#146 in 63,137 T&E(3) min-expands

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#146 in 63,137 T&E(3) min-expands

Postby denis_berthier » Tue Jan 24, 2023 5:14 am

.
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.
denis_berthier
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Re: #146 in 63,137 T&E(3) min-expands

Postby 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.
Cenoman
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Re: #146 in 63,137 T&E(3) min-expands

Postby denis_berthier » Wed Jan 25, 2023 4:30 am

.
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.
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Re: #146 in 63,137 T&E(3) min-expands

Postby denis_berthier » Wed Jan 25, 2023 4:40 am

.
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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