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by **shye** » Fri Jun 06, 2025 7:36 pm

Code: Select all: +-------+-------+-------+ | 9 8 . | . 7 6 | . . . | | 5 . 6 | 8 . 4 | 7 . . | | . 7 4 | 9 5 . | 8 . . | +-------+-------+-------+ | . 9 8 | . . . | 6 . . | | 7 . 5 | . . . | . . . | | 6 4 . | . . 8 | . . . | +-------+-------+-------+ | . 6 9 | 4 . . | . 7 8 | | . . . | . . . | 9 . 3 | | . . . | . . . | 4 2 . | +-------+-------+-------+ 98..76...5.68.47...7495.8...98...6..7.5......64...8....694...78......9.3......42.

estimated rating: 10.1
another monster for people to sink their teeth into :3

by **denis_berthier** » Sat Jun 07, 2025 12:17 am

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Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 9 8 123 ! 123 7 6 ! 1235 1345 1245 ! ! 5 123 6 ! 8 123 4 ! 7 139 129 ! ! 123 7 4 ! 9 5 123 ! 8 136 126 ! +----------------------+----------------------+----------------------+ ! 123 9 8 ! 12357 1234 12357 ! 6 1345 12457 ! ! 7 123 5 ! 1236 123469 1239 ! 123 8 1249 ! ! 6 4 123 ! 12357 1239 8 ! 1235 1359 12579 ! +----------------------+----------------------+----------------------+ ! 123 6 9 ! 4 123 1235 ! 15 7 8 ! ! 4 125 127 ! 12567 8 1257 ! 9 156 3 ! ! 8 135 137 ! 13567 1369 13579 ! 4 2 156 ! +----------------------+----------------------+----------------------+ 170 candidates

A double tridagon with the same digits in the same blocks (something not rare when there are several guardians):

Code: Select all: Trid-OR4-relation for digits 2, 3 and 1 in blocks: b1, with cells (marked #): r1c3, r2c2, r3c1 b2, with cells (marked #): r1c4, r2c5, r3c6 b4, with cells (marked #): r6c3, r5c2, r4c1 b5, with cells (marked #): r6c4, r5c6, r4c5 with 4 guardians (in cells marked @): n4r4c5 n9r5c6 n5r6c4 n7r6c4 +-------------------------+-------------------------+-------------------------+ ! 9 8 123# ! 123# 7 6 ! 1235 1345 1245 ! ! 5 123# 6 ! 8 123# 4 ! 7 139 129 ! ! 123# 7 4 ! 9 5 123# ! 8 136 126 ! +-------------------------+-------------------------+-------------------------+ ! 123# 9 8 ! 12357 1234#@ 12357 ! 6 1345 12457 ! ! 7 123# 5 ! 1236 123469 1239#@ ! 123 8 1249 ! ! 6 4 123# ! 12357#@ 1239 8 ! 1235 1359 12579 ! +-------------------------+-------------------------+-------------------------+ ! 123 6 9 ! 4 123 1235 ! 15 7 8 ! ! 4 125 127 ! 12567 8 1257 ! 9 156 3 ! ! 8 135 137 ! 13567 1369 13579 ! 4 2 156 ! +-------------------------+-------------------------+-------------------------+ Trid-OR4-relation for digits 2, 3 and 1 in blocks: b1, with cells (marked #): r1c3, r2c2, r3c1 b2, with cells (marked #): r1c4, r2c5, r3c6 b4, with cells (marked #): r6c3, r5c2, r4c1 b5, with cells (marked #): r6c5, r5c4, r4c6 with 4 guardians (in cells marked @): n5r4c6 n7r4c6 n6r5c4 n9r6c5 +-------------------------+-------------------------+-------------------------+ ! 9 8 123# ! 123# 7 6 ! 1235 1345 1245 ! ! 5 123# 6 ! 8 123# 4 ! 7 139 129 ! ! 123# 7 4 ! 9 5 123# ! 8 136 126 ! +-------------------------+-------------------------+-------------------------+ ! 123# 9 8 ! 12357 1234 12357#@ ! 6 1345 12457 ! ! 7 123# 5 ! 1236#@ 123469 1239 ! 123 8 1249 ! ! 6 4 123# ! 12357 1239#@ 8 ! 1235 1359 12579 ! +-------------------------+-------------------------+-------------------------+ ! 123 6 9 ! 4 123 1235 ! 15 7 8 ! ! 4 125 127 ! 12567 8 1257 ! 9 156 3 ! ! 8 135 137 ! 13567 1369 13579 ! 4 2 156 ! +-------------------------+-------------------------+-------------------------+

z-chain[4]: r5n4{c5 c9} - r5n9{c9 c6} - c5n9{r6 r9} - c5n6{r9 .} ==> r5c5≠1, r5c5≠3, r5c5≠2
t-whip[4]: r5n4{c9 c5} - c5n6{r5 r9} - r9n9{c5 c6} - r5n9{c6 .} ==> r5c9≠1, r5c9≠2
t-whip[4]: c5n6{r9 r5} - r5n4{c5 c9} - r5n9{c9 c6} - c5n9{r6 .} ==> r9c5≠1, r9c5≠3

The simplest way to use the tridagons is as a guide for eleven's replacement:

Code: Select all: +-------------------+-------------------+-------------------+ ! 9 8 123 ! 123 7 6 ! 1235 1345 1245 ! ! 5 123 6 ! 8 123 4 ! 7 139 129 ! ! 123 7 4 ! 9 5 123 ! 8 136 126 ! +-------------------+-------------------+-------------------+ ! 123 9 8 ! 12357 1234 12357 ! 6 1345 12457 ! ! 7 123 5 ! 1236 469 1239 ! 123 8 49 ! ! 6 4 123 ! 12357 1239 8 ! 1235 1359 12579 ! +-------------------+-------------------+-------------------+ ! 123 6 9 ! 4 123 1235 ! 15 7 8 ! ! 4 125 127 ! 12567 8 1257 ! 9 156 3 ! ! 8 135 137 ! 13567 69 13579 ! 4 2 156 ! +-------------------+-------------------+-------------------+

***** STARTING ELEVEN_S REPLACEMENT TECHNIQUE *****
RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.
Trying in block 4

AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to 3 digits 1, 2 and 3 in 3 cells r6c3, r5c2 and r4c1,
the resolution state is:

Code: Select all: +----------------------+----------------------+----------------------+ ! 9 8 123 ! 123 7 6 ! 1235 12345 12345 ! ! 5 123 6 ! 8 123 4 ! 7 1239 1239 ! ! 123 7 4 ! 9 5 123 ! 8 1236 1236 ! +----------------------+----------------------+----------------------+ ! 3 9 8 ! 12357 1234 12357 ! 6 12345 123457 ! ! 7 2 5 ! 1236 469 1239 ! 123 8 49 ! ! 6 4 1 ! 12357 1239 8 ! 1235 12359 123579 ! +----------------------+----------------------+----------------------+ ! 123 6 9 ! 4 123 1235 ! 1235 7 8 ! ! 4 1235 1237 ! 123567 8 12357 ! 9 12356 123 ! ! 8 1235 1237 ! 123567 69 123579 ! 4 123 12356 ! +----------------------+----------------------+----------------------+ THIS IS THE PUZZLE THAT WILL NOW BE SOLVED. RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.

Note that this new puzzle has several tridagons for the same digits 1,2,3 but in blocks b2,b3,b8,b9 and with many guardians.
They don't need to be used, as there's a solution in W6.

t-whip[6]: c1n1{r3 r7} - c2n1{r9 r2} - c5n1{r2 r4} - c5n4{r4 r5} - c5n6{r5 r9} - c9n6{r9 .} ==> r3c9≠1
t-whip[6]: r2c2{n3 n1} - c1n1{r3 r7} - c5n1{r7 r4} - c5n4{r4 r5} - r5c9{n4 n9} - c8n9{r6 .} ==> r2c8≠3
whip[6]: c1n1{r7 r3} - r2c2{n1 n3} - c5n3{r2 r6} - r5n3{c6 c7} - c7n1{r5 r1} - b2n1{r1c4 .} ==> r7c5≠1
biv-chain[4]: c5n1{r2 r4} - b5n4{r4c5 r5c5} - r5c9{n4 n9} - b3n9{r2c9 r2c8} ==> r2c8≠1
z-chain[4]: r5n1{c6 c7} - r7n1{c7 c1} - b1n1{r3c1 r2c2} - c5n1{r2 .} ==> r4c6≠1
t-whip[4]: r7c5{n3 n2} - r7c1{n2 n1} - c2n1{r9 r2} - r2c5{n1 .} ==> r6c5≠3
biv-chain[3]: r7c5{n3 n2} - r6c5{n2 n9} - b8n9{r9c5 r9c6} ==> r9c6≠3
z-chain[4]: c9n6{r3 r9} - r9c5{n6 n9} - r6c5{n9 n2} - r2n2{c5 .} ==> r3c9≠2
z-chain[5]: r7n3{c6 c7} - r5c7{n3 n1} - r5c6{n1 n9} - r6c5{n9 n2} - r7c5{n2 .} ==> r8c6≠3
z-chain[5]: c5n3{r7 r2} - c5n1{r2 r4} - r5c6{n1 n9} - r6c5{n9 n2} - r7c5{n2 .} ==> r7c6≠3
biv-chain[4]: r9c5{n6 n9} - c6n9{r9 r5} - c6n3{r5 r3} - r3c9{n3 n6} ==> r9c9≠6
singles ==> r8c8=6, r3c9=6
z-chain[5]: c5n3{r7 r2} - r2c2{n3 n1} - r9c2{n1 n5} - b9n5{r9c9 r7c7} - r7n3{c7 .} ==> r9c4≠3
t-whip[5]: c5n1{r4 r2} - b1n1{r2c2 r3c1} - b1n2{r3c1 r1c3} - b2n2{r1c4 r3c6} - c6n3{r3 .} ==> r5c6≠1
biv-chain[4]: r1n4{c8 c9} - r5c9{n4 n9} - r5c6{n9 n3} - r3n3{c6 c8} ==> r1c8≠3
z-chain[4]: r2n2{c9 c5} - r6c5{n2 n9} - r5c6{n9 n3} - r3n3{c6 .} ==> r3c8≠2
biv-chain[3]: r3c8{n1 n3} - c6n3{r3 r5} - r5c7{n3 n1} ==> r1c7≠1, r4c8≠1
biv-chain[3]: r3c8{n3 n1} - r3c1{n1 n2} - r1c3{n2 n3} ==> r1c7≠3, r1c9≠3
z-chain[3]: c7n3{r6 r7} - c5n3{r7 r2} - b3n3{r2c9 .} ==> r6c8≠3
z-chain[3]: r2c2{n1 n3} - r1n3{c3 c4} - r1n1{c4 .} ==> r2c9≠1
biv-chain[4]: r5c7{n1 n3} - c6n3{r5 r3} - r3c8{n3 n1} - c1n1{r3 r7} ==> r7c7≠1
hidden-single-in-a-column ==> r5c7=1
whip[1]: b6n3{r6c9 .} ==> r6c4≠3
biv-chain[4]: r1c7{n2 n5} - r7n5{c7 c6} - r7n1{c6 c1} - c1n2{r7 r3} ==> r1c3≠2
stte

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