n1r7c5 n1r8c6 n4r8c6

t-whip[4]: c5n8{r2 r5} - r6c4{n8 n5} - r9c4{n5 n9} - c5n9{r9 .} ==> r2c5≠6, r2c5≠5, r2c5≠3, r2c5≠1
biv-chain[3]: r2n1{c4 c6} - r5n1{c6 c5} - c5n8{r5 r2} ==> r2c4≠8
EL13c290-OR3-ctr-whip[4]: c4n7{r3 r2} - r2n1{c4 c6} - r5n1{c6 c5} - OR3{{n1r8c5 n1r7c5 n7r2c2 | .}} ==> r3c4≠5
EL13c290-OR3-ctr-whip[4]: c4n7{r3 r2} - r2n1{c4 c6} - r5n1{c6 c5} - OR3{{n1r8c5 n1r7c5 n7r2c2 | .}} ==> r3c4≠9
naked-single ==> r3c4=7
EL13c290-OR3-whip[4]: r5n1{c6 c5} - OR3{{n1r7c5 n1r8c5 | n7r2c2}} - r5c2{n7 n3} - r5c1{n3 .} ==> r5c6≠5
biv-chain[5]: r4n9{c6 c7} - r4n8{c7 c3} - c3n4{r4 r5} - c3n7{r5 r2} - b1n9{r2c3 r3c3} ==> r3c6≠9
hidden-single-in-a-row ==> r3c3=9

At least one candidate of a previous EL15c64-OR5-relation between candidates n8r1c8 n9r3c6 n1r7c5 n1r8c6 n4r8c6 has just been eliminated.
There remains an EL15c64-OR4-relation between candidates: n8r1c8 n1r7c5 n1r8c6 n4r8c6

z-chain[4]: r6c4{n5 n8} - c5n8{r5 r2} - r2n9{c5 c6} - r2n1{c6 .} ==> r2c4≠5
whip[4]: b6n3{r6c7 r5c9} - r5c1{n3 n5} - r3n5{c1 c6} - r4n5{c6 .} ==> r6c7≠5
Trid-OR2-whip[4]: r5n7{c3 c2} - OR2{{n7r2c2 | n8r1c8}} - c4n8{r1 r6} - r6n5{c4 .} ==> r5c3≠5
z-chain[5]: c5n8{r5 r2} - c5n9{r2 r9} - c4n9{r9 r2} - r2n1{c4 c6} - r5n1{c6 .} ==> r5c5≠5
naked-triplets-in-a-row: r5{c5 c6 c7}{n8 n1 n9} ==> r5c3≠8
biv-chain[3]: b5n5{r4c6 r6c4} - c4n8{r6 r1} - b2n4{r1c4 r1c6} ==> r1c6≠5
biv-chain[4]: r2n8{c8 c5} - b5n8{r5c5 r6c4} - b5n5{r6c4 r4c6} - r4c8{n5 n6} ==> r2c8≠6
t-whip[5]: r5c6{n1 n9} - r5c7{n9 n8} - c5n8{r5 r2} - b2n9{r2c5 r2c4} - r2n1{c4 .} ==> r8c6≠1

At least one candidate of a previous EL15c64-OR4-relation between candidates n8r1c8 n1r7c5 n1r8c6 n4r8c6 has just been eliminated.
There remains an EL15c64-OR3-relation between candidates: n8r1c8 n1r7c5 n4r8c6

EL15c64-OR3-whip[4]: c4n4{r1 r8} - OR3{{n4r8c6 n8r1c8 | n1r7c5}} - r7c3{n1 n2} - c4n2{r7 .} ==> r1c4≠8
singles ==> r2c5=8, r1c8=8, r6c4=8, r4c3=8, r5c3=4, r5c2=7, r2c3=7, r4c9=4, r5c7=8, r4c7=9, r4c6=5, r4c8=6, r6c7=3, r5c9=5, r5c1=3, r3c6=3, r1c3=3, r8c2=3

At least one candidate of a previous EL13c290-OR3-relation between candidates n1r8c5 n1r7c5 n7r2c2 has just been eliminated.
There remains an EL13c290-OR2-relation between candidates: n1r8c5 n1r7c5

whip[1]: r1n6{c6 .} ==> r2c6≠6
EL13c290-OR2-whip[1]: OR2{{n1r7c5 n1r8c5 | .}} ==> r8c4≠1, r5c5≠1, r7c4≠1
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