The New Sudoku Players' Forum

[denis_berthier]

.
Continuing with the series of min-expands, as I find interesting cases.

I'm still busy analysing mith's 63,137 database.
(At the same time, I'm testing my ORk-chain rules on this large database, and improving their merging with the previously existing rules, before I publish any update to CSP-Rules).

Puzzles in this database can have very different behaviours wrt the anti-tridagon vs "normal" chain rules: for each kind of rules, a puzzle may require only easy vs very hard ones, and this for each of the two kinds independently. And the hard "normal" chain rules may have to be before and/or in between and/or after the tridagon chain rules. The positive side of it, it makes the tridagon and "normal" chain rules naturally blend in the same global framework. In particular, the "simplest-first" strategy works in exactly the same way with the Tridagon rules.

As I find puzzles with different behaviours, I propose them here.

This puzzle is much easier than the previous #19828.

Code: Select all

+-------+-------+-------+
! . . . ! . 5 6 ! 7 . 9 !
! 4 . . ! 1 8 9 ! . . . !
! . . . ! 3 . . ! . . . !
+-------+-------+-------+
! 2 . 4 ! . . 5 ! 6 9 . !
! . . 5 ! . . . ! . 7 4 !
! . . . ! . . . ! 5 . 2 !
+-------+-------+-------+
! 5 . . ! . 6 . ! 9 2 . !
! 6 . 9 ! 5 2 . ! 4 . 7 !
! . 4 2 ! . . . ! . . . !
+-------+-------+-------+
....567.94..189......3.....2.4..569...5....74......5.25...6.92.6.952.4.7.42......;7164;135727
SER = 10.9

[Cenoman]

Code: Select all

 +---------------------------+--------------------------+-----------------------+
 |  138     1238     138     |  24      5       6       |  7     1348    9      |
 |  4       23567    367     |  1       8       9       |  23    356     356    |
 |  189     125689   168     |  3       47      247     |  128   14568   1568   |
 +---------------------------+--------------------------+-----------------------+
 |  2       1378*    4       |  78      137     5       |  6     9       138*   |
 |  138-9*  69-138   5       |  2689    139     1238    |  138*  7       4      |
 |  13789   13689-7  138-67* |  46789   13479   13478   |  5     138*    2      |
 +---------------------------+--------------------------+-----------------------+
 |  5       1378     1378*   |  478     6       13478   |  9     2       138*   |
 |  6       138*     9       |  5       2       138     |  4     138*    7      |
 |  1378*   4        2       |  789     1379    1378    |  138*  56      56     |
 +---------------------------+--------------------------+-----------------------+

TH(138)b4679 has six guardians (7r4c2, 9r5c1, 67r6c3, 7r7c3, 7r9c1). Four preliminary steps eliminate three of them, and ease the subsequent use of the three others.
1. (6)r5c2 = (6-2)r5c4 = r1c4 - (2=318)r1c123 - (1|8=6)r3c3 => -6 r6c3
2. (6)r5c2 = (6-2)r5c4 = r1c4 - (2=7138)r1478c2 => -138 r5c2
3. (9)r5c5 = [(4=789)r479c4 - r9c5 *=* (9-4)r6c5 = (4)r3c5] - (4=2)r1c4 - (2=318)r1c123 - (1|8=9)r3c1 => -9 r5c1
4. Kraken column (7)r3679c6
XW(7)c16\r69
||
[(2=4)r1c4 - r7c4 = (4-7)r7c6 *=* (7-2)r3c6 = (2)r1c4] - (2=1387)r1478c2^ - r2c2 = (7)r2c3
=> -7 r6c2^, r6c3

Code: Select all

 +-------------------------+--------------------------+-----------------------+
 |  138     1238    138    |  24      5       6       |  7     1348    9      |
 |  4       2357    367    |  1       8       9       |  23    356     356    |
 |  189     12589   168    |  3       47      247     |  128   14568   1568   |
 +-------------------------+--------------------------+-----------------------+
 |  2       1378*   4      |  78      13-7    5       |  6     9       138*   |
 |  138*    69      5      |  269-8   139     1238    |  138*  7       4      |
 |  13789   13689   138*   |  4679-8  13479   13478   |  5     138*    2      |
 +-------------------------+--------------------------+-----------------------+
 |  5       1378    1378*  |  478     6       13478   |  9     2       138*   |
 |  6       138*    9      |  5       2       138     |  4     138*    7      |
 |  1378*   4       2      |  9-78    139-7    1378   |  138*  56      56     |
 +-------------------------+--------------------------+-----------------------+

TH(138)b4679 has now three guardians: 7r4c2, 7r7c3, 7r9c1. Note that 7r4c2 and 7r9c1 have the same parity (both conjugates of 7r6c1) => only one chain needed for guardians 7r4c2 and 7r9c1.
5. (8=7)r4c4 - r4c2 == r7c3 - r2c3 = r2c2 - (7=1382)r1478c2 - (2=478)r147c4 => -8 r569c4
6. (7)r9c1 = r6c1 - *r4c2 == r7c3 - r2c3 = r2c2 - (7=1382)r1478c2 - (2)r4c1 = (487)r147c4 & (47)b2p18 => -7 r4c5*, r9c45; lcls, 13 placements

Code: Select all

 +------------------------+-------------------+---------------------+
 |  138     2      138    |  4    5    6      |  7     138   9      |
 |  4       357    367    |  1    8    9      |  2     56^   56-3^  |
 |  189     1589   168    |  3    7    2      |  18    4     1568   |
 +------------------------+-------------------+---------------------+
 |  2      e1378   4      | d78  a13*  5      |  6     9     138    |
 |  138     6      5      |  2    9    138    |  138   7     4      |
 |  13789   1389   138    |  6    4    1378   |  5     138   2      |
 +------------------------+-------------------+---------------------+
 |  5      a1378* b1378   | c78   6    4      |  9     2     138    |
 |  6      a138*   9      |  5    2    138    |  4     138   7      |
 | a1378*   4      2      |  9   a13*  1378   |  138   56^   56^    |
 +------------------------+-------------------+---------------------+

7. Almost RP(13)r78c2, r4c5 (r9c1-r9c5)
[RP(13)r78c2,r4c5] = (1|3-7)r7c3 = r7c4 - r4c4 = (7)r4c2 => -13 r4c2
8. UR(56)r29c89 using single internal =>+3 r2c9; ste

[DEFISE]

After basics :

Code: Select all

|--------------------------------------------------------------------|
| 138    1238   138    | 24     5      6      | 7      1348   9      |
| 4      23567  367    | 1      8      9      | 23     356    356    |
| 189    125689 168    | 3      47     247    | 128    14568  1568   |
|--------------------------------------------------------------------|
| 2      1378   4      | 78     137    5      | 6      9      138    |
| 1389   13689  5      | 2689   139    1238   | 138    7      4      |
| 13789  136789 13678  | 46789  13479  13478  | 5      138    2      |
|--------------------------------------------------------------------|
| 5      1378   1378   | 478    6      13478  | 9      2      138    |
| 6      138    9      | 5      2      138    | 4      138    7      |
| 1378   4      2      | 789    1379   1378   | 138    56     56     |
|--------------------------------------------------------------------|

Tridagon (1,3,8) in b4p249, b6p348, b7p357, b9p357
with 6 guardians: 7r4c2,9r5c1,6r6c3,7r6c3,7r7c3,7r9c1

whip[8]: r3n9{c1 c2}- c2n5{r3 r2}- c2n2{r2 r1}- r1c4{n2 n4}- c5n4{r3 r6}- r6n9{c5 c4}- c4n6{r6 r5}- c4n2{r5 .} => -9r5c1

5 guardians remaining : 7r4c2, 6r6c3, 7r6c3, 7r7c3, 7r9c1

whip[4]: r5n6{c2 c4}- c4n2{r5 r1}- c2n2{r1 r3}- c2n5{r3 .} => -6r2c2
whip[4]: r5n6{c2 c4}- c4n2{r5 r1}- c2n2{r1 r2}- c2n5{r2 .} => -6r3c2
Box/Line: 6c2b4 => -6r6c3
4 guardians remaining: 7r4c2,7r6c3,7r7c3,7r9c1

OR4-contrad-whip[2]: r2n7{c2 c3}- OR4{all guardians |.} => -7r7c2

Then the puzzle is solvable in W7 without using the tridagon.
Here is the end in W8 to reduce the number of steps a bit :

whip[6]: r3n7{c5 c6}- r7n7{c6 c3}- r2n7{c3 c2}- r2n2{c2 c7}- r3n2{c7 c2}- c2n5{r3 .} => -7r9c5
whip[6]: c1n7{r9 r6}- c2n7{r6 r2}- c2n5{r2 r3}- c2n2{r3 r1}- b2n2{r1c4 r3c6}- c6n7{r3 .} => -7r9c4
whip[8]: c4n7{r4 r7}- c6n7{r7 r3}- b2n2{r3c6 r1c4}- c4n4{r1 r6}- r6n6{c4 c2}- c2n7{r6 r2}- c2n2{r2 r3}- c2n5{r3 .} => -7r4c5
Hidden pairs: 47c5r36 => -1r6c5 -3r6c5 -9r6c5
whip[8]: r6n6{c4 c2}- r5n6{c2 c4}- c4n2{r5 r1}- c4n4{r1 r7}- c4n7{r7 r4}- c2n7{r4 r2}- c2n2{r2 r3}- c2n5{r3 .} => -9r6c4
Box/Line: 9r6b4 => -9r5c2
whip[5]: c2n9{r6 r3}- c2n5{r3 r2}- c2n2{r2 r1}- c4n2{r1 r5}- r5n6{c4 .} => -6r6c2
Single(s): 6r6c4, 6r5c2
whip[8]: c2n9{r6 r3}- c2n5{r3 r2}- r2n2{c2 c7}- r3n2{c7 c6}- r1c4{n2 n4}- r7n4{c4 c6}- c6n7{r7 r9}- c1n7{r9 .} => -7r6c2
whip[5]: c4n7{r7 r4}- c2n7{r4 r2}- c2n5{r2 r3}- c2n2{r3 r1}- r1c4{n2 .} => -4r7c4
Single(s): 4r7c6, 4r6c5, 7r3c5, 2r3c6, 4r1c4, 2r1c2, 2r2c7, 4r3c8, 2r5c4, 9r5c5, 9r9c4
Hidden pairs: 56c8r29 => -3r2c8
Xwing in columns: 7c16r69 => -7r6c3
whip[5]: c7n3{r9 r5}- r4n3{c9 c2}- r4n7{c2 c4}- r7n7{c4 c3}- b7n3{r7c3 .} => -3r9c5
Single(s): 1r9c5, 3r4c5
whip[3]: r4n7{c2 c4}- r7n7{c4 c3}- b7n1{r7c3 .} => -1r4c2
Single(s): 1r4c9, 1r3c7, 1r8c8
whip[5]: r8n3{c6 c2}- r7n3{c2 c9}- r2n3{c9 c3}- c3n7{r2 r7}- r9n7{c1 .} => -3r9c6
STTE

[denis_berthier]

.
As you have found, there are several steps in W8 for reducing the numbers of guardians.
Finally, a single OR3-whip[1] - or OR3-forcing-whip[1], which is the same thing - is enough to make the puzzle solvable in W5.

Code: Select all

hidden-pairs-in-a-row: r9{n5 n6}{c8 c9} ==> r9c9≠8, r9c9≠3, r9c9≠1, r9c8≠8, r9c8≠3, r9c8≠1
   +----------------------+----------------------+----------------------+
   ! 138    1238   138    ! 24     5      6      ! 7      1348   9      !
   ! 4      23567  367    ! 1      8      9      ! 23     356    356    !
   ! 189    125689 168    ! 3      47     247    ! 128    14568  1568   !
   +----------------------+----------------------+----------------------+
   ! 2      1378   4      ! 78     137    5      ! 6      9      138    !
   ! 1389   13689  5      ! 2689   139    1238   ! 138    7      4      !
   ! 13789  136789 13678  ! 46789  13479  13478  ! 5      138    2      !
   +----------------------+----------------------+----------------------+
   ! 5      1378   1378   ! 478    6      13478  ! 9      2      138    !
   ! 6      138    9      ! 5      2      138    ! 4      138    7      !
   ! 1378   4      2      ! 789    1379   1378   ! 138    56     56     !
   +----------------------+----------------------+----------------------+

OR6-anti-tridagon[12] for digits 1, 3 and 8 in blocks:
        b4, with cells: r4c2, r5c1, r6c3
        b6, with cells: r4c9, r5c7, r6c8
        b7, with cells: r8c2, r9c1, r7c3
        b9, with cells: r8c8, r9c7, r7c9
with 6 guardians: n7r4c2 n9r5c1 n6r6c3 n7r6c3 n7r7c3 n7r9c1

Code: Select all

z-chain[4]: c2n5{r2 r3} - c2n2{r3 r1} - c4n2{r1 r5} - r5n6{c4 .} ==> r2c2≠6
z-chain[4]: c2n5{r3 r2} - c2n2{r2 r1} - c4n2{r1 r5} - r5n6{c4 .} ==> r3c2≠6
whip[1]: c2n6{r6 .} ==> r6c3≠6

At least one candidate of a previous Trid-OR6-relation has just been eliminated.
There remains a Trid-OR5-relation between candidates: n7r4c2 n9r5c1 n7r6c3 n7r7c3 n7r9c1

   +----------------------+----------------------+----------------------+
   ! 138    1238   138    ! 24     5      6      ! 7      1348   9      !
   ! 4      2357   367    ! 1      8      9      ! 23     356    356    !
   ! 189    12589  168    ! 3      47     247    ! 128    14568  1568   !
   +----------------------+----------------------+----------------------+
   ! 2      1378   4      ! 78     137    5      ! 6      9      138    !
   ! 1389   13689  5      ! 2689   139    1238   ! 138    7      4      !
   ! 13789  136789 1378   ! 46789  13479  13478  ! 5      138    2      !
   +----------------------+----------------------+----------------------+
   ! 5      1378   1378   ! 478    6      13478  ! 9      2      138    !
   ! 6      138    9      ! 5      2      138    ! 4      138    7      !
   ! 1378   4      2      ! 789    1379   1378   ! 138    56     56     !
   +----------------------+----------------------+----------------------+

Code: Select all

z-chain[4]: c6n2{r5 r3} - c7n2{r3 r2} - c7n3{r2 r9} - c5n3{r9 .} ==> r5c6≠3
t-whip[6]: r2n6{c9 c3} - r2n7{c3 c2} - c2n5{r2 r3} - c2n2{r3 r1} - r1c4{n2 n4} - r3n4{c6 .} ==> r3c8≠6
whip[6]: r5n6{c2 c4} - r6n6{c4 c2} - c2n9{r6 r3} - c2n5{r3 r2} - c2n2{r2 r1} - c4n2{r1 .} ==> r5c2≠8
whip[6]: r5n6{c2 c4} - r6n6{c4 c2} - c2n9{r6 r3} - c2n5{r3 r2} - c2n2{r2 r1} - c4n2{r1 .} ==> r5c2≠3
whip[6]: r5n6{c2 c4} - r6n6{c4 c2} - c2n9{r6 r3} - c2n5{r3 r2} - c2n2{r2 r1} - c4n2{r1 .} ==> r5c2≠1
whip[6]: c4n6{r6 r5} - r5c2{n6 n9} - b5n9{r5c4 r6c5} - c5n4{r6 r3} - c5n7{r3 r9} - c1n7{r9 .} ==> r6c4≠7
whip[7]: c1n7{r6 r9} - c4n7{r9 r7} - r7n4{c4 c6} - r6n4{c6 c4} - c4n6{r6 r5} - b5n9{r5c4 r5c5} - r5c2{n9 .} ==> r6c5≠7
whip[8]: r2n7{c3 c2} - c2n5{r2 r3} - c2n2{r3 r1} - r1c4{n2 n4} - r3c5{n4 n7} - r4n7{c5 c4} - r7n7{c4 c6} - r7n4{c6 .} ==> r6c3≠7

At least one candidate of a previous Trid-OR5-relation has just been eliminated.
There remains a Trid-OR4-relation between candidates: n7r4c2 n9r5c1 n7r7c3 n7r9c1

   +----------------------+----------------------+----------------------+
   ! 138    1238   138    ! 24     5      6      ! 7      1348   9      !
   ! 4      2357   367    ! 1      8      9      ! 23     356    356    !
   ! 189    12589  168    ! 3      47     247    ! 128    1458   1568   !
   +----------------------+----------------------+----------------------+
   ! 2      1378   4      ! 78     137    5      ! 6      9      138    !
   ! 1389   69     5      ! 2689   139    128    ! 138    7      4      !
   ! 13789  136789 138    ! 4689   1349   13478  ! 5      138    2      !
   +----------------------+----------------------+----------------------+
   ! 5      1378   1378   ! 478    6      13478  ! 9      2      138    !
   ! 6      138    9      ! 5      2      138    ! 4      138    7      !
   ! 1378   4      2      ! 789    1379   1378   ! 138    56     56     !
   +----------------------+----------------------+----------------------+

Code: Select all

whip[8]: r9n7{c6 c1} - r6n7{c1 c2} - c2n6{r6 r5} - c2n9{r5 r3} - c2n5{r3 r2} - c2n2{r2 r1} - r1c4{n2 n4} - r7n4{c4 .} ==> r7c6≠7
whip[6]: c1n7{r6 r9} - b8n7{r9c6 r7c4} - r4n7{c4 c5} - r3c5{n7 n4} - c4n4{r1 r6} - r6n6{c4 .} ==> r6c2≠7
whip[8]: r5c2{n9 n6} - c4n6{r5 r6} - r6n9{c4 c5} - c2n9{r6 r3} - c2n5{r3 r2} - c2n2{r2 r1} - r1c4{n2 n4} - c5n4{r3 .} ==> r5c1≠9

At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n7r4c2 n7r7c3 n7r9c1

   +-------------------+-------------------+-------------------+
   ! 138   1238  138   ! 24    5     6     ! 7     1348  9     !
   ! 4     2357  367   ! 1     8     9     ! 23    356   356   !
   ! 189   12589 168   ! 3     47    247   ! 128   1458  1568  !
   +-------------------+-------------------+-------------------+
   ! 2     1378  4     ! 78    137   5     ! 6     9     138   !
   ! 138   69    5     ! 2689  139   128   ! 138   7     4     !
   ! 13789 13689 138   ! 4689  1349  13478 ! 5     138   2     !
   +-------------------+-------------------+-------------------+
   ! 5     1378  1378  ! 478   6     1348  ! 9     2     138   !
   ! 6     138   9     ! 5     2     138   ! 4     138   7     !
   ! 1378  4     2     ! 789   1379  1378  ! 138   56    56    !
   +-------------------+-------------------+-------------------+

Trid-OR3-whp[3]: {{n7r4c2 n7r7c3 n7r9c1 | . }} ==> r7c2≠7

Code: Select all

t-whip[5]: c2n7{r4 r2} - c2n5{r2 r3} - c2n2{r3 r1} - r1c4{n2 n4} - r3c5{n4 .} ==> r4c5≠7
biv-chain[2]: c1n7{r9 r6} - b5n7{r6c6 r4c4} ==> r9c4≠7
t-whip[5]: r7n7{c4 c3} - r2n7{c3 c2} - c2n5{r2 r3} - c2n2{r3 r1} - r1c4{n2 .} ==> r7c4≠4
hidden-single-in-a-block ==> r7c6=4
naked-pairs-in-a-column: c4{r4 r7}{n7 n8} ==> r9c4≠8, r6c4≠8, r5c4≠8
naked-single ==> r9c4=9
biv-chain[4]: c2n7{r2 r4} - b5n7{r4c4 r6c6} - r3c6{n7 n2} - b3n2{r3c7 r2c7} ==> r2c2≠2
hidden-single-in-a-row ==> r2c7=2
biv-chain[4]: r3n2{c6 c2} - b1n5{r3c2 r2c2} - c2n7{r2 r4} - b5n7{r4c4 r6c6} ==> r3c6≠7
end in S3Fin + Z4