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Website · by **denis_berthier** » Wed Sep 06, 2023 6:26 pm

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Code: Select all: +-------+-------+-------+ ! . . . ! . . . ! . . 1 ! ! . . 1 ! . 2 3 ! . 4 5 ! ! . . 5 ! 1 . 6 ! 3 2 . ! +-------+-------+-------+ ! . . . ! . 3 2 ! . . . ! ! . 7 . ! . . . ! . 5 . ! ! . 8 9 ! . . . ! . 6 . ! +-------+-------+-------+ ! . . . ! . 6 4 ! 2 . . ! ! 4 . . ! 2 . 5 ! . . 6 ! ! 6 . . ! 3 1 . ! 5 . 4 ! +-------+-------+-------+ ........1..1.23.45..51.632.....32....7.....5..89....6.....642..4..2.5..66..31.5.4 SER = PGX = 7.1

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 23789 23469 234678 ! 45789 45789 789 ! 6789 789 1 ! ! 789 69 1 ! 789 2 3 ! 6789 4 5 ! ! 789 49 5 ! 1 4789 6 ! 3 2 789 ! +----------------------+----------------------+----------------------+ ! 15 1456 46 ! 456789 3 2 ! 14789 1789 789 ! ! 123 7 2346 ! 4689 489 189 ! 1489 5 2389 ! ! 1235 8 9 ! 457 457 17 ! 147 6 237 ! +----------------------+----------------------+----------------------+ ! 135789 1359 378 ! 789 6 4 ! 2 13789 789 ! ! 4 139 378 ! 2 789 5 ! 1789 13789 6 ! ! 6 29 278 ! 3 1 789 ! 5 789 4 ! +----------------------+----------------------+----------------------+ 182 candidates.

by **marek stefanik** » Thu Sep 07, 2023 9:30 am

Code: Select all: ,-----------------------,-------------------,-------------------, | 23789 23469 234678 | 45789 45789 789 | 6789 789 1 | | 789 69 1 | 789 2 3 | 6789 4 5 | | 789 49 5 | 1 4789 6 | 3 2 789 | :-----------------------+-------------------+-------------------: |a15 b*1456 46 | 46789 3 2 | 14789 789–1 789 | | 123 7 2346 | 4689 489 189 | 1489 5 23 | | 23 8 9 | 457 457 17 | 147 6 23 | :-----------------------+-------------------+-------------------: | 135789 c1359 378 | 789 6 4 | 2 *13789 789 | | 4 *139 378 | 2 789 5 |*1789 *13789 6 | | 6 29 278 | 3 1 789 | 5 789 4 | '-----------------------'-------------------'-------------------'

(1=5)r4c1 – 5r4c2 = (5–1)r7c2 = err 1c2b9\r48c8 => –1r4c8

Code: Select all: ,-----------------------,-------------------,-----------------, | 23–789 2346–9 2346–78| 45789 45789 #789 | 6789 #789 1 | | 789 69 1 |#789 2 3 |#6789 4 5 | | 789 49 5 | 1 #4789 6 | 3 2 #789 | :-----------------------+-------------------+-----------------: | 15 1456 46 | 46789 3 2 | 14789 789 789 | | 123 7 2346 | 4689 489 189 | 1489 5 23 | | 23 8 9 | 457 457 17 | 147 6 23 | :-----------------------+-------------------+-----------------: | 135789 1359 378 |#789 6 4 | 2 13 #789 | | 4 139 378 | 2 #789 5 |#789 13 6 | | 6 29 278 | 3 1 #789 | 5 #789 4 | '-----------------------'-------------------'-----------------'

TH 789# needs an external in r1c457 which creates a triple with r1c68 => –789r1c123

Code: Select all: ,------------,-----------------,----------------, | 2 3 46 | 45 45789 789 | 6789 789 1 | | 78 69 1 | 789 2 3 | 6789 4 5 | | 78 49 5 | 1 789–4 6 | 3 2 789 | :------------+-----------------+----------------: | 5 46 46 |#789 3 2 | 1 #789 789 | | 1 7 2 | 6 489 89 | 489 5 3 | | 3 8 9 | 45 457 1 | 47 6 2 | :------------+-----------------+----------------: | 9 5 3 |#78 6 4 | 2 1 #78 | | 4 1 78 | 2 789 5 | 789 3 6 | | 6 2 78 | 3 1 789 | 5 #789 4 | '------------'-----------------'----------------'

789: The digit in r4c9 must prevent bivalue oddagon in # by appearing in r7c4 or r9c8, hence each of 789 appears in r47c4 or r49c8.
Then the digit in r1c8 appears in r47c4 and in b2 is forced into r3c5 => –4r3c5, stte

Marek

by **Cenoman** » Thu Sep 07, 2023 3:59 pm

Code: Select all: +----------------------------+------------------------+------------------------+ | 23789 23469 234678 | 45789 45789 789 | 6789 789 1 | | 789 69 1 | 789 2 3 | 6789 4 5 | | 789 49 5 | 1 4789 6 | 3 2 789 | +----------------------------+------------------------+------------------------+ | 15 1456 46 | 46789 3 2 | 14789 1789 789 | | 23+1* 7 2346 | 4689 489 189 | 1489 5 23* | | 23* 8 9 | 457 457 17 | 147 6 23* | +----------------------------+------------------------+------------------------+ | 135789 1359 378 | 789 6 4 | 2 13789 789 | | 4 139 378 | 2 789 5 | 1789 13789 6 | | 6 29 278 | 3 1 789 | 5 789 4 | +----------------------------+------------------------+------------------------+

1. UR(23)r56c19 having a single internal => +1 r5c1; lcls, 18 placements
Added 2, slightly different TH move:

Hidden Text: Show

Code: Select all: +-----------------+----------------------+----------------------+ | 2 3 a46 | 45 45789 789 | a6789 789 1 | | 78 b69 1 | B789 2 3 | A6789 4 5 | | 78 4-9 5 | 1 4789 6 | 3 2 zE789 | +-----------------+----------------------+----------------------+ | 5 46 46 |xC789* 3 2 | 1 x789* yD789 | | 1 7 2 | 6 B489 B89 | A489 5 3 | | 3 8 9 | 45 457 1 | 47 6 2 | +-----------------+----------------------+----------------------+ | 9 5 3 | 78* 6 4 | 2 1 78* | | 4 1 78 | 2 789 5 | v789 3 6 | | 6 2 78 | 3 1 789 | 5 w789* 4 | +-----------------+----------------------+----------------------+

Look at the 5-link bivalue oddagon (78)r47, c48, b9 (*) [Marek's], having three guardians => (9)r4c48 = (9)r9c8
Then:
2. Kraken column (9)r1258c7
(96)r1c37 - (6=9)r2c2
(9)r2c7|r5c7 - r2c4&r5c56 = r4c4 - r4c9 = (9)r3c9
(9)r8c7 - r9c8 == r4c48 - r4c9 = (9)r3c9
=> -9 r3c2; ste

Added 1, finish with AIC's:

Hidden Text: Show

by **eleven** » Fri Sep 08, 2023 2:21 pm

The puzzle is not hard, can be solved with normal advanced techniques and a short chain.
However, after the UR 23 and basics there is a 15 cell impossible pattern:

Code: Select all: +---------------------+-----------------------+-----------------------+ | 2 3 46 | 45 45789 #789 | 6789 #789 1 | | 78 69 1 | #789 2 3 | #789+6 4 5 | | 78 49 5 | 1 4789 6 | 3 2 #789 | +---------------------+-----------------------+-----------------------+ | 5 46 46 | #789 3 2 | 1 #789 #789 | | 1 7 2 | 6 489 #89 | 489 5 3 | | 3 8 9 | 45 457 1 | 47 6 2 | +---------------------+-----------------------+-----------------------+ | 9 5 3 | #78 6 4 | 2 1 #78 | | 4 1 78 | 2 #789 5 | #789 3 6 | | 6 2 78 | 3 1 #789 | 5 #789 4 | +---------------------+-----------------------+-----------------------+

=> 6r2c7, stte

Hidden Text: Show

Code: Select all: *-------------------------------------------------* | . . . | . . 789 | . 789 . | | . . . | 789 . - | 789 . . | | . . . | . . . | . . 789 | |------------+------------------+-----------------| | . . . | 789 . . | . 789 789 | | . . . | . . 789 | . . . | | . . . | . . . | . . . | |------------+------------------+-----------------| | . . . | 789 . . | . . 789 | | . . . | . 789 . | 789 . . | | . . . | . . 789 | . 789 . | *-------------------------------------------------*

7 8 9

Code: Select all: *-------------------------------------------------* | . . . | . . 89 | . 7 . | | . . . |*789 . - | 9-8 . . | | . . . | . . . | . . *89 | |------------+------------------+-----------------| | . . . | 79 . . | . 8 79 | | . . . | . . 789 | . . . | | . . . | . . . | . . . | |------------+------------------+-----------------| | . . . |*789 . . | . . *78 | | . . . | . 789 . | 78 . . | | . . . | . . 78 | . 9 . | *-------------------------------------------------*

skyscraper 8 c49 -> 9r2c6

Code: Select all: *-------------------------------------------------* | . . . | . . 89 | . 7 . | | . . . | 8 . - | 9 . . | | . . . | . . . | . . 8 | |------------+------------------+-----------------| | . . . | 7 . . | . 8 9 | | . . . | . . 89 | . . . | | . . . | . . . | . . . | |------------+------------------+-----------------| | . . . | 89 . . | . . 7 | | . . . | . 789 . | 8 . . | | . . . | . . 7 | . 9 . | *-------------------------------------------------*

-> no 8 in b8

Website · by **denis_berthier** » Fri Sep 08, 2023 2:39 pm

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Thanks for your solutions. Mine uses similar rules with a tridagon.
I chose this puzzle (in mith's collection) as an example of one with the tridagon pattern but with low SER (7.1).

eleven wrote:The puzzle is not hard, can be solved with normal advanced techniques and a short chain.
However, after the UR 23 and basics there is a 15 cell impossible pattern:
Code: Select all
+---------------------+-----------------------+-----------------------+ | 2 3 46 | 45 45789 #789 | 6789 #789 1 | | 78 69 1 | #789 2 3 | #789+6 4 5 | | 78 49 5 | 1 4789 6 | 3 2 #789 | +---------------------+-----------------------+-----------------------+ | 5 46 46 | #789 3 2 | 1 #789 #789 | | 1 7 2 | 6 489 #89 | 489 5 3 | | 3 8 9 | 45 457 1 | 47 6 2 | +---------------------+-----------------------+-----------------------+ | 9 5 3 | #78 6 4 | 2 1 #78 | | 4 1 78 | 2 #789 5 | #789 3 6 | | 6 2 78 | 3 1 #789 | 5 #789 4 | +---------------------+-----------------------+-----------------------+

=> 6r2c7, stte

I hadn't thought of looking for other impossible patterns.

Website · by **denis_berthier** » Sat Sep 09, 2023 4:25 am

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I forgot to give my solution

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 23789 23469 234678 ! 45789 45789 789 ! 6789 789 1 ! ! 789 69 1 ! 789 2 3 ! 6789 4 5 ! ! 789 49 5 ! 1 4789 6 ! 3 2 789 ! +----------------------+----------------------+----------------------+ ! 15 1456 46 ! 456789 3 2 ! 14789 1789 789 ! ! 123 7 2346 ! 4689 489 189 ! 1489 5 2389 ! ! 1235 8 9 ! 457 457 17 ! 147 6 237 ! +----------------------+----------------------+----------------------+ ! 135789 1359 378 ! 789 6 4 ! 2 13789 789 ! ! 4 139 378 ! 2 789 5 ! 1789 13789 6 ! ! 6 29 278 ! 3 1 789 ! 5 789 4 ! +----------------------+----------------------+----------------------+ 182 candidates.

Code: Select all: hidden-pairs-in-a-row: r6{n2 n3}{c1 c9} ==> r6c9≠7, r6c1≠5, r6c1≠1 whip[1]: r6n5{c5 .} ==> r4c4≠5 hidden-pairs-in-a-column: c9{n2 n3}{r5 r6} ==> r5c9≠9, r5c9≠8

There's a tridagon with 3 guardians, but one will disappear before it is used.

Code: Select all: Trid-OR3-relation for digits 7, 8 and 9 in blocks: b2, with cells (marked #): r1c6, r2c4, r3c5 b3, with cells (marked #): r1c8, r2c7, r3c9 b8, with cells (marked #): r9c6, r7c4, r8c5 b9, with cells (marked #): r9c8, r7c9, r8c7 with 3 guardians (in cells marked @): n6r2c7 n4r3c5 n1r8c7 +----------------------+----------------------+----------------------+ ! 23789 23469 234678 ! 45789 45789 789# ! 6789 789# 1 ! ! 789 69 1 ! 789# 2 3 ! 6789#@ 4 5 ! ! 789 49 5 ! 1 4789#@ 6 ! 3 2 789# ! +----------------------+----------------------+----------------------+ ! 15 1456 46 ! 46789 3 2 ! 14789 1789 789 ! ! 123 7 2346 ! 4689 489 189 ! 1489 5 23 ! ! 23 8 9 ! 457 457 17 ! 147 6 23 ! +----------------------+----------------------+----------------------+ ! 135789 1359 378 ! 789# 6 4 ! 2 13789 789# ! ! 4 139 378 ! 2 789# 5 ! 1789#@ 13789 6 ! ! 6 29 278 ! 3 1 789# ! 5 789# 4 ! +----------------------+----------------------+----------------------+ z-chain[4]: r4c1{n1 n5} - c2n5{r4 r7} - c2n1{r7 r8} - c7n1{r8 .} ==> r4c8≠1 whip[1]: c8n1{r8 .} ==> r8c7≠1 At least one candidate of a previous Trid-OR3-relation between candidates n6r2c7 n4r3c5 n1r8c7 has just been eliminated. There remains a Trid-OR2-relation between candidates: n6r2c7 n4r3c5

hidden-pairs-in-a-block: b9{n1 n3}{r7c8 r8c8} ==> r8c8≠9, r8c8≠8, r8c8≠7, r7c8≠9, r7c8≠8, r7c8≠7
Trid-OR2-whip[3]: r3c2{n9 n4} - OR2{{n4r3c5 | n6r2c7}} - r2c2{n6 .} ==> r7c2≠9
Trid-OR2-whip[3]: r3c2{n9 n4} - OR2{{n4r3c5 | n6r2c7}} - r2c2{n6 .} ==> r8c2≠9
naked-pairs-in-a-row: r8{c2 c8}{n1 n3} ==> r8c3≠3
z-chain[3]: b7n9{r7c1 r9c2} - c2n2{r9 r1} - c2n3{r1 .} ==> r7c1≠3
Trid-OR2-whip[3]: r3c2{n9 n4} - OR2{{n4r3c5 | n6r2c7}} - r2c2{n6 .} ==> r9c2≠9

The end is easy:

Code: Select all: singles ==> r9c2=2, r7c1=9, r7c2=5, r4c1=5, r5c1=1, r8c2=1, r8c8=3, r7c8=1, r7c3=3, r6c1=3, r6c9=2, r5c9=3, r5c3=2, r1c1=2, r5c4=6, r1c2=3, r4c7=1, r6c6=1 whip[1]: c1n7{r3 .} ==> r1c3≠7 whip[1]: c1n8{r3 .} ==> r1c3≠8 whip[1]: b4n4{r4c3 .} ==> r4c4≠4 hidden-pairs-in-a-column: c4{n4 n5}{r1 r6} ==> r6c4≠7, r1c4≠9, r1c4≠8, r1c4≠7 finned-x-wing-in-columns: n9{c4 c9}{r4 r2} ==> r2c7≠9 finned-x-wing-in-columns: n9{c9 c4}{r4 r3} ==> r3c5≠9 swordfish-in-columns: n9{c2 c4 c9}{r3 r2 r4} ==> r4c8≠9 biv-chain[2]: r8n9{c5 c7} - c8n9{r9 r1} ==> r1c5≠9 biv-chain[3]: b2n9{r1c6 r2c4} - r2c2{n9 n6} - b3n6{r2c7 r1c7} ==> r1c7≠9 x-wing-in-columns: n9{c5 c7}{r5 r8} ==> r5c6≠9 naked-single ==> r5c6=8 biv-chain[3]: r1c6{n7 n9} - c4n9{r2 r4} - b5n7{r4c4 r6c5} ==> r1c5≠7, r3c5≠7 hidden-pairs-in-a-block: b2{n7 n9}{r1c6 r2c4} ==> r2c4≠8 singles ==> r7c4=8, r7c9=7, r3c1=7, r2c1=8 finned-x-wing-in-rows: n7{r2 r4}{c4 c7} ==> r6c7≠7 stte

One thing I wanted to show with this example is, even though it has a low SER, without using uniqueness or tridagon (or another impossible pattern), the puzzle would be quite hard to solve. Unfortunately, all the other solutions use uniqueness.

Two more points to notice are, this puzzle is in T&E(3) ; and, if we exclude uniqueness from SER, then it's rating becomes 11.7 instead of 7.1.

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