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Website · by **denis_berthier** » Tue Jan 24, 2023 5:14 am

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This slightly harder puzzle has two different impossible patterns (with guardians, of course).
I wonder if RT can also be used to bypass the second pattern.

Code: Select all: +-------+-------+-------+ ! . . . ! . . . ! 7 8 . ! ! 4 . 7 ! . . . ! . 3 6 ! ! 8 . 6 ! . . . ! 5 . 4 ! +-------+-------+-------+ ! 2 . . ! . 3 . ! . . . ! ! . . . ! . . . ! 3 5 8 ! ! 9 . . ! . . 5 ! . . . ! +-------+-------+-------+ ! . . . ! . 6 7 ! 8 . 5 ! ! . . . ! 8 4 . ! . 7 3 ! ! . . . ! 5 . 3 ! 4 6 . ! +-------+-------+-------+ ......78.4.7....368.6...5.42...3..........3589....5.......678.5...84..73...5.346.;63;576 SER = 11.6

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 135 12359 12359 ! 123469 1259 12469 ! 7 8 129 ! ! 4 1259 7 ! 129 12589 1289 ! 129 3 6 ! ! 8 1239 6 ! 12379 1279 129 ! 5 129 4 ! +----------------------+----------------------+----------------------+ ! 2 145678 1458 ! 1467 3 1468 ! 169 149 179 ! ! 167 1467 14 ! 124679 1279 12469 ! 3 5 8 ! ! 9 134678 1348 ! 1467 178 5 ! 126 124 127 ! +----------------------+----------------------+----------------------+ ! 13 12349 12349 ! 129 6 7 ! 8 129 5 ! ! 156 12569 1259 ! 8 4 129 ! 129 7 3 ! ! 17 12789 1289 ! 5 129 3 ! 4 6 129 ! +----------------------+----------------------+----------------------+ 199 candidates.

by **Cenoman** » Tue Jan 24, 2023 4:57 pm

denis_berthier wrote:I wonder if RT can also be used to bypass the second pattern.

With all my bad will, I find the answer is 'yes'

Code: Select all: +-------------------------+---------------------------+--------------------+ | 135 12359 12359 | 46 1259* 46 | 7 8 129* | | 4 1259 7 | 129* 12589 1289 | 129* 3 6 | | 8 129 6 | 3 7 129* | 5 129* 4 | +-------------------------+---------------------------+--------------------+ | 2 145678 1458 | 1467 3 1468 | 169 149 179 | | 167 1467 14 | 124679 129 12469 | 3 5 8 | | 9 134678 1348 | 1467 18 5 | 126 124 127 | +-------------------------+---------------------------+--------------------+ | 13 12349 12349 | 129* 6 7 | 8 129* 5 | | 156 12569 1259 | 8 4 129* | 129* 7 3 | | 17 12789 1289 | 5 129* 3 | 4 6 129* | +-------------------------+---------------------------+--------------------+

1. TH(129)b2389 having a single guardian (5r1c5), at a vertex of a rectangle r19c59 => +5 r1c5 AND RT(129)r1c9, r9c5, r9c9

Resolution state after +5r1c5

Code: Select all: +-----------------------+------------------------+--------------------+ | 13 1239 1239 | 46 5 46 | 7 8 a129 | | 4 5 7 | 129 1289 1289 | 129 3 6 | | 8 a129* 6 | 3 7 129 | 5 129 4 | +-----------------------+------------------------+--------------------+ | 2 1478 5 | 1467 3 1468 | 169 149 179 | | 6 147 14 | 12479 129 1249 | 3 5 8 | | 9 13478 1348 | 1467 18 5 | 126 124 127 | +-----------------------+------------------------+--------------------+ | 13 12349 12349 | 129 6 7 | 8 129 5 | | 5 6 129 | 8 4 129 | 129 7 3 | | 7 8-129 1289 | 5 129* 3 | 4 6 129* | +-----------------------+------------------------+--------------------+

2. ER b1 for each digit 1,2,9 leads to same digit a at r1c9 and r3c2 => RT(129)r3c2, r9c59 => -129 r9c2; ste

I have noticed that such association RT + ER is encountered frequently, when RT is effective (ER is not surprising, as givens are restricted for TH digits).

Nevertheless, I have to learn how to spot the other impossible patterns. Pretty sure you will find puzzles w/o RT.

Website · by **denis_berthier** » Wed Jan 25, 2023 4:30 am

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The two patterns are closely related:
o's are for the only cell that can have 123 as givens.
We can see that eleven's pattern almost requires the tridagon pattern:
- the pre-condition on givens are much more restrictive for eleven's pattern.
- I say "almost" because one cell may not satisfy the tridagon conditions. Starting from Tridagon, delete one of its cells, say x, (done by the simplest tridagon elimination rule) and add 2 cells as shown in each of the two blocks in the two bands.

Code: Select all: Tridagon: +-------+-------+-------+ ! . . . ! . . X ! . . X ! ! . . . ! . x . ! . X . ! ! . . . ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! X . . ! . . X ! ! . . . ! . X . ! . X . ! ! . . . ! . . X ! X . . ! +-------+-------+-------+ ! o o o ! . . . ! . . . ! ! o o o ! . . . ! . . . ! ! o o o ! . . . ! . . . ! +-------+-------+-------+ eleven's pattern #97 (ex #37) +-------+-------+-------+ ! . . . ! . . X ! . . X ! ! . . X ! . . . ! . X . ! ! . X . ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! X . . ! . . X ! ! . X . ! . X . ! . X . ! ! . . X ! . . X ! X . . ! +-------+-------+-------+ ! o . . ! . . . ! . . . ! ! o . . ! . . . ! . . . ! ! o . . ! . . . ! . . . ! +-------+-------+-------+

In most of the cases I've found with eleven's pattern (and in the 3 I've proposed here), it is indeed the simplest tridagon elimination rule that gets applied.
It seems in this case the RT is more efficient at eliminating candidates that lead to a solution. I'll nevertheless give the solution without it below.

Website · by **denis_berthier** » Wed Jan 25, 2023 4:40 am

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Code: Select all: hidden-pairs-in-a-row: r1{n4 n6}{c4 c6} ==> r1c6≠9, r1c6≠2, r1c6≠1, r1c4≠9, r1c4≠3, r1c4≠2, r1c4≠1 singles ==> r3c4=3, r3c5=7 +----------------------+----------------------+----------------------+ ! 135 12359 12359 ! 46 1259 46 ! 7 8 129 ! ! 4 1259 7 ! 129 12589 1289 ! 129 3 6 ! ! 8 129 6 ! 3 7 129 ! 5 129 4 ! +----------------------+----------------------+----------------------+ ! 2 145678 1458 ! 1467 3 1468 ! 169 149 179 ! ! 167 1467 14 ! 124679 129 12469 ! 3 5 8 ! ! 9 134678 1348 ! 1467 18 5 ! 126 124 127 ! +----------------------+----------------------+----------------------+ ! 13 12349 12349 ! 129 6 7 ! 8 129 5 ! ! 156 12569 1259 ! 8 4 129 ! 129 7 3 ! ! 17 12789 1289 ! 5 129 3 ! 4 6 129 ! +----------------------+----------------------+----------------------+

Tridagon elimination:

Code: Select all: tridagon for digits 1, 2 and 9 in blocks: b2, with cells: r1c5 (target cell), r3c6, r2c4 b3, with cells: r1c9, r3c8, r2c7 b8, with cells: r9c5, r8c6, r7c4 b9, with cells: r9c9, r8c7, r7c8 ==> r1c5≠1,2,9

Detection of eleven's pattern with 4 guardians:

Code: Select all: singles ==> r1c5=5, r2c2=5, r4c3=5, r8c1=5, r8c2=6, r5c1=6, r9c1=7 +-------------------+-------------------+-------------------+ ! 13 1239 1239 ! 46 5 46 ! 7 8 129 ! ! 4 5 7 ! 129 1289 1289 ! 129 3 6 ! ! 8 129 6 ! 3 7 129 ! 5 129 4 ! +-------------------+-------------------+-------------------+ ! 2 1478 5 ! 1467 3 1468 ! 169 149 179 ! ! 6 147 14 ! 12479 129 1249 ! 3 5 8 ! ! 9 13478 1348 ! 1467 18 5 ! 126 124 127 ! +-------------------+-------------------+-------------------+ ! 13 12349 12349 ! 129 6 7 ! 8 129 5 ! ! 5 6 129 ! 8 4 129 ! 129 7 3 ! ! 7 1289 1289 ! 5 129 3 ! 4 6 129 ! +-------------------+-------------------+-------------------+ OR4-anti-eleven#97[15] for digits 2, 9 and 1 anti-block: 4 anti-column: 1 3-cell blocks: b3, with cells: r2c7, r1c9, r3c8 b8, with cells: r8c6, r9c5, r7c4 b9, with cells: r8c7, r9c9, r7c8 2-cell blocks: b1, with cells: r1c3, r3c2 b2, with cells: r2c4, r3c6 b7, with cells: r9c2, r7c3 with 4 guardians: n3r1c3 n3r7c3 n4r7c3 n8r9c2

Use the OR4 relation:

whip[3]: r4n8{c2 c6} - r6c5{n8 n1} - b6n1{r6c7 .} ==> r4c2≠1
El97-OR4-whip[4]: r6n3{c2 c3} - c3n8{r6 r9} - OR4{{n8r9c2 n3r1c3 n3r7c3 | n4r7c3}} - r5c3{n4 .} ==> r6c2≠1
t-whip[5]: r4n8{c2 c6} - c6n6{r4 r1} - c6n4{r1 r5} - r5c3{n4 n1} - r5c2{n1 .} ==> r4c2≠7
biv-chain[3]: r5c3{n1 n4} - r4c2{n4 n8} - b7n8{r9c2 r9c3} ==> r9c3≠1
El97-OR4-ctr-whip[4]: r5c3{n1 n4} - r4c2{n4 n8} - r6c3{n8 n3} - OR4{{n3r1c3 n3r7c3 n4r7c3 n8r9c2 | .}} ==> r1c3≠1, r7c3≠1, r8c3≠1

Easy end in W6:

Code: Select all: whip[1]: c3n1{r6 .} ==> r5c2≠1 finned-x-wing-in-rows: n1{r8 r2}{c7 c6} ==> r3c6≠1 whip[1]: b2n1{r2c6 .} ==> r2c7≠1 biv-chain[2]: r3n1{c8 c2} - c1n1{r1 r7} ==> r7c8≠1 whip[6]: r3c6{n9 n2} - r3c2{n2 n1} - b3n1{r3c8 r1c9} - r9n1{c9 c5} - b8n2{r9c5 r7c4} - r7c8{n2 .} ==> r3c8≠9 z-chain[3]: b3n9{r1c9 r2c7} - r8n9{c7 c6} - r3n9{c6 .} ==> r1c3≠9 whip[1]: c3n9{r9 .} ==> r7c2≠9, r9c2≠9 z-chain[4]: c5n8{r2 r6} - c3n8{r6 r9} - r9n9{c3 c9} - b3n9{r1c9 .} ==> r2c5≠9 t-whip[5]: r1n9{c2 c9} - r2c7{n9 n2} - b2n2{r2c4 r3c6} - r8n2{c6 c3} - r1c3{n2 .} ==> r1c2≠3 z-chain[4]: r7n4{c2 c3} - r7n3{c3 c1} - r1n3{c1 c3} - c3n2{r1 .} ==> r7c2≠2 t-whip[5]: r1n9{c2 c9} - r2c7{n9 n2} - b2n2{r2c4 r3c6} - r8n2{c6 c3} - r1n2{c3 .} ==> r1c2≠1 t-whip[5]: c2n3{r6 r7} - r7c1{n3 n1} - r1n1{c1 c9} - r9n1{c9 c5} - r6c5{n1 .} ==> r6c2≠8 whip[5]: r3c8{n1 n2} - r7c8{n2 n9} - r9c9{n9 n2} - c2n2{r9 r1} - r1n9{c2 .} ==> r1c9≠1 singles ==> r3c8=1, r1c1=1, r7c1=3, r6c2=3, r5c2=7, r1c3=3 whip[1]: c3n2{r9 .} ==> r9c2≠2 biv-chain[3]: r9c2{n1 n8} - c3n8{r9 r6} - r6c5{n8 n1} ==> r9c5≠1 biv-chain[3]: r5c3{n1 n4} - c2n4{r4 r7} - r7n1{c2 c4} ==> r5c4≠1 biv-chain[4]: r6c5{n1 n8} - b4n8{r6c3 r4c2} - r9c2{n8 n1} - b9n1{r9c9 r8c7} ==> r6c7≠1 biv-chain[2]: b8n1{r7c4 r8c6} - c7n1{r8 r4} ==> r4c4≠1 biv-chain[4]: r6c5{n1 n8} - b4n8{r6c3 r4c2} - r9c2{n8 n1} - r7n1{c2 c4} ==> r6c4≠1 naked-triplets-in-a-column: c4{r1 r4 r6}{n6 n4 n7} ==> r5c4≠4 biv-chain[4]: c4n1{r2 r7} - c2n1{r7 r9} - c2n8{r9 r4} - c6n8{r4 r2} ==> r2c6≠1 biv-chain[4]: b2n1{r2c5 r2c4} - r7n1{c4 c2} - r7n4{c2 c3} - r5c3{n4 n1} ==> r5c5≠1 naked-pairs-in-a-block: b5{r5c4 r5c5}{n2 n9} ==> r5c6≠9, r5c6≠2 naked-pairs-in-a-column: c5{r5 r9}{n2 n9} ==> r2c5≠2 biv-chain[4]: r7c8{n2 n9} - r4c8{n9 n4} - c2n4{r4 r7} - r7n1{c2 c4} ==> r7c4≠2 biv-chain[3]: b8n2{r9c5 r8c6} - r3n2{c6 c2} - r1n2{c2 c9} ==> r9c9≠2 biv-chain[3]: c7n1{r4 r8} - r9c9{n1 n9} - b3n9{r1c9 r2c7} ==> r4c7≠9 finned-x-wing-in-columns: n9{c7 c6}{r8 r2} ==> r2c4≠9 whip[1]: b2n9{r3c6 .} ==> r8c6≠9 biv-chain[3]: r4c7{n1 n6} - r6n6{c7 c4} - r6n7{c4 c9} ==> r6c9≠1 whip[1]: b6n1{r4c9 .} ==> r4c6≠1 hidden-pairs-in-a-row: r6{n1 n8}{c3 c5} ==> r6c3≠4 biv-chain[3]: r7c4{n9 n1} - r8c6{n1 n2} - r8c3{n2 n9} ==> r7c3≠9 biv-chain[3]: b8n1{r8c6 r7c4} - r7n9{c4 c8} - b9n2{r7c8 r8c7} ==> r8c7≠1, r8c6≠2 stte

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