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Website · by **denis_berthier** » Wed Aug 13, 2025 4:58 am

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Code: Select all: +-------+-------+-------+ ! 1 . 3 ! . . . ! 7 8 . ! ! . 5 . ! . . . ! . . 3 ! ! 7 8 . ! 1 . . ! . 6 5 ! +-------+-------+-------+ ! . . . ! . 9 . ! 6 5 . ! ! . 6 . ! . 4 2 ! . . . ! ! . . 7 ! 8 . . ! . . . ! +-------+-------+-------+ ! 6 7 . ! . . 5 ! 8 3 . ! ! . 3 1 ! . . . ! . . . ! ! . . . ! . . . ! . 1 . ! +-------+-------+-------+ 1.3...78..5......378.1...65....9.65..6..42.....78.....67...583..31.............1.

[Edit]: added for clarity

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1 249 3 ! 24569 256 469 ! 7 8 249 ! ! 249 5 6 ! 2479 278 4789 ! 1 249 3 ! ! 7 8 249 ! 1 23 349 ! 249 6 5 ! +----------------------+----------------------+----------------------+ ! 234 124 24 ! 37 9 137 ! 6 5 8 ! ! 3589 6 589 ! 35 4 2 ! 39 7 1 ! ! 359 19 7 ! 8 356 136 ! 2349 249 249 ! +----------------------+----------------------+----------------------+ ! 6 7 249 ! 249 1 5 ! 8 3 249 ! ! 24589 3 1 ! 24679 2678 46789 ! 2459 249 24679 ! ! 24589 249 24589 ! 234679 23678 346789 ! 2459 1 24679 ! +----------------------+----------------------+----------------------+ 169 candidates.

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by **Cenoman** » Wed Aug 13, 2025 9:53 pm

Code: Select all: +----------------------+----------------------------+---------------------+ | 1 249* 3 | 24569 256 469 | 7 8 249* | | 249* 5 6 | 2479 278 4789 | 1 249* 3 | | 7 8 249* | 1 23 349 | 249* 6 5 | +----------------------+----------------------------+---------------------+ | 234 124 24 | 37 9 137 | 6 5 8 | | 3589 6 58 | 35 4 2 | 39 7 1 | | 359 19 7 | 8 356 136 | 2349 249 249 | +----------------------+----------------------------+---------------------+ | 6 7 249* | 249 1 5 | 8 3 249* | | 58-249* 3 1 | 24679 2678 46789 | 2459 249* 67 | | 249-58 249* 58 | 234679 23678 346789 | 2459* 1 67 | +----------------------+----------------------------+---------------------+

1. Tridagon (249)b1379 having 3 guardians: 58r8c1, 5r9c7
Note that 5r8c1 & 5r9c7 have the same valence (both conjugates of 5r8c7), therefore (5,8)r8c1 is an effective set of guardians => -249 r8c1; NP(58)b7p49 => -58 r9c1

A bunch of classical steps:

Code: Select all: +---------------------+----------------------------+---------------------+ | 1 249 3 | 24569 56-2 469 | 7 8 249 | | 24-9 5 6 | 247-9 278 4789 | 1 249 3 | | 7 8 249 | 1 23 349 | 24-9 6 5 | +---------------------+----------------------------+---------------------+ | 234 124 24 | 37 9 137 | 6 5 8 | | 3589 6 58 | 35 4 2 | 39 7 1 | | 59-3 19 7 | 8 356 136 | 2349 24-9 249 | +---------------------+----------------------------+---------------------+ | 6 7 249 | 249 1 5 | 8 3 249 | | 58 3 1 | 24679 2678 46789 | 245-9 249 67 | | 249 249 58 | 23467-9 23678 34678-9 | 2459 1 67 | +---------------------+----------------------------+---------------------+

2. (5)r1c5 = (5-6)r1c4 = r89c4 - r89c5 = (65)r16c5 => -2 r1c5
3. (3)r4c1 = r4c46 - (3=5)r5c4 - r6c5 = (5)r6c1 => -3 r6c1
4. (9)r5c7 = (9-8)r5c1 = (8-5)r8c1 = (5)r8c7 => -9 r8c7

5. Kraken cell (2459)r9c7 =>-9r2c1
(2|4)r9c7 - (24)r9c12 = (2|4-9)r7c3 = (9)r3c3
(5)r9c7 - r9c3 = r5c3 - (5=3)r5c4 - r4c46 = r4c1 - (3=589)r568c1
(9)r9c7-r5c7=(9)r5c1

6. Kraken cell (2459)r9c7 =>-9r3c7
(2|4)r9c7 - (24)r9c12 = (2|4-9)r7c3 = (9)r3c3
(5)r9c7 - r9c3 - (5=3)r5c4 - (3=9)r5c7
(9)r9c7

7. Subsequent X-chains:
#(9)r8c46 = r8c8 - ^r2c8 = r2c46 - *r3c6 = r3c3 - r7c3 = r9c12*# - r9c7 = r56c7^ => -9 r9c6*, r6c8^, r9c4#
(9)r123c6 = r8c6 - r8c8 = r2c8 => -9 r2c4

A uniqueness step:

Code: Select all: +---------------------+--------------------------+---------------------+ | 1 249 3 | 24569 56 469 | 7 8 249 | | 24 5 6 | 247 278* 4789* | 1 249 3 | | 7 8 249 | 1 23 349 | 24 6 5 | +---------------------+--------------------------+---------------------+ | 234 124 24 | 37 9 137 | 6 5 8 | | 3589 6 58 | 35 4 2 | 39 7 1 | | 59 19 7 | 8 56-3 136 | 2349 24 249 | +---------------------+--------------------------+---------------------+ | 6 7 249 | 249 1 5 | 8 3 249 | | 58 3 1 | 24679 #278-6* 46789 | 245 249 67* | | 249 249 58 | 23467 2378-6* 34678* | 2459 1 67* | +---------------------+--------------------------+---------------------+

8. MUG(678)r2c56, r8c59, r9c569 using mixed internals-external
(23)r389c5
(3)r9c6 - r3c6 = (3)r3c5
(4)r9c6 - (4=295)r9c127 - r9c3 = r5c3 - (5=3)r5c4
(7)r2c4 - (7=3)r4c4
=> -3 r6c5; NP(56)r16c5 (-6r89c5)

The closure of the dragon's mouth:

Code: Select all: +---------------------+-------------------------+---------------------+ | 1 249 3 | 24569 56 469 | 7 8 249 | | 24 5 6 | 247 278 4789 | 1 249 3 | | 7 8 249 | 1 23 349 | 24 6 5 | +---------------------+-------------------------+---------------------+ | 234 124 24 | 37 9 137 | 6 5 8 | | 3589 6 8-5 | 35 4 2 | 39 7 1 | | 59 19 7 | 8 56 136 | 2349 24 249 | +---------------------+-------------------------+---------------------+ | 6 7 249 | 249 1 5 | 8 3 249 | | 8-5 3 1 | 24679 278 46789 | 5-24 249 67 | | 249 249 5-8 | 367-24 378-2 3678-4 | 2459 1 67 | +---------------------+-------------------------+---------------------+

9. Triple Kraken cells (249)r9c1, (247)r2c4, (249)r7c4 =>
(2)r9c1* - [(2=4)r2c1 - (4=7)r2c4 - (7=3)r4c4 - (3=24)r24c1] = (2)r2c4 - [(2=9)r7c4 - r8c46 = r8c8 - (9=24)r2c18] = (4)r7c4^
(4)r9c1^ - [(4=2)r2c1 - (2=7)r2c4 - (7=3)r4c4 - (3=24)r24c1] = (4)r2c4 - [(4=9)r7c4 - r8c46 = r8c8 - (9=24)r2c18] = (2)r7c4*
(9)r9c1 - (9=5)r6c1 - r8c1 = r9c3 - (5=249)r9c127*^
=> -2 r9c45*, -4r9c46^; NQ(3678)r9c4569 => -8 r9c3, 4 placements (+5r9c3, +8r8c1, +8r5c3, +5r8c7)

Code: Select all: +--------------------+-----------------------+---------------------+ | 1 249 3 | 24569 56 469 | 7 8 249 | | a24* 5 6 | 7-24 278 4789 | 1 a249* 3 | | 7 8 249 | 1 23 349 | 24 6 5 | +--------------------+-----------------------+---------------------+ | 234 124 24 | 37 9 137 | 6 5 8 | | 359 6 8 | 35 4 2 | 39 7 1 | | 59 19 7 | 8 56 136 | 2349 24 249 | +--------------------+-----------------------+---------------------+ | 6 7 249 | e249 1 5 | 8 3 249 | | 8 3 1 | d24679 d27 d4679 | 5 c249* 67 | | 249 249 5 | 367 378 3678 | 249 1 67 | +--------------------+-----------------------+---------------------+

10. The original tridagon has eventually only one guardian true => (249)r2c18, r8c8 are Remote Triple
Together with empty rectangle (2,4)b8: (2,4)r2c18 == r8c8 - r8c456 = r7c4 => -24 r2c4; 20 placements

End with a remote pair:

Code: Select all: +-----------------+-------------------+-------------------+ | 1 24* 3 | 249 5 6 | 7 8 9-24 | | 24 5 6 | 7 28 489 | 1 249 3 | | 7 8 9 | 1 23 34 | 24* 6 5 | +-----------------+-------------------+-------------------+ | 24 1 24 | 3 9 7 | 6 5 8 | | 3 6 8 | 5 4 2 | 9 7 1 | | 5 9 7 | 8 6 1 | 3 24 24 | +-----------------+-------------------+-------------------+ | 6 7 24 | 249 1 5 | 8 3 249 | | 8 3 1 | 249 7 49 | 5 249 6 | | 9 24* 5 | 6 38 38 | 24* 1 7 | +-----------------+-------------------+-------------------+

11. (2)r1c2 = r9c2 - r9c7 = r3c7 => -24 r1c9; ste

Website · by **denis_berthier** » Fri Aug 15, 2025 5:10 am

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Same start:

hidden-pairs-in-a-column: c3{n5 n8}{r5 r9} ==> r9c3≠9, r9c3≠4, r9c3≠2, r5c3≠9

Code: Select all: Trid-OR3-relation for digits 2, 4 and 9 in blocks: b1, with cells (marked #): r1c2, r2c1, r3c3 b3, with cells (marked #): r1c9, r2c8, r3c7 b7, with cells (marked #): r9c2, r8c1, r7c3 b9, with cells (marked #): r9c7, r8c8, r7c9 with 3 guardians (in cells marked @): n5r8c1 n8r8c1 n5r9c7 +-------------------------+-------------------------+-------------------------+ ! 1 249# 3 ! 24569 256 469 ! 7 8 249# ! ! 249# 5 6 ! 2479 278 4789 ! 1 249# 3 ! ! 7 8 249# ! 1 23 349 ! 249# 6 5 ! +-------------------------+-------------------------+-------------------------+ ! 234 124 24 ! 37 9 137 ! 6 5 8 ! ! 3589 6 58 ! 35 4 2 ! 39 7 1 ! ! 359 19 7 ! 8 356 136 ! 2349 249 249 ! +-------------------------+-------------------------+-------------------------+ ! 6 7 249# ! 249 1 5 ! 8 3 249# ! ! 24589#@ 3 1 ! 24679 2678 46789 ! 2459 249# 67 ! ! 24589 249# 58 ! 234679 23678 346789 ! 2459#@ 1 67 ! +-------------------------+-------------------------+-------------------------+

Trid-OR3-relation between candidates n5r8c1, n8r8c1 and n5r9c7
+ same valence for candidates n5r9c7 and n5r8c1 via c-chain[2]: n5r9c7,n5r8c7,n5r8c1
==> Trid-OR3-relation can be split into two Trid-OR2-relations with respective lists of guardians:
n5r8c1 n8r8c1 and n8r8c1 n5r9c7 .

Trid-OR2-whip[1]: OR2{{n8r8c1 n5r8c1 | .}} ==> r8c1≠9, r8c1≠2, r8c1≠4

The rest is standard chains in gW8

Code: Select all: naked-pairs-in-a-block: b7{r8c1 r9c3}{n5 n8} ==> r9c1≠8, r9c1≠5 biv-chain[3]: r6n5{c1 c5} - r5c4{n5 n3} - b6n3{r5c7 r6c7} ==> r6c1≠3 biv-chain[3]: r8n5{c7 c1} - r6c1{n5 n9} - r5n9{c1 c7} ==> r8c7≠9 z-chain[4]: c5n5{r1 r6} - r6n6{c5 c6} - r1n6{c6 c4} - r1n5{c4 .} ==> r1c5≠2 whip[8]: b7n9{r9c2 r7c3} - r3n9{c3 c7} - r5n9{c7 c1} - c1n8{r5 r8} - r9n8{c3 c5} - r9n3{c5 c4} - r9n6{c4 c9} - r9n7{c9 .} ==> r9c6≠9 g-whip[5]: c3n9{r3 r7} - r9n9{c2 c4} - c4n3{r9 r456} - r6n3{c5 c7} - r5c7{n3 .} ==> r3c7≠9 z-chain[3]: b3n9{r2c8 r1c9} - c2n9{r1 r9} - c7n9{r9 .} ==> r6c8≠9 finned-x-wing-in-columns: n9{c8 c6}{r8 r2} ==> r2c4≠9 g-whip[8]: r8n9{c6 c8} - b3n9{r2c8 r1c9} - c2n9{r1 r6} - r6c1{n9 n5} - r8n5{c1 c7} - r8n2{c7 c456} - r7c4{n2 n4} - r8n4{c4 .} ==> r9c4≠9 z-chain[3]: b4n9{r6c1 r6c2} - r9n9{c2 c7} - r5n9{c7 .} ==> r2c1≠9 whip[8]: r2c1{n2 n4} - r1c2{n4 n9} - r6c2{n9 n1} - c6n1{r6 r4} - r4n7{c6 c4} - r2c4{n7 n2} - r7n2{c4 c9} - r1n2{c9 .} ==> r3c3≠2 whip[5]: c7n3{r6 r5} - c7n9{r5 r9} - b7n9{r9c1 r7c3} - r3c3{n9 n4} - r3c7{n4 .} ==> r6c7≠2 whip[8]: r3c3{n9 n4} - r2c1{n4 n2} - r9c1{n2 n4} - r4c1{n4 n3} - r4c4{n3 n7} - r2c4{n7 n4} - r7n4{c4 c9} - r1n4{c9 .} ==> r7c3≠9 hidden-single-in-a-column ==> r3c3=9 whip[1]: b7n9{r9c2 .} ==> r9c7≠9 whip[1]: c7n9{r6 .} ==> r6c9≠9 naked-pairs-in-a-block: b6{r6c8 r6c9}{n2 n4} ==> r6c7≠4 whip[7]: r2c1{n2 n4} - r4c1{n4 n3} - r4c4{n3 n7} - r2c4{n7 n2} - r7n2{c4 c9} - b9n9{r7c9 r8c8} - r2c8{n9 .} ==> r9c1≠2 biv-chain[3]: b1n4{r1c2 r2c1} - c1n2{r2 r4} - r4c3{n2 n4} ==> r4c2≠4 t-whip[5]: c2n4{r1 r9} - c2n9{r9 r6} - r6c1{n9 n5} - r8n5{c1 c7} - c7n4{r8 .} ==> r1c9≠4 biv-chain[3]: b6n2{r6c8 r6c9} - c9n4{r6 r7} - b9n9{r7c9 r8c8} ==> r8c8≠2 finned-x-wing-in-rows: n2{r3 r8}{c7 c5} ==> r9c5≠2 biv-chain[3]: r2c1{n2 n4} - b3n4{r2c8 r3c7} - r3n2{c7 c5} ==> r2c4≠2, r2c5≠2 biv-chain[3]: r3c6{n3 n4} - r2c4{n4 n7} - b5n7{r4c4 r4c6} ==> r4c6≠3 biv-chain[3]: r3c6{n3 n4} - r2c4{n4 n7} - r4c4{n7 n3} ==> r6c6≠3 biv-chain[5]: r8n5{c7 c1} - r6n5{c1 c5} - b2n5{r1c5 r1c4} - b2n2{r1c4 r3c5} - r3c7{n2 n4} ==> r8c7≠4 z-chain[5]: r8n2{c5 c7} - r8n5{c7 c1} - r6c1{n5 n9} - r9n9{c1 c2} - b7n2{r9c2 .} ==> r7c4≠2 z-chain[3]: r7c4{n4 n9} - b9n9{r7c9 r8c8} - r8n4{c8 .} ==> r9c6≠4, r9c4≠4 t-whip[3]: r4n7{c6 c4} - r2c4{n7 n4} - c6n4{r3 .} ==> r8c6≠7 biv-chain[4]: r2n9{c6 c8} - c9n9{r1 r7} - r7c4{n9 n4} - r2c4{n4 n7} ==> r2c6≠7 biv-chain[4]: r6n6{c5 c6} - b5n1{r6c6 r4c6} - c6n7{r4 r9} - r9c9{n7 n6} ==> r9c5≠6 z-chain[4]: r7c4{n4 n9} - c9n9{r7 r1} - r2c8{n9 n2} - r2c1{n2 .} ==> r2c4≠4 stte

I chose this puzzle as an example of what one gets by the systematic procedure of generating T&E(1) puzzles with high B rating form T&E(3) puzzles.
It is in B12 (very far from the highest one (B24) obtained by the procedure). See this thread: http://forum.enjoysudoku.com/1-brt-expansion-paths-within-t-e-n-and-beyond-t45647.html.

After the single guardian case, this is the simplest way a tridagon can be slain: a Trid-O2-whip[1].
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