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by **shye** » Fri Jun 06, 2025 10:36 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

Website · by **denis_berthier** » Sat Jun 07, 2025 3:17 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

by **shye** » Fri Jun 13, 2025 12:04 am

thanks for solving! had a feeling a tridagon path might be possible
i used a different impossible pattern though

Code: Select all: +-------+-------+-------+ | . . X | X . . | X . . | | . X . | . X . | . . . | | X . . | . . X | . . . | +-------+-------+-------+ | X . . | . . . | . . . | | . X . | . . . | X . . | | . . X | . X . | X . . | +-------+-------+-------+ | X . . | . X X | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | +-------+-------+-------+

proof of impossibility:
let r6c5 be A
5 instances of A to be placed in b124, r7, c7
4 of which are covered by c16, r15, leaving Ar2c2
the rest of the placements for A all go in as singles
a bivalue oddagon on BC is left in r3c1, r3c6, r2c5, r7c5, r7c1

in this puzzle the pattern appears twice and is linked by 9b5
the remaining guardians are 5r1c7, 5r6c7, 5r7c6
-5r7c7

this reduces the puzzle to 8.3

the rest of my solve: Show

by **Cenoman** » Fri Jun 13, 2025 4:35 pm

FWIW, here is a tridagon path, that I barely dare post

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 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 | +--------------------+---------------------------+------------------------+

1. (6)r9c5 = (64-9)r5c59 = r5c6 - r9c6 = (9)r9c5 loop => -13 r9c5, -12 r5c9, -123r5c5

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 | +--------------------+-------------------------+------------------------+

2. Tridagon 1: (123)b1245, with cells b5p348 (*), having 4 guardians (57r4c6, 6r5c4, 9r6c5)
Tridagon 2: (123)b1245, with cells b5p348 (^), having 4 guardians (57r6c4, 4r4c5, 9r5c6)

Looking at links between those guardians, it appears that:
4r4c5 and 9r6c5 are conjugates: SL (4=1239)r2467c5, WL (4)r4c5 - r5c5 = (4-9)r5c9 = r5c56 - (9)r6c5
6r5c4 and 9r5c6 are conjugates: SL (6=1239)r5c2467, WL (6)r5c4 - r5c5 = (6-9)r9c5 = r56c5 - (9)r5c6
9r5c6 and 9r6c5 are conflicting
At least one guardian of each tridagon has to be True.
If 9r6c5 is True, then 4r4c5 and 9r5c6 are False, 6r5c4 is True, 5|7r6c4 has to be true (otherwise Trid.2 has no guardian).
If 9r6c5 is False, then 4r4c5 is True, with several configurations of other guardians.

Adding 4r4c5 to puzzle givens yields a no solution puzzle.
Subsequently, +6r5c4, +9r6c5, +57r6c4 (-123 r5c4, r6c45); 7 placements

Code: Select all: +--------------------+--------------------------+------------------------+ | 9 8 123 | 123 7 6 | 1235 1345 1245 | | 5 123 6 | 8 123 4 | 7 9 12 | | 123* 7 4 | 9 5 123* | 8 13 6 | +--------------------+--------------------------+------------------------+ | d13-2 9 8 |ca12357* ca123* ca12357* | 6 45-13 2457-1 | | 7 123 5 | 6 4 b13-2* | 123 8 9 | | 6 4 123 | 57 9 8 | 1235 135 1257 | +--------------------+--------------------------+------------------------+ | 123 6 9 | 4 123 1235 | 15 7 8 | | 4 125 127 | 1257 8 1257 | 9 6 3 | | 8 135 137 | 1357 6 9 | 4 2 15 | +--------------------+--------------------------+------------------------+

3. ER (2)r4c456 = r5c6 - r3c6 = r3c1 => -2 r4c1
4. (1)r4c456 = (1-3)r5c6 = r4c456 - (3=1)r4c1 loop => -1 r4c89, -2 r5c6, -3 r4c8

Code: Select all: +--------------------+------------------------+-----------------------+ | 9 8 123 | 123 7 6 | 1235 1345 1245 | | 5 123 6 | 8 123 4 | 7 9 12 | | 123 7 4 | 9 5 123 | 8 13 6 | +--------------------+------------------------+-----------------------+ | 13 9 8 | 12357 123 12357 | 6 45 457 | | 7 123 5 | 6 4 13 | 123 8 9 | | 6 4 123 | 57 9 8 | 1235 135 1257 | +--------------------+------------------------+-----------------------+ | 123 6 9 | 4 123 1235 | 15 7 8 | | 4 125 127 | 1257 8 157 | 9 6 3 | | 8 135 137 | 1357 6 9 | 4 2 15 | +--------------------+------------------------+-----------------------+

5. Tridagon #2 has one guardian True at a vertex of a rectangle (*) => Remote Triple at r1c3, r1c4, r6c3
With the Empty Rectangle at b6 for digits (1,2,3): (1,2,3)r5c7 = r6c789 - r6c3 =RT=r1c34 => -123 r1c7
End with a skyscraper:

Code: Select all: +--------------------+-----------------------+-------------------+ | 9 8 123 | 123 7 6 | 5 134 124 | | 5 123* 6 | 8 123* 4 | 7 9 12 | | 123 7 4 | 9 5 123 | 8 13 6 | +--------------------+-----------------------+-------------------+ | 13 9 8 | 12357 123 1237 | 6 45 47 | | 7 123 5 | 6 4 13 | 23 8 9 | | 6 4 23 | 57 9 8 | 23 15 17 | +--------------------+-----------------------+-------------------+ | 23* 6 9 | 4 23* 5 | 1 7 8 | | 4 5 127 | 127 8 17 | 9 6 3 | | 8 13 137 | 137 6 9 | 4 2 5 | +--------------------+-----------------------+-------------------+

6. (3)r2c2 = r2c5 - r7c5 = r7c1 => -3 r9c2, r3c1; ste

by **eleven** » Fri Jun 13, 2025 8:03 pm

shye wrote:i used a different impossible pattern though

Nice! Could not catch that.

by **pjb** » Sat Jun 14, 2025 1:31 am

Too hard for me, but interestingly Cenoman's first step is achieved using Shye's fireworks:
Fireworks triple of 469 at r5c5, r5c9 and r9c5 => -13 r9c5, -12 r5c9, -123r5c5
Phil

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