#18913 in mith's 63137 T&E(3) min-expands

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#18913 in mith's 63137 T&E(3) min-expands

Postby denis_berthier » Fri Dec 23, 2022 6:43 am

.
That's a hard one again.
I continue writing the SER, though it is now clear it doesn't mean much when anti-tridagon rules can be applied.

As I said in a previous post, I've now coded the full ORk-g-whips (soon to be published on GitHub). What I observe is, they don't add much resolution power to ORk-whips. This puzzle is one more of the few cases where they do.

Code: Select all
+-------+-------+-------+
! . . . ! . . . ! 7 8 . !
! . . . ! 7 8 9 ! 1 . 2 !
! . . . ! . . . ! . 6 4 !
+-------+-------+-------+
! 2 . 5 ! 9 1 . ! 8 . . !
! . . . ! . 2 8 ! . . . !
! 8 . . ! 5 . 7 ! 2 . . !
+-------+-------+-------+
! . 7 1 ! 8 9 . ! . . 5 !
! 5 . . ! . . . ! . . . !
! 9 . 2 ! . 7 5 ! . . . !
+-------+-------+-------+
......78....7891.2.......642.591.8......28...8..5.72...7189...55........9.2.75...;4024;280382
SER = 10.4

Code: Select all
Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 13     12359  39     ! 12346  3456   12346  ! 7      8      39     !
   ! 346    3456   346    ! 7      8      9      ! 1      35     2      !
   ! 137    123589 3789   ! 123    35     123    ! 359    6      4      !
   +----------------------+----------------------+----------------------+
   ! 2      346    5      ! 9      1      346    ! 8      347    367    !
   ! 13467  13469  34679  ! 346    2      8      ! 34569  13459  1369   !
   ! 8      13469  3469   ! 5      346    7      ! 2      1349   1369   !
   +----------------------+----------------------+----------------------+
   ! 346    7      1      ! 8      9      2346   ! 346    234    5      !
   ! 5      3468   3468   ! 12346  346    12346  ! 3469   123479 136789 !
   ! 9      3468   2      ! 1346   7      5      ! 346    134    1368   !
   +----------------------+----------------------+----------------------+
193 candidates.
denis_berthier
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby totuan » Fri Dec 23, 2022 1:58 pm

denis_berthier wrote:.
That's a hard one again.
Code: Select all
+-------+-------+-------+
! . . . ! . . . ! 7 8 . !
! . . . ! 7 8 9 ! 1 . 2 !
! . . . ! . . . ! . 6 4 !
+-------+-------+-------+
! 2 . 5 ! 9 1 . ! 8 . . !
! . . . ! . 2 8 ! . . . !
! 8 . . ! 5 . 7 ! 2 . . !
+-------+-------+-------+
! . 7 1 ! 8 9 . ! . . 5 !
! 5 . . ! . . . ! . . . !
! 9 . 2 ! . 7 5 ! . . . !
+-------+-------+-------+
......78....7891.2.......642.591.8......28...8..5.72...7189...55........9.2.75...;4024;280382
SER = 10.4

Code: Select all
Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 13     12359  39     ! 12346  3456   12346  ! 7      8      39     !
   ! 346    3456   346    ! 7      8      9      ! 1      35     2      !
   ! 137    123589 3789   ! 123    35     123    ! 359    6      4      !
   +----------------------+----------------------+----------------------+
   ! 2      346    5      ! 9      1      346    ! 8      347    367    !
   ! 13467  13469  34679  ! 346    2      8      ! 34569  13459  1369   !
   ! 8      13469  3469   ! 5      346    7      ! 2      1349   1369   !
   +----------------------+----------------------+----------------------+
   ! 346    7      1      ! 8      9      2346   ! 346    234    5      !
   ! 5      3468   3468   ! 12346  346    12346  ! 3469   123479 136789 !
   ! 9      3468   2      ! 1346   7      5      ! 346    134    1368   !
   +----------------------+----------------------+----------------------+
193 candidates.

Hmmm...!
Don't know how this one is hard??? :roll:
All guardians of Tridagon (346) lead to r5c8=5 then... nothing :lol:

Thanks for the puzzle!
totuan
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby Cenoman » Fri Dec 23, 2022 10:28 pm

Code: Select all
 +-----------------------+------------------------+-----------------------------+
 |  1     25     b39     |  246     456   246     |   7       8       c39       |
 |  346   3456    346    |  7       8     9       |   1       35       2        |
 |  7     2589   B89     |  123     35    123     |   5-9     6        4        |
 +-----------------------+------------------------+-----------------------------+
 |  2     346*    5      |  9       1     346*    |   8       347      367      |
 |  346*  13469   7      |  346*    2     8       |   34569   13459    1369     |
 |  8     13469  a3469*  |  5       346*  7       |   2       1349     1369     |
 +-----------------------+------------------------+-----------------------------+
 |  346*  7       1      |  8       9    F2346*   |zGg346    G234      5        |
 |  5     3468   A3468*  |  12346   346*  12346   |zGg3469    123479   136789   |
 |  9    f3468*   2      | y1346*   7     5       |zGg346   zg134     g1368     |
 +-----------------------+------------------------+-----------------------------+

1. TH(346)b4578 having five guardians: 9r6c3, 8r8c3, 8r9c2, 2r7c6, 1r9c4
(9)r6c3 - r1c3 = (9)r1c9
(8)r8c3 - (8=9)r3c3
(8)r9c2 - (8=13469)b9p14789
(2)r7c6 - (2=3469)b9p1247
(1)r9c4 - (1=3469)b9p1478
=> -9 r3c7; 9 placements

Code: Select all
 +----------------------+-----------------------+-------------------------+
 |  1     2       3     |  46      5    46      |  7      8       9       |
 |  46    5       46    |  7       8    9       |  1      3       2       |
 |  7     89      89    |  12*     3    12*     |  5      6       4       |
 +----------------------+-----------------------+-------------------------+
 |  2     346     5     |  9       1    346     |  8     b47      367     |
 |  346   13469   7     |  346     2    8       |  3469   5       136     |
 |  8     13469   469   |  5       46   7       |  2      149     136     |
 +----------------------+-----------------------+-------------------------+
 |  346   7       1     |  8       9    2346    |  346    24      5       |
 |  5     3468    468   |  12346*  46   12346*  |  3469  c12-479 d3678-1  |
 |  9     3468    2     |  1346    7    5       |  346   a14     d368-1   |
 +----------------------+-----------------------+-------------------------+

2. (1=4)r9c8 - (4=7)r4c8 - r8c8 = (78)r89c9 => -1 r89c9
3. UR(12)r48c46 using externals => +12 r8c8

Code: Select all
 +------------------+-----------------------+-----------------+
 |  1    2     3    |  46      5    46      |  7    8    9    |
 |  46   5     46   |  7       8    9       |  1    3    2    |
 |  7    8     9    |  12      3    12      |  5    6    4    |
 +------------------+-----------------------+-----------------+
 |  2   d34*   5    |  9       1   d34*     |  8    7    6    |
 | c36   9     7    | b36      2    8       |  4    5    1    |
 |  8    1     46   |  5      a46   7       |  2    9    3    |
 +------------------+-----------------------+-----------------+
 | d34*  7     1    |  8       9  fd2346*   | e36#  24   5    |
 |  5    346   8    |  12346   4-6  12346   |  9    12   7    |
 |  9    346   2    |  1346    7    5       |  36   14   8    |
 +------------------+-----------------------+-----------------+

Almost kite:
4. (6)r6c5 = r5c4 - (6=3)r5c1 - [r4c2 = r4c6 - r7c6 = r7c1] = (3-6)r7c7 = (6)r7c6 => -6 r8c5; ste
Cenoman
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby eleven » Fri Dec 23, 2022 10:46 pm

Very similar. 5 guardians => 5r3c7
9r6c3 - (9=3)r1c3 - r1c9 = (3-5)r2c8 = 5r3c7
2r7c6 - r7c8 = (2-7)r8c8 = 79r8c97 - (9=5)r3c7
8r8c3 - (8=9)r3c3 - (9=5)r3c7
8r9c2 - r9c9 = 879r8c7 - (9=5)r3c7
1r9c4 - r9c89 = 179r8c897 - (9=5)r3c7

Then there is a UR type 3 (again!). Without it it would be harder!
Code: Select all
+----------------------+-----------------------+----------------------+
| 1      2      3      | 46      5      46     | 7      8      9      |
| 46     5      46     | 7       8      9      | 1      3      2      |
| 7      89     89     |#12      3     #12     | 5      6      4      |
+----------------------+------- ----------------+----------------------+
| 2      346    5      | 9       1      346    | 8      47     367    |
| 346    13469  7      | 346     2      8      | 3469   5      136    |
| 8      13469  469    | 5       46     7      | 2      149    136    |
+----------------------+-----------------------+----------------------+
| 346    7      1      | 8       9      2346   | 346    24     5      |
| 5     *3468  *468    |#12+346 *46    #12+346 | 9-346  1279   178-36 |
| 9      3468   2      | 1346    7      5      | 346    14     1368   |
+----------------------+-----------------------+----------------------+

UR 12 r38c46: quad 3468r8c23456 => -346r7c7,-36r7c9

singles and locked 6c2b7.
Code: Select all
+----------------------+----------------------+----------------------+
| 1      2      3      | 46     5      46     | 7      8      9      |
| 46     5      46     | 7      8      9      | 1      3      2      |
| 7      8      9      | 12     3      12     | 5      6      4      |
+----------------------+----------------------+----------------------+
| 2      34     5      | 9      1     c34     | 8      7      6      |
|e36     9      7      | 36     2      8      | 4      5      1      |
| 8      1     a46     | 5     b46     7      | 2      9      3      |
+----------------------+----------------------+----------------------+
|e34     7      1      | 8      9     d2346   | 36    d24     5      |
| 5      346    8      | 12346 c46     12346  | 9      12     7      |
| 9      346    2      | 1346   7      5      | 36     14     8      |
+----------------------+----------------------+----------------------+

4r6c3 = r6c5 - (4=36)r4c6,r8c5 - (3|6=24)r7c68 - (4=36)r5c1 => -6r6c3, stte
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby marek stefanik » Fri Dec 23, 2022 11:53 pm

After the same start, avoiding the UR:
Code: Select all
.-----------------.------------------.--------------------.
| 1    2      3   | 46     5   46    | 7     8      9     |
| 46   5      46  | 7      8   9     | 1     3      2     |
| 7    89    *89  | 12     3   12    | 5     6      4     |
:-----------------+------------------+--------------------:
| 2   #346    5   | 9      1  #346   | 8     47     367   |
|#346  13469  7   |#346    2   8     | 3469  5      136   |
| 8    13469 #46+9| 5     #46  7     | 2     149    136   |
:-----------------+------------------+--------------------:
|#346  7      1   | 8      9 α#346+2 | 346  β24     5     |
| 5    3468  #46+8| 12346 #46  12346 | 3469 *127–49*178–36|
| 9  a#346+8  2   |A#346+1 7   5     | 346  B14   bB1368  |
'-----------------'------------------'--------------------'
TH 346#, 3 has been eliminated from the rectangle => no RTs
Rectangle guardians (9r6c3 and 8r8c3) are linked via r3c3 => mutually exclusive
Hence another guardian is needed (8r9c2 or 1r9c4 or 2r7c6).
Code: Select all
|8r9c2 – 8r9c9 = 78r8c89
|1r9c4 – 1r9c89 = 17r8c89
|2r7c6 – 2r7c8 = 27r8c89
=> –49r8c8, -36r8c9
Reaching the same state as with the UR.

Marek
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby eleven » Sat Dec 24, 2022 1:17 am

marek stefanik wrote:Hence another guardian is needed (8r9c2 or 1r9c4 or 2r7c6).

So this pattern is impossible
Code: Select all
 *-----------------------------------*
 | .     123  .     | .    .    123  |
 | 123   .    .     | 123  .    .    |
 | .     .    12    | .    12   .    |
 |------------------+----------------|
 | 123   .    .     | .    .    123  |
 | .     .    .     | .    12   .    |
 | .     123  .     | 123  .    .    |
 *-----------------------------------*

I need 2 cases to prove it (r1c2=12 or 3). is that a "hard" pattern ? [Added: sure now, no]
eleven
 
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby marek stefanik » Sat Dec 24, 2022 2:06 am

Code: Select all
 *-----------------------------------*
 | .    A123  .     | .    .   A123  |
 |B123   .    .     |B123  .    .    |
 | .     .   C12    | .   C12   .    |
 |------------------+----------------|
 | 123   .    .     | .    .   A123  |
 | .     .    .     | .   C12   .    |
 | .     123  .     |B123  .    .    |
 *-----------------------------------*
Yeah, it's not that hard. I like the parity-based proof a bit more:
The permutations in boxes (in order ABC) all have the same parity and C-marked cells in b15 contain the same digit.
Hence the permutations in b15 are identical and the digits in the A- and B-marked cells are both eliminated from b4p18, i.e. contra.

Marek
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby denis_berthier » Sat Dec 24, 2022 4:37 am

eleven wrote:So this pattern is impossible
Code: Select all
 *-----------------------------------*
 | .     123  .     | .    .    123  |
 | 123   .    .     | 123  .    .    |
 | .     .    12    | .    12   .    |
 |------------------+----------------|
 | 123   .    .     | .    .    123  |
 | .     .    .     | .    12   .    |
 | .     123  .     | 123  .    .    |
 *-----------------------------------*

is that a "hard" pattern ? [Added: sure now, no]

If you mean hard in the sense of what's needed to prove it contradictory:
- it doesn't require T&E(3)
- it doesn't even require T&E(2), not even some T&E(Bp, 1), p>0
- it requires only T&E(1)

Indeed, it can be proven contradictory using only whips[≤ 10]. So, not hard compared to the T&E(3) patterns, but not easy when considered as a T&E(1) pattern.

In CSP-Rules, this proof can be done by using function "solve-k-digit-pattern-string" with the string ".1...1...1..1.......1.1....1....1.......1.....1.1....." corresponding to the above pattern. This function supposes that all the cells in the pattern have exactly k digits 1, 2 ... k; but I've already shown how to do for the "incomplete" cells:
Code: Select all
 (bind ?*simulated-eliminations* (create$
   (nrc-to-label 3 3 3)
   (nrc-to-label 3 3 5 )
   (nrc-to-label 3 5 5)
))
(solve-k-digit-pattern-string 3 ".1...1...1..1.......1.1....1....1.......1.....1.1.....")

(Some day, I may write a more general function solve-kl-digit-patterns.)

Code: Select all
Resolution state after Singles and whips[1]:
   +-------------------------------+-------------------------------+-------------------------------+
   ! 123456789 123       123456789 ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
   ! 123       123456789 123456789 ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 12        ! 123456789 12        123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+
   ! 123       123456789 123456789 ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123456789 ! 123456789 12        123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123       123456789 ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+
   ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+

660 candidates.

biv-chain[2]: r3c3{n1 n2} - r3c5{n2 n1} ==> r3c1≠1, r3c2≠1, r3c7≠1, r3c8≠1, r3c9≠1, r3c4≠1, r3c6≠1
biv-chain[2]: r3c3{n2 n1} - r3c5{n1 n2} ==> r3c1≠2, r3c2≠2, r3c7≠2, r3c8≠2, r3c9≠2, r3c4≠2, r3c6≠2
biv-chain[2]: r3c5{n1 n2} - r5c5{n2 n1} ==> r1c5≠1, r2c5≠1, r7c5≠1, r8c5≠1, r9c5≠1, r4c5≠1, r6c5≠1
biv-chain[2]: r3c5{n2 n1} - r5c5{n1 n2} ==> r1c5≠2, r2c5≠2, r7c5≠2, r8c5≠2, r9c5≠2, r4c5≠2, r6c5≠2
whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r1c1≠1
whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r1c1≠2
whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r1c3≠1
whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r1c3≠2
whip[3]: r3c5{n1 n2} - r1c6{n2 n3} - r2c4{n3 .} ==> r1c4≠1
whip[3]: r3c5{n2 n1} - r1c6{n1 n3} - r2c4{n3 .} ==> r1c4≠2
whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r2c2≠1
whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r2c2≠2
whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r2c3≠1
whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r2c3≠2
whip[3]: r3c5{n1 n2} - r1c6{n2 n3} - r2c4{n3 .} ==> r2c6≠1
whip[3]: r3c5{n2 n1} - r1c6{n1 n3} - r2c4{n3 .} ==> r2c6≠2
whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r4c4≠1
whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r4c4≠2
whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r5c4≠1
whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r5c4≠2
whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r5c6≠1
whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r6c6≠1
whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r5c6≠2
whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r6c6≠2
whip[10]: r5c5{n1 n2} - r3n2{c5 c3} - b1n1{r3c3 r1c2} - r2c1{n1 n3} - r4c1{n3 n2} - r6c2{n2 n3} - r6c4{n3 n1} - r2c4{n1 n2} - r1c6{n2 n3} - r4c6{n3 .} ==> r5c1≠1
whip[10]: r5c5{n2 n1} - r3n1{c5 c3} - b1n2{r3c3 r1c2} - r2c1{n2 n3} - r4c1{n3 n1} - r6c2{n1 n3} - r6c4{n3 n2} - r2c4{n2 n1} - r1c6{n1 n3} - r4c6{n3 .} ==> r5c1≠2
whip[10]: r5c5{n1 n2} - r3n2{c5 c3} - b1n1{r3c3 r2c1} - r1c2{n1 n3} - r6c2{n3 n2} - r4c1{n2 n3} - r4c6{n3 n1} - r1c6{n1 n2} - r2c4{n2 n3} - r6c4{n3 .} ==> r5c2≠1
whip[10]: r5c5{n2 n1} - r3n1{c5 c3} - b1n2{r3c3 r2c1} - r1c2{n2 n3} - r6c2{n3 n1} - r4c1{n1 n3} - r4c6{n3 n2} - r1c6{n2 n1} - r2c4{n1 n3} - r6c4{n3 .} ==> r5c2≠2
whip[10]: r3c3{n1 n2} - c5n2{r3 r5} - b5n1{r5c5 r6c4} - r4c6{n1 n3} - r4c1{n3 n2} - r6c2{n2 n3} - r1c2{n3 n1} - r1c6{n1 n2} - r2c4{n2 n3} - r2c1{n3 .} ==> r4c3≠1
whip[10]: r3c3{n1 n2} - c5n2{r3 r5} - b5n1{r5c5 r4c6} - r6c4{n1 n3} - r6c2{n3 n2} - r4c1{n2 n3} - r2c1{n3 n1} - r2c4{n1 n2} - r1c6{n2 n3} - r1c2{n3 .} ==> r6c3≠1
whip[10]: r3c3{n2 n1} - c5n1{r3 r5} - b5n2{r5c5 r6c4} - r4c6{n2 n3} - r4c1{n3 n1} - r6c2{n1 n3} - r1c2{n3 n2} - r1c6{n2 n1} - r2c4{n1 n3} - r2c1{n3 .} ==> r4c3≠2
whip[10]: r3c3{n2 n1} - c5n1{r3 r5} - b5n2{r5c5 r4c6} - r6c4{n2 n3} - r6c2{n3 n1} - r4c1{n1 n3} - r2c1{n3 n2} - r2c4{n2 n1} - r1c6{n1 n3} - r1c2{n3 .} ==> r6c3≠2
whip[10]: r3n1{c3 c5} - r5c5{n1 n2} - r3n2{c5 c3} - r2c1{n2 n3} - r2c4{n3 n2} - r1c6{n2 n3} - r4c6{n3 n1} - r4c1{n1 n2} - r6c2{n2 n3} - r6c4{n3 .} ==> r1c2≠1
t-whip[3]: r1c2{n3 n2} - r3c3{n2 n1} - r2c1{n1 .} ==> r3c2≠3, r3c1≠3, r2c3≠3, r2c2≠3, r1c3≠3, r1c1≠3
z-chain[4]: r3c5{n2 n1} - r1c6{n1 n3} - r1c2{n3 n2} - r3n2{c3 .} ==> r2c4≠2
z-chain[3]: r2c4{n3 n1} - r1c6{n1 n2} - r1c2{n2 .} ==> r1c5≠3, r1c4≠3
t-whip[3]: r2c4{n3 n1} - r3c5{n1 n2} - r1c6{n2 .} ==> r3c6≠3, r3c4≠3, r2c6≠3, r2c5≠3
whip[1]: r3n3{c9 .} ==> r1c7≠3, r1c8≠3, r1c9≠3, r2c7≠3, r2c8≠3, r2c9≠3
z-chain[4]: r5c5{n1 n2} - r6c4{n2 n3} - r2c4{n3 n1} - c5n1{r3 .} ==> r4c6≠1
z-chain[3]: r4c6{n3 n2} - r6c4{n2 n1} - r2c4{n1 .} ==> r5c4≠3, r4c4≠3
t-whip[3]: r4c6{n3 n2} - r5c5{n2 n1} - r6c4{n1 .} ==> r6c6≠3, r6c5≠3, r5c6≠3, r4c5≠3
whip[1]: c5n3{r9 .} ==> r7c4≠3, r7c6≠3, r8c4≠3, r8c6≠3, r9c4≠3, r9c6≠3
biv-chain[2]: b5n3{r6c4 r4c6} - r1n3{c6 c2} ==> r6c2≠3
biv-chain[2]: r2n3{c1 c4} - b5n3{r6c4 r4c6} ==> r4c1≠3
biv-chain[2]: r4c1{n2 n1} - r6c2{n1 n2} ==> r4c2≠2, r5c3≠2, r6c1≠2
biv-chain[2]: r6c2{n1 n2} - r4c1{n2 n1} ==> r4c2≠1, r5c3≠1, r6c1≠1
biv-chain[3]: c6n3{r4 r1} - r1c2{n3 n2} - b4n2{r6c2 r4c1} ==> r4c6≠2
singles ==> r4c6=3, r2c4=3, r1c2=3
biv-chain[2]: r6c4{n2 n1} - r6c2{n1 n2} ==> r6c7≠2, r6c8≠2, r6c9≠2
biv-chain[2]: r6c2{n1 n2} - r6c4{n2 n1} ==> r6c7≠1, r6c8≠1, r6c9≠1
biv-chain[2]: r2c1{n2 n1} - r4c1{n1 n2} ==> r7c1≠2, r8c1≠2, r9c1≠2
biv-chain[2]: r4c1{n1 n2} - r2c1{n2 n1} ==> r7c1≠1, r8c1≠1, r9c1≠1
biv-chain[3]: b4n2{r6c2 r4c1} - b1n2{r2c1 r3c3} - c5n2{r3 r5} ==> r6c4≠2
singles ==> r6c4=1, r5c5=2, r3c5=1, r1c6=2, r3c3=2, r2c1=1
PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR BN-CELL b4n1


[Edit]: There's a simpler proof if you allow Subsets, but it is NOT in T&E(1):
Code: Select all
naked-pairs-in-a-column: c5{r3 r5}{n1 n2} ==> r9c5≠2, r9c5≠1, r8c5≠2, r8c5≠1, r7c5≠2, r7c5≠1, r6c5≠2, r6c5≠1, r4c5≠2, r4c5≠1, r2c5≠2, r2c5≠1, r1c5≠2, r1c5≠1
naked-pairs-in-a-row: r3{c3 c5}{n1 n2} ==> r3c9≠2, r3c9≠1, r3c8≠2, r3c8≠1, r3c7≠2, r3c7≠1, r3c6≠2, r3c6≠1, r3c4≠2, r3c4≠1, r3c2≠2, r3c2≠1, r3c1≠2, r3c1≠1
naked-triplets-in-a-block: b5{r4c6 r5c5 r6c4}{n3 n2 n1} ==> r6c6≠3, r6c6≠2, r6c6≠1, r6c5≠3, r5c6≠3, r5c6≠2, r5c6≠1, r5c4≠3, r5c4≠2, r5c4≠1, r4c5≠3, r4c4≠3, r4c4≠2, r4c4≠1
naked-triplets-in-a-block: b2{r1c6 r2c4 r3c5}{n2 n3 n1} ==> r3c6≠3, r3c4≠3, r2c6≠3, r2c6≠2, r2c6≠1, r2c5≠3, r1c5≠3, r1c4≠3, r1c4≠2, r1c4≠1
whip[1]: c5n3{r9 .} ==> r7c4≠3, r7c6≠3, r8c4≠3, r8c6≠3, r9c4≠3, r9c6≠3
naked-triplets-in-a-block: b1{r1c2 r2c1 r3c3}{n2 n3 n1} ==> r3c2≠3, r3c1≠3, r2c3≠3, r2c3≠2, r2c3≠1, r2c2≠3, r2c2≠2, r2c2≠1, r1c3≠3, r1c3≠2, r1c3≠1, r1c1≠3, r1c1≠2, r1c1≠1
whip[1]: r3n3{c9 .} ==> r1c7≠3, r1c8≠3, r1c9≠3, r2c7≠3, r2c8≠3, r2c9≠3
biv-chain[2]: b1n3{r2c1 r1c2} - c6n3{r1 r4} ==> r4c1≠3
biv-chain[2]: c4n3{r6 r2} - b1n3{r2c1 r1c2} ==> r6c2≠3
naked-pairs-in-a-block: b4{r4c1 r6c2}{n1 n2} ==> r6c3≠2, r6c3≠1, r6c1≠2, r6c1≠1, r5c3≠2, r5c3≠1, r5c2≠2, r5c2≠1, r5c1≠2, r5c1≠1, r4c3≠2, r4c3≠1, r4c2≠2, r4c2≠1
whip[6]: r1n3{c2 c6} - r2n3{c4 c1} - b1n1{r2c1 r3c3} - c5n1{r3 r5} - r4c6{n1 n2} - b4n2{r4c1 .} ==> r1c2≠2
biv-chain[3]: b4n2{r6c2 r4c1} - b1n2{r2c1 r3c3} - c5n2{r3 r5} ==> r6c4≠2
biv-chain[3]: b4n1{r4c1 r6c2} - r6c4{n1 n3} - r2n3{c4 c1} ==> r2c1≠1
biv-chain[3]: b4n2{r6c2 r4c1} - r2c1{n2 n3} - r1c2{n3 n1} ==> r6c2≠1
singles ==> r6c2=2, r4c1=1
biv-chain[3]: c5n2{r3 r5} - r4c6{n2 n3} - b2n3{r1c6 r2c4} ==> r2c4≠2
whip[1]: c4n2{r9 .} ==> r7c6≠2, r8c6≠2, r9c6≠2
naked-pairs-in-a-column: c4{r2 r6}{n1 n3} ==> r9c4≠1, r8c4≠1, r7c4≠1
whip[1]: b8n1{r9c6 .} ==> r1c6≠1
biv-chain[3]: r3c5{n2 n1} - r2c4{n1 n3} - r2c1{n3 n2} ==> r3c3≠2
singles ==> r3c3=1, r1c2=3, r1c6=2
PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RN-CELL r3n2
denis_berthier
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby denis_berthier » Sat Dec 24, 2022 5:26 am

denis_berthier wrote:(Some day, I may write a more general function solve-kl-digit-patterns.)

Done.
The above proof can now be called by:
Code: Select all
(solve-kl-digit-pattern-string 2 3 ".3...3...3..3.......2.2....3....3.......2.....3.3.....")

(Dots can also be 0s).

Note that k and l in the pattern string stand respectively for candidates 1...k and 1...l in the corresponding cell.
As was the case for function solve-k-digit-pattern-string, "incomplete" strings (i.e. shorter than 81) are automatically complete with 0s.
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby denis_berthier » Fri Dec 30, 2022 7:24 am

.
I realise I've forgotten to give my solution to this puzzle. I'll give two, one based on ORk-g-whips (next post) and one based on ORk-forcing-g-whips (second next post).

Starting from the RS after whips[1], both have the same start:

Code: Select all
naked-pairs-in-a-row: r1{c3 c9}{n3 n9} ==> r1c6≠3, r1c5≠3, r1c4≠3, r1c2≠9, r1c2≠3, r1c1≠3
naked-single ==> r1c1=1
whip[1]: b2n3{r3c6 .} ==> r3c1≠3, r3c2≠3, r3c3≠3, r3c7≠3
singles ==> r3c1=7, r5c3=7
147 g-candidates, 693 csp-glinks and 384 non-csp glinks
   +----------------------+----------------------+----------------------+
   ! 1      25     39     ! 246    456    246    ! 7      8      39     !
   ! 346    3456   346    ! 7      8      9      ! 1      35     2      !
   ! 7      2589   89     ! 123    35     123    ! 59     6      4      !
   +----------------------+----------------------+----------------------+
   ! 2      346    5      ! 9      1      346    ! 8      347    367    !
   ! 346    13469  7      ! 346    2      8      ! 34569  13459  1369   !
   ! 8      13469  3469   ! 5      346    7      ! 2      1349   1369   !
   +----------------------+----------------------+----------------------+
   ! 346    7      1      ! 8      9      2346   ! 346    234    5      !
   ! 5      3468   3468   ! 12346  346    12346  ! 3469   123479 136789 !
   ! 9      3468   2      ! 1346   7      5      ! 346    134    1368   !
   +----------------------+----------------------+----------------------+

OR5-anti-tridagon[12] for digits 4, 6 and 3 in blocks:
        b4, with cells: r4c2, r5c1, r6c3
        b5, with cells: r4c6, r5c4, r6c5
        b7, with cells: r9c2, r7c1, r8c3
        b8, with cells: r9c4, r7c6, r8c5
with 5 guardians: n9r6c3 n2r7c6 n8r8c3 n8r9c2 n1r9c4

z-chain[3]: r2c8{n3 n5} - c7n5{r3 r5} - c7n3{r5 .} ==> r7c8≠3, r9c8≠3, r8c8≠3
t-whip[3]: r9c8{n1 n4} - c7n4{r9 r5} - r5n5{c7 .} ==> r5c8≠1
t-whip[6]: r9c8{n1 n4} - c7n4{r9 r5} - r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - b6n9{r6c9 .} ==> r6c8≠1
whip[1]: c8n1{r9 .} ==> r8c9≠1, r9c9≠1
whip[5]: c2n1{r6 r5} - b4n9{r5c2 r6c3} - r6c8{n9 n3} - b3n3{r2c8 r1c9} - r1n9{c9 .} ==> r6c2≠4
t-whip[8]: c7n6{r9 r5} - r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - b6n9{r6c9 r6c8} - c3n9{r6 r3} - r3n8{c3 c2} - r9n8{c2 .} ==> r9c9≠6
t-whip[8]: r9c9{n8 n3} - c7n3{r9 r5} - r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - b6n9{r6c9 r6c8} - c3n9{r6 r3} - c3n8{r3 .} ==> r8c9≠8, r9c2≠8

At least one candidate of a previous Trid-OR5-relation has just been eliminated.
There remains a Trid-OR4-relation between candidates: n9r6c3 n2r7c6 n8r8c3 n1r9c4

   +-------------------+-------------------+-------------------+
   ! 1     25    39    ! 246   456   246   ! 7     8     39    !
   ! 346   3456  346   ! 7     8     9     ! 1     35    2     !
   ! 7     2589  89    ! 123   35    123   ! 59    6     4     !
   +-------------------+-------------------+-------------------+
   ! 2     346   5     ! 9     1     346   ! 8     347   367   !
   ! 346   13469 7     ! 346   2     8     ! 34569 3459  1369  !
   ! 8     1369  3469  ! 5     346   7     ! 2     349   1369  !
   +-------------------+-------------------+-------------------+
   ! 346   7     1     ! 8     9     2346  ! 346   24    5     !
   ! 5     3468  3468  ! 12346 346   12346 ! 3469  12479 3679  !
   ! 9     346   2     ! 1346  7     5     ! 346   14    38    !
   +-------------------+-------------------+-------------------+

hidden-single-in-a-row ==> r9c9=8


RESOLUTION STATE RS2, common starting point for the two solutions:
Code: Select all
+-------------------+-------------------+-------------------+
! 1     25    39    ! 246   456   246   ! 7     8     39    !
! 346   3456  346   ! 7     8     9     ! 1     35    2     !
! 7     2589  89    ! 123   35    123   ! 59    6     4     !
+-------------------+-------------------+-------------------+
! 2     346   5     ! 9     1     346   ! 8     347   367   !
! 346   13469 7     ! 346   2     8     ! 34569 3459  1369  !
! 8     1369  3469  ! 5     346   7     ! 2     349   1369  !
+-------------------+-------------------+-------------------+
! 346   7     1     ! 8     9     2346  ! 346   24    5     !
! 5     3468  3468  ! 12346 346   12346 ! 3469  12479 3679  !
! 9     346   2     ! 1346  7     5     ! 346   14    8     !
+-------------------+-------------------+-------------------+
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Re: #18913 in mith's 63137 T&E(3) min-expands

Postby denis_berthier » Fri Dec 30, 2022 7:30 am

.
Solution with ORk-g-whips:

The ORk-part in gWW8+OR4gW8:
whip[8]: c2n1{r6 r5} - b4n9{r5c2 r6c3} - r1n9{c3 c9} - b3n3{r1c9 r2c8} - b3n5{r2c8 r3c7} - r3c5{n5 n3} - r6n3{c5 c9} - r6n1{c9 .} ==> r6c2≠6
Trid-OR4-whip[8]: r3n2{c6 c2} - r3n8{c2 c3} - b1n9{r3c3 r1c3} - OR4{{n9r6c3 n2r7c6 n8r8c3 | n1r9c4}} - r9c8{n1 n4} - c7n4{r9 r5} - c7n5{r5 r3} - r3n9{c7 .} ==> r1c6≠2
whip[5]: c6n1{r8 r3} - c6n2{r3 r7} - r7c8{n2 n4} - r4n4{c8 c2} - b7n4{r8c2 .} ==> r8c6≠4
whip[6]: c6n1{r3 r8} - c6n2{r8 r7} - r7c8{n2 n4} - b6n4{r4c8 r5c7} - c7n5{r5 r3} - r3c5{n5 .} ==> r3c6≠3
Trid-OR4-gwhip[8]: r3n8{c2 c3} - b1n9{r3c3 r1c3} - r3n9{c2 c7} - c7n5{r3 r5} - b6n4{r5c7 r456c8} - r9c8{n4 n1} - OR4{{n1r9c4 n8r8c3 n9r6c3 | n2r7c6}} - r7c8{n2 .} ==> r3c2≠2
singles ==> r1c2=2, r1c5=5, r3c5=3
whip[7]: r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - r5n9{c9 c2} - c2n1{r5 r6} - r6n3{c2 c3} - r1c3{n3 .} ==> r5c7≠3
whip[1]: c7n3{r9 .} ==> r8c9≠3
whip[7]: r1n6{c6 c4} - r1n4{c4 c6} - r4c6{n4 n3} - r7c6{n3 n2} - r7c8{n2 n4} - b6n4{r4c8 r5c7} - r5c4{n4 .} ==> r8c6≠6
whip[8]: r8c5{n4 n6} - r6c5{n6 n4} - r4n4{c6 c8} - c8n7{r4 r8} - r8c9{n7 n9} - r1n9{c9 c3} - r3c3{n9 n8} - r8n8{c3 .} ==> r8c2≠4
Trid-OR4-whip[8]: b3n9{r1c9 r3c7} - r3c3{n9 n8} - r3c2{n8 n5} - c2n9{r3 r6} - c8n9{r6 r8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r5c9≠9
Trid-OR4-whip[8]: b3n9{r1c9 r3c7} - r3c3{n9 n8} - r3c2{n8 n5} - c2n9{r3 r5} - c8n9{r5 r8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r6c9≠9
Trid-OR4-whip[6]: r1c3{n3 n9} - c9n9{r1 r8} - r8n7{c9 c8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r8c3≠3
Trid-OR4-whip[7]: r3n8{c2 c3} - b1n9{r3c3 r1c3} - c9n9{r1 r8} - r8n7{c9 c8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r3c2≠5


The end is easy:
Code: Select all
singles ==> r2c2=5, r2c8=3, r1c9=9, r1c3=3, r3c7=5, r5c8=5
hidden-pairs-in-a-row: r6{n1 n3}{c2 c9} ==> r6c9≠6, r6c2≠9
finned-x-wing-in-columns: n3{c1 c4}{r5 r7} ==> r7c6≠3
biv-chain[3]: c5n4{r8 r6} - r6c8{n4 n9} - b9n9{r8c8 r8c7} ==> r8c7≠4
biv-chain[3]: r2c3{n4 n6} - r6n6{c3 c5} - c5n4{r6 r8} ==> r8c3≠4
z-chain[3]: b7n4{r7c1 r9c2} - r4n4{c2 c8} - r8n4{c8 .} ==> r7c6≠4
biv-chain[3]: r7c8{n4 n2} - r7c6{n2 n6} - r8c5{n6 n4} ==> r8c8≠4
whip[1]: r8n4{c5 .} ==> r9c4≠4
biv-chain[3]: r1c4{n6 n4} - b8n4{r8c4 r8c5} - c5n6{r8 r6} ==> r5c4≠6
biv-chain[3]: r5c4{n4 n3} - c1n3{r5 r7} - b7n4{r7c1 r9c2} ==> r5c2≠4
biv-chain[4]: r8c3{n6 n8} - r3c3{n8 n9} - c2n9{r3 r5} - c7n9{r5 r8} ==> r8c7≠6
z-chain[4]: r8c3{n6 n8} - r8c2{n8 n3} - c6n3{r8 r4} - b5n6{r4c6 .} ==> r8c5≠6
singles ==> r8c5=4, r6c5=6
biv-chain[2]: r6n4{c8 c3} - c2n4{r4 r9} ==> r9c8≠4
naked-single ==> r9c8=1

At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n9r6c3 n2r7c6 n8r8c3
   +----------------+----------------+----------------+
   ! 1    2    3    ! 46   5    46   ! 7    8    9    !
   ! 46   5    46   ! 7    8    9    ! 1    3    2    !
   ! 7    89   89   ! 12   3    12   ! 5    6    4    !
   +----------------+----------------+----------------+
   ! 2    346  5    ! 9    1    34   ! 8    47   367  !
   ! 346  1369 7    ! 34   2    8    ! 469  5    136  !
   ! 8    13   49   ! 5    6    7    ! 2    49   13   !
   +----------------+----------------+----------------+
   ! 346  7    1    ! 8    9    26   ! 346  24   5    !
   ! 5    368  68   ! 1236 4    123  ! 39   1279 67   !
   ! 9    346  2    ! 36   7    5    ! 346  1    8    !
   +----------------+----------------+----------------+

hidden-pairs-in-a-column: c4{n1 n2}{r3 r8} ==> r8c4≠6, r8c4≠3
finned-x-wing-in-columns: n3{c4 c1}{r5 r9} ==> r9c2≠3
finned-x-wing-in-rows: n6{r4 r8}{c9 c2} ==> r9c2≠6
naked-single ==> r9c2=4
biv-chain[3]: c1n3{r7 r5} - c4n3{r5 r9} - b8n6{r9c4 r7c6} ==> r7c1≠6
naked-single ==> r7c1=3
whip[1]: b7n6{r8c3 .} ==> r8c9≠6
stte
denis_berthier
2010 Supporter
 
Posts: 4237
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Location: Paris

Re: #18913 in mith's 63137 T&E(3) min-expands

Postby denis_berthier » Fri Dec 30, 2022 7:35 am

.
Solution with ORk-forcing-g-whips:

The ORk part:
Code: Select all
OR4-forcing-gwhip-elim[7]:
   || n9r6c3 -
   || n1r9c4 - partial-gwhip[2]: b9n1{r9c8 r8c8} - c8n9{r8 r456} -
   || n2r7c6 - partial-gwhip[2]: r8n2{c6 c8} - c8n9{r8 r456} -
   || n8r8c3 - partial-whip[2]: r3c3{n8 n9} - r1n9{c3 c9} -
 ==> r6c9≠9

whip[8]: c2n1{r6 r5} - b4n9{r5c2 r6c3} - r1n9{c3 c9} - b3n3{r1c9 r2c8} - b3n5{r2c8 r3c7} - r3c5{n5 n3} - r6n3{c5 c9} - r6n1{c9 .} ==> r6c2≠6

Trid-OR4-forcing-whip-elim[8]:
   || n9r6c3 - partial-whip[1]: r1n9{c3 c9} -
   || n1r9c4 - partial-whip[2]: b9n1{r9c8 r8c8} - r8n7{c8 c9} -
   || n2r7c6 - partial-whip[2]: r8n2{c6 c8} - r8n7{c8 c9} -
   || n8r8c3 - partial-whip[2]: r3c3{n8 n9} - r1n9{c3 c9} -
 ==> r8c9≠9

biv-chain[3]: r8n9{c8 c7} - r3c7{n9 n5} - b6n5{r5c7 r5c8} ==> r5c8≠9
biv-chain[4]: r2c8{n3 n5} - r3c7{n5 n9} - r8n9{c7 c8} - c8n7{r8 r4} ==> r4c8≠3

Trid-OR4-forcing-whip-elim[7]:
   || n8r8c3 - partial-whip[1]: r3c3{n8 n9} -
   || n9r6c3 - partial-whip[1]: c2n9{r6 r3} -
   || n1r9c4 - partial-whip[2]: b9n1{r9c8 r8c8} - r8n9{c8 c7} -
   || n2r7c6 - partial-whip[2]: r8n2{c6 c8} - r8n9{c8 c7} -
 ==> r3c7≠9


The end is easy:
Code: Select all
singles ==> r3c7=5, r2c8=3, r1c9=9, r1c3=3, r3c5=3, r5c8=5, r2c2=5, r1c2=2, r1c5=5
hidden-pairs-in-a-row: r6{n1 n3}{c2 c9} ==> r6c9≠6, r6c2≠9
finned-x-wing-in-columns: n3{c1 c4}{r5 r7} ==> r7c6≠3
biv-chain[3]: r6n3{c9 c2} - c2n1{r6 r5} - r5n9{c2 c7} ==> r5c7≠3
whip[1]: c7n3{r9 .} ==> r8c9≠3
biv-chain[3]: c5n4{r8 r6} - r6c8{n4 n9} - b9n9{r8c8 r8c7} ==> r8c7≠4
biv-chain[3]: r2c3{n4 n6} - r6n6{c3 c5} - c5n4{r6 r8} ==> r8c3≠4
biv-chain[4]: r8c3{n6 n8} - r3c3{n8 n9} - c2n9{r3 r5} - c7n9{r5 r8} ==> r8c7≠6
z-chain[4]: b7n4{r9c2 r7c1} - r7n3{c1 c7} - r8c7{n3 n9} - r5n9{c7 .} ==> r5c2≠4
biv-chain[5]: r6c9{n3 n1} - b4n1{r6c2 r5c2} - r5n9{c2 c7} - r8c7{n9 n3} - c6n3{r8 r4} ==> r4c9≠3
naked-pairs-in-a-column: c9{r4 r8}{n6 n7} ==> r5c9≠6
finned-x-wing-in-columns: n6{c9 c5}{r8 r4} ==> r4c6≠6
biv-chain[3]: r1c4{n4 n6} - b5n6{r5c4 r6c5} - c5n4{r6 r8} ==> r8c4≠4, r9c4≠4
z-chain[3]: b8n4{r8c6 r7c6} - r4n4{c6 c2} - r9n4{c2 .} ==> r8c8≠4
whip[4]: c5n4{r8 r6} - r4c6{n4 n3} - r4c2{n3 n6} - r6n6{c3 .} ==> r8c2≠4
whip[1]: r8n4{c6 .} ==> r7c6≠4
z-chain[4]: b7n4{r9c2 r7c1} - r7n3{c1 c7} - b9n6{r7c7 r8c9} - r4n6{c9 .} ==> r9c2≠6
finned-x-wing-in-columns: n6{c5 c2}{r8 r6} ==> r6c3≠6
singles ==> r6c5=6, r8c5=4
biv-chain[2]: r6n4{c8 c3} - c2n4{r4 r9} ==> r9c8≠4
naked-single ==> r9c8=1

At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n9r6c3 n2r7c6 n8r8c3
   +----------------+----------------+----------------+
   ! 1    2    3    ! 46   5    46   ! 7    8    9    !
   ! 46   5    46   ! 7    8    9    ! 1    3    2    !
   ! 7    89   89   ! 12   3    12   ! 5    6    4    !
   +----------------+----------------+----------------+
   ! 2    346  5    ! 9    1    34   ! 8    47   67   !
   ! 346  1369 7    ! 34   2    8    ! 469  5    13   !
   ! 8    13   49   ! 5    6    7    ! 2    49   13   !
   +----------------+----------------+----------------+
   ! 346  7    1    ! 8    9    26   ! 346  24   5    !
   ! 5    368  68   ! 1236 4    1236 ! 39   1279 67   !
   ! 9    34   2    ! 36   7    5    ! 346  1    8    !
   +----------------+----------------+----------------+

hidden-pairs-in-a-column: c4{n1 n2}{r3 r8} ==> r8c4≠6, r8c4≠3
finned-x-wing-in-columns: n3{c4 c1}{r5 r9} ==> r9c2≠3
naked-single ==> r9c2=4
biv-chain[3]: c1n3{r7 r5} - r5c4{n3 n4} - c7n4{r5 r7} ==> r7c7≠3
hidden-single-in-a-row ==> r7c1=3
whip[1]: b7n6{r8c3 .} ==> r8c6≠6, r8c9≠6
stte
denis_berthier
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Posts: 4237
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