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by **champagne** » Sun May 04, 2025 5:36 am

Hope this puzzle has not yet been posted
It comes from mith's file and the skf rating is 11.6

.6435....5.7....1.8.9...3....526..31...1.......2.43.......16.45.....2....4..3..2.

After singles, I am here

Code: Select all: 12 6 4 |3 5 1789 |2789 789 2789 5 3 7 |689 289 489 |24689 1 24689 8 12 9 |67 27 147 |3 5 2467 -------------------------------------------------- 479 789 5 |2 6 789 |4789 3 1 34679 789 368 |1 789 5789 |2456789 6789 246789 1679 1789 2 |5789 4 3 |56789 6789 6789 -------------------------------------------------- 2379 2789 38 |789 1 6 |789 4 5 679 5 168 |4 789 2 |16789 6789 3 679 4 168 |5789 3 5789 |16789 2 6789

and the solution grid has the loki pattern

Hidden Text: Show

Website · by **denis_berthier** » Sun May 04, 2025 6:32 am

.
I can't see anything special with this puzzle. Tens of similar ones have been published in this section.

Generally speaking, the number of guardians is undefined if you don't specify at what moment in resolution you consider it.
Here, it's 7, be it after Singles or after whips[1]:

Code: Select all: Resolution state after Singles and whips[1]: +-------------------------+-------------------------+-------------------------+ ! 12 6 4 ! 3 5 1789 ! 2789 789 2789 ! ! 5 3 7 ! 689 289 489 ! 24689 1 24689 ! ! 8 12 9 ! 67 27 147 ! 3 5 2467 ! +-------------------------+-------------------------+-------------------------+ ! 479 789 5 ! 2 6 789 ! 4789 3 1 ! ! 34679 789 368 ! 1 789 5789 ! 2456789 6789 246789 ! ! 1679 1789 2 ! 5789 4 3 ! 56789 6789 6789 ! +-------------------------+-------------------------+-------------------------+ ! 2379 2789 38 ! 789 1 6 ! 789 4 5 ! ! 679 5 168 ! 4 789 2 ! 16789 6789 3 ! ! 679 4 168 ! 5789 3 5789 ! 16789 2 6789 ! +-------------------------+-------------------------+-------------------------+ 179 candidates.

Code: Select all: Trid-OR7-relation for digits 7, 8 and 9 in blocks: b5, with cells (marked #): r4c6, r5c5, r6c4 b6, with cells (marked #): r4c7, r5c8, r6c9 b8, with cells (marked #): r9c6, r8c5, r7c4 b9, with cells (marked #): r9c9, r8c8, r7c7 with 7 guardians (in cells marked @): n4r4c7 n6r5c8 n5r6c4 n6r6c9 n6r8c8 n5r9c6 n6r9c9 +-------------------------+-------------------------+-------------------------+ ! 12 6 4 ! 3 5 1789 ! 2789 789 2789 ! ! 5 3 7 ! 689 289 489 ! 24689 1 24689 ! ! 8 12 9 ! 67 27 147 ! 3 5 2467 ! +-------------------------+-------------------------+-------------------------+ ! 479 789 5 ! 2 6 789# ! 4789#@ 3 1 ! ! 34679 789 368 ! 1 789# 5789 ! 2456789 6789#@ 246789 ! ! 1679 1789 2 ! 5789#@ 4 3 ! 56789 6789 6789#@ ! +-------------------------+-------------------------+-------------------------+ ! 2379 2789 38 ! 789# 1 6 ! 789# 4 5 ! ! 679 5 168 ! 4 789# 2 ! 16789 6789#@ 3 ! ! 679 4 168 ! 5789 3 5789#@ ! 16789 2 6789#@ ! +-------------------------+-------------------------+-------------------------+

But this number is immediately reduced, thrice, by applying elementary short chains:

Code: Select all: biv-chain[3]: r3c5{n7 n2} - b1n2{r3c2 r1c1} - r1n1{c1 c6} ==> r1c6≠7 whip[1]: r1n7{c9 .} ==> r3c9≠7 z-chain[3]: c1n3{r5 r7} - r7n2{c1 c2} - c2n7{r7 .} ==> r5c1≠7 z-chain[3]: c1n3{r5 r7} - r7n2{c1 c2} - c2n9{r7 .} ==> r5c1≠9 z-chain[4]: c1n4{r4 r5} - c1n3{r5 r7} - r7n2{c1 c2} - c2n7{r7 .} ==> r4c1≠7 z-chain[4]: c1n4{r4 r5} - c1n3{r5 r7} - r7n2{c1 c2} - c2n9{r7 .} ==> r4c1≠9 naked-single ==> r4c1=4 At least one candidate of a previous Trid-OR6-relation between candidates n4r4c7 n6r5c8 n5r6c4 n6r6c9 n6r8c8 n6r9c9 has just been eliminated. There remains a Trid-OR5-relation between candidates: n6r5c8 n5r6c4 n6r6c9 n6r8c8 n6r9c9 At least one candidate of a previous Trid-OR6-relation between candidates n4r4c7 n6r5c8 n6r6c9 n6r8c8 n5r9c6 n6r9c9 has just been eliminated. There remains a Trid-OR5-relation between candidates: n6r5c8 n6r6c9 n6r8c8 n5r9c6 n6r9c9 hidden-pairs-in-a-block: b6{n2 n4}{r5c7 r5c9} ==> r5c9≠9, r5c9≠8, r5c9≠7, r5c9≠6, r5c7≠9, r5c7≠8, r5c7≠7, r5c7≠6, r5c7≠5 singles ==> r6c7=5, r5c6=5, r9c4=5 At least one candidate of a previous Trid-OR5-relation between candidates n6r5c8 n6r6c9 n6r8c8 n5r9c6 n6r9c9 has just been eliminated. There remains a Trid-OR4-relation between candidates: n6r5c8 n6r6c9 n6r8c8 n6r9c9 biv-chain[4]: r5c9{n2 n4} - r3n4{c9 c6} - r3n1{c6 c2} - b1n2{r3c2 r1c1} ==> r1c9≠2 z-chain[4]: c2n2{r7 r3} - r3c5{n2 n7} - c4n7{r3 r6} - c1n7{r6 .} ==> r7c2≠7 whip[1]: b7n7{r9c1 .} ==> r6c1≠7

Code: Select all: +-------------------+-------------------+-------------------+ ! 12 6 4 ! 3 5 189 ! 2789 789 789 ! ! 5 3 7 ! 689 289 489 ! 24689 1 24689 ! ! 8 12 9 ! 67 27 147 ! 3 5 246 ! +-------------------+-------------------+-------------------+ ! 4 789 5 ! 2 6 789 ! 789 3 1 ! ! 36 789 368 ! 1 789 5 ! 24 6789 24 ! ! 169 1789 2 ! 789 4 3 ! 5 6789 6789 ! +-------------------+-------------------+-------------------+ ! 2379 289 38 ! 789 1 6 ! 789 4 5 ! ! 679 5 168 ! 4 789 2 ! 16789 6789 3 ! ! 679 4 168 ! 5 3 789 ! 16789 2 6789 ! +-------------------+-------------------+-------------------+ 124 candidates

That was a lot of classical rules applying before the tridagon woke up and used its 3 mouths to reduce the puzzle to BC3 ashes.
Note that the tridaagon hasn't degenerated in the process (which anyway wouldn't change anything in what follows).

Trid-OR4-ctr-whip[6]: r3n6{c4 c9} - r3n4{c9 c6} - c6n1{r3 r1} - c1n1{r1 r6} - r6n6{c1 c8} - OR4{{n6r9c9 n6r6c9 n6r8c8 n6r5c8 | .}} ==> r3c4≠7
naked-single ==> r3c4=6
naked-pairs-in-a-column: c9{r3 r5}{n2 n4} ==> r2c9≠4, r2c9≠2
Trid-OR4-ctr-whip[8]: r3n7{c5 c6} - r3n1{c6 c2} - r6n1{c2 c1} - r1c1{n1 n2} - b3n2{r1c7 r2c7} - r2n6{c7 c9} - r6n6{c9 c8} - OR4{{n6r9c9 n6r6c9 n6r8c8 n6r5c8 | .}} ==> r3c5≠2
singles ==> r3c5=7, r2c5=2
Trid-OR4-ctr-whip[6]: r1c1{n1 n2} - c7n2{r1 r5} - c7n4{r5 r2} - r2n6{c7 c9} - r6n6{c9 c8} - OR4{{n6r5c8 n6r6c9 n6r8c8 n6r9c9 | .}} ==> r6c1≠1

Easy end in BC3:

Code: Select all: singles ==> r6c2=1, r3c2=2, r1c1=1, r3c9=4, r3c6=1, r5c9=2, r5c7=4, r1c7=2, r2c6=4, r7c1=2, r7c3=3, r5c1=3 finned-x-wing-in-rows: n7{r7 r6}{c4 c7} ==> r4c7≠7 whip[1]: c7n7{r9 .} ==> r8c8≠7, r9c9≠7 biv-chain[3]: r5n7{c2 c8} - r5n6{c8 c3} - r6c1{n6 n9} ==> r5c2≠9 biv-chain[3]: r8c5{n8 n9} - r5n9{c5 c8} - r4c7{n9 n8} ==> r8c7≠8 biv-chain[3]: r5n9{c5 c8} - r5n7{c8 c2} - r4n7{c2 c6} ==> r4c6≠9 biv-chain[3]: r8c5{n8 n9} - b5n9{r5c5 r6c4} - c4n7{r6 r7} ==> r7c4≠8 finned-swordfish-in-rows: n8{r4 r7 r1}{c6 c2 c7} ==> r2c7≠8 biv-chain[3]: b5n7{r4c6 r6c4} - r7c4{n7 n9} - c2n9{r7 r4} ==> r4c2≠7 singles ==> r5c2=7, r4c6=7, r7c4=7 naked-pairs-in-a-column: c7{r4 r7}{n8 n9} ==> r9c7≠9, r9c7≠8, r8c7≠9, r2c7≠9 naked-single ==> r2c7=6 finned-x-wing-in-columns: n6{c9 c1}{r6 r9} ==> r9c3≠6 biv-chain[2]: r2n9{c9 c4} - c6n9{r1 r9} ==> r9c9≠9 biv-chain[2]: c6n8{r9 r1} - r2n8{c4 c9} ==> r9c9≠8 naked-single ==> r9c9=6 naked-pairs-in-a-row: r8{c5 c8}{n8 n9} ==> r8c3≠8, r8c1≠9 x-wing-in-rows: n9{r5 r8}{c5 c8} ==> r6c8≠9, r1c8≠9 whip[1]: b3n9{r2c9 .} ==> r6c9≠9 finned-x-wing-in-columns: n8{c3 c5}{r5 r9} ==> r9c6≠8 stte

by **champagne** » Sun May 04, 2025 6:45 am

denis_berthier wrote:.
Generally speaking, the number of guardians is undefined if you don't specify at what moment in resolution you consider it.
Here, it's 7, be it after Singles or after whips[1]:

Nice to read this after recent posts

by **P.O.** » Sun May 04, 2025 7:27 am

why not start the resolution process of a grid in T&E(N,Singles) with the resolution state after T&E(N-1,Singles)

Code: Select all: after T&E(2,Singles) 12 6 4 3 5 18 2789 789 789 5 3 7 689 289 489 246 1 2468 8 12 9 67 27 147 3 5 246 4 789 5 2 6 789 789 3 1 36 789 36 1 789 5 24 789 24 19 1789 2 789 4 3 5 6 789 2379 29 38 789 1 6 789 4 5 679 5 168 4 789 2 167 789 3 679 4 168 5 3 789 16 2 6789 124 candidates. 37 values. .6435....537....1.8.9...35.4.526..31...1.5.....2.4356.....16.45.5.4.2..3.4.53..2.

basics:

Hidden Text: Show

tridagon pattern in b5689 with one guardian n6r9c9 => r9c9 = 6
basics:

Hidden Text: Show

Code: Select all: 1 6 4 3 5 89 2 789 789 5 3 7 89 2 4 6 1 89 8 2 9 6 7 1 3 5 4 4 789 5 2 6 789 789 3 1 3 789 6 1 89 5 4 789 2 79 1 2 789 4 3 5 6 789 2 79 3 789 1 6 89 4 5 6 5 18 4 89 2 1789 789 3 79 4 18 5 3 789 189 2 6 65 candidates. 56 values. 7r69c1 => r4c2 r6c4 <> 7 r9c1=7 - c6n7{r9 r4} ste.

by **champagne** » Sun May 04, 2025 7:38 am

P.O. wrote:why not start the resolution process of a grid in T&E(N,Singles) with the resolution state after T&E(N-1,Singles)

In my code, I start from T&E(1,Singles), so yes, this can be done to give indication of the hardness, but AFAIK, the T&E rule is not in the bag of manual solvers, so our experts should come with something else if there are nice paths.

your T&E(2,singles) let's hope to have eliminations of all guardians but one.
I proposed this puzzle just because I had still 6 guardians with my "T&E(1) filter.

by **P.O.** » Sun May 04, 2025 7:44 am

the resolution state after T&E(1,Singles) and T&E(2,Singles) are the same

by **champagne** » Sun May 04, 2025 7:53 am

P.O. wrote:the resolution state after T&E(1,Singles) and T&E(2,Singles) are the same

Thanks for this, I have to check what is wrong on my side.
I checked the skfr path, the "one guardian" appears in the path, after a long and boring sequence. Difficult to compare eliminations with a T&E view

Website · by **denis_berthier** » Sun May 04, 2025 8:11 am

P.O. wrote:the resolution state after T&E(1,Singles) and T&E(2,Singles) are the same

Confirmed:

Code: Select all: +----------------+----------------+----------------+ ! 12 6 4 ! 3 5 18 ! 2789 789 789 ! ! 5 3 7 ! 689 289 489 ! 246 1 2468 ! ! 8 12 9 ! 67 27 147 ! 3 5 246 ! +----------------+----------------+----------------+ ! 4 789 5 ! 2 6 789 ! 789 3 1 ! ! 36 789 36 ! 1 789 5 ! 24 789 24 ! ! 19 1789 2 ! 789 4 3 ! 5 6 789 ! +----------------+----------------+----------------+ ! 2379 29 38 ! 789 1 6 ! 789 4 5 ! ! 679 5 168 ! 4 789 2 ! 167 789 3 ! ! 679 4 168 ! 5 3 789 ! 16 2 6789 ! +----------------+----------------+----------------+

Only 1 guardian remaining.

Website · by **denis_berthier** » Sun May 04, 2025 8:14 am

champagne wrote: the T&E rule is not in the bag of manual solvers.

There's no T&E "rule"; it's a procedure.
I would say it's the only way of solving that the vast majority of manual solvers know. Most Sudoku players don't even know what a Pair is.
.

Website · by **denis_berthier** » Sun May 04, 2025 8:17 am

P.O. wrote:why not start the resolution process of a grid in T&E(N,Singles) with the resolution state after T&E(N-1,Singles)

Because it's absurd. The T&E(N-1) part can be much harder than applying the tridagon before it.
.

by **yzfwsf** » Sun May 04, 2025 8:21 am

Code: Select all: Hidden Single: 3 in r2 => r2c2=3 Hidden Single: 5 in r3 => r3c8=5 Hidden Single: 4 in r8 => r8c4=4 Hidden Single: 5 in r8 => r8c2=5 Hidden Single: 3 in c9 => r8c9=3 Locked Candidates 2 (Claiming): 1 in c3 => r8c1<>1,r9c1<>1 M3-Wing: (7=2)r3c5 - r3c2 = (2-1)r1c1 = 1r1c6 => r1c6<>7 Locked Candidates 2 (Claiming): 7 in r1 => r3c9<>7 Grouped AIC Type 2: 4r4c1 = (4-3)r5c1 = (3-2)r7c1 = (2-7)r7c2 = 7r456c2 => r4c1<>7 Grouped Discontinuous Nice Loop: 4r4c1 = (4-3)r5c1 = (3-2)r7c1 = (2-9)r7c2 = r456c2 - (9=4)r4c1 => r4c1=4 Hidden Pair: 24 in r5c7,r5c9 => r5c7<>56789,r5c9<>6789 Hidden Single: 5 in r5 => r5c6=5 Hidden Single: 5 in r6 => r6c7=5 Hidden Single: 5 in r9 => r9c4=5 Uniqueness Test 7: 24 in r25c79; 2*biCell + 1*conjugate pairs(4c7) => r2c9 <> 2 AIC Type 1: 2r1c1 = (2-1)r3c2 = (1-4)r3c6 = r3c9 - (4=2)r5c9 => r1c9<>2 Grouped L3-Wing: 3r5c1 = (3-2)r7c1 = (2-7)r7c2 = 7r456c2 => r5c1<>7 Grouped L3-Wing: 3r5c1 = (3-2)r7c1 = (2-9)r7c2 = 9r456c2 => r5c1<>9 Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r1c1<>2 6r5c8 - 6r6c89 = 6r6c1 - (6=3792)r5789c1 6r6c9 - 6r23c9 = 6r2c7 - (6=17892)r14789c7 6r8c8 - (6=17892)r14789c7 6r9c9 - (6=17892)r14789c7

After the above steps, the difficulty of the puzzle is reduced to skfr 7.6.
4 guardians forcing chain.

by **Cenoman** » Sun May 04, 2025 3:30 pm

After basics (5 singles, locked 1r89c3)

Code: Select all: +-----------------------+----------------------+----------------------------+ | 12 6 4 | 3 5 1789 | 2789 789 2789 | | 5 3 7 | 689 289 489 | 24689 1 24689 | | 8 12 9 | 67 27 147 | 3 5 2467 | +-----------------------+----------------------+----------------------------+ | a479 d789 5 | 2 6 789 | 4789 3 1 | | a34679 d789 368 | 1 789 5789 | 2456789 6789 246789 | | 1679 d1789 2 | 5789 4 3 | 56789 6789 6789 | +-----------------------+----------------------+----------------------------+ | b2379 c2789 38 | 789 1 6 | 789 4 5 | | 679 5 168 | 4 789 2 | 16789 6789 3 | | 679 4 168 | 5789 3 5789 | 16789 2 6789 | +-----------------------+----------------------+----------------------------+

1. (43)r45c1 = (3-2)r7c1 = (2-7|9)r7c2 = (79)r456c2 => -79 r45c1; lcls, 4 placements

Code: Select all: +----------------------+---------------------+-------------------------+ | a1-2 6 4 | 3 5 1789 | h2789 789 2789 | | 5 3 7 | 689 289 489 | g24689 1 24689 | | 8 12 9 | 67 27 147 | 3 5 2467 | +----------------------+---------------------+-------------------------+ | 4 789 5 | 2 6 789* | 789* 3 1 | | 36 789 368 | 1 789* 5 | g24 d6789* 24 | | b1679 1789 2 | 789* 4 3 | 5 c6789 c6789* | +----------------------+---------------------+-------------------------+ | 2379 2789 38 | 789* 1 6 | 789* 4 5 | | 679 5 168 | 4 789* 2 | f16789 e6789* 3 | | 679 4 168 | 5 3 789* | f16789 2 e6789* | +----------------------+---------------------+-------------------------+

2. Tridagon (789)b5689 having four guardians: 6b69p59
Note that 6r6c9 and 6r8c8 have the same valence (both conjugates of 6r56c8) => considering three guardians is enough, e.g. (6)r5c8 = r8c8 = r9c9
(1)r1c1 = (1-6)r6c1 = r6c89 - (6)r5c8 == (6)b9p59 - r89c7 = (64-2)r25c7 (2)r1c7 => -2 r1c1; 15 placements

End with four rather simple steps:

Code: Select all: +--------------------+-------------------+------------------------+ | 1 6 4 | 3 5 b89 | 2 a789 789 | | 5 3 7 | c89 2 4 | 689 1 689 | | 8 2 9 | 6 7 1 | 3 5 4 | +--------------------+-------------------+------------------------+ | 4 z789 5 | 2 6 789^ | 9-78 3 1 | | 3 y789 y68 | 1 x89 5 | 4 A6789 2 | | 679* 1 2 | d78-9* 4 3 | 5 Ae6789 e6789 | +--------------------+-------------------+------------------------+ | 2 89-7 3 | 789*^ 1 6 | 789^ 4 5 | | 679* 5 168 | 4 w89 2 | 16789 v6789 3 | | 679* 4 168 | 5 3 789^ | 16789 2 6789 | +--------------------+-------------------+------------------------+

3. (7)r7c4 = r6c4 - r6c1 = r89c1 => -7 r7c2 (locked 7r89c1 => -7 r6c1)
4. (9=6)r6c1 - (6=8)r5c3 - (8=9)r5c5 => -9 r6c4
5. (7)r7c7 = r7c4 - r9c6 = r4c6 => -7 r4c7
6. Kraken column (8)r1568c8
(8)r1c8 - r1c6 = r2c4 - r6c4 = (8)r6c89
(8)r56c8
(8)r8c8 - r8c5 = r5c5 - r5c23 = (8)r4c2
=>-8r4c7; ste

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