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[denis_berthier]

.

Intended in particular for eleven.

Code: Select all

+-------+-------+-------+
! . . . ! 4 . 6 ! 7 8 . !
! . . 7 ! 1 8 . ! 2 . . !
! . . . ! . 7 2 ! . 4 1 !
+-------+-------+-------+
! . 9 . ! . . . ! 4 . . !
! 5 3 . ! . . . ! 1 . 2 !
! . . 6 ! 2 . 1 ! . . . !
+-------+-------+-------+
! . . . ! . . . ! 6 2 . !
! 8 7 . ! . . . ! . 1 . !
! . . . ! . . 4 ! . . 7 !
+-------+-------+-------+
...4.678...718.2......72.41.9....4..53....1.2..62.1.........62.87.....1......4..7;4184;89540
SER = 11.1

Code: Select all

Resolution state after Singles and whips[1]:
   +-------------------+-------------------+-------------------+
   ! 1239  125   12359 ! 4     359   6     ! 7     8     359   !
   ! 349   45    7     ! 1     8     359   ! 2     3569  3569  !
   ! 369   568   3589  ! 359   7     2     ! 359   4     1     !
   +-------------------+-------------------+-------------------+
   ! 127   9     128   ! 35678 356   3578  ! 4     3567  3568  !
   ! 5     3     48    ! 6789  469   789   ! 1     679   2     !
   ! 47    48    6     ! 2     3459  1     ! 3589  3579  3589  !
   +-------------------+-------------------+-------------------+
   ! 1349  145   13459 ! 35789 1359  35789 ! 6     2     34589 !
   ! 8     7     23459 ! 3569  23569 359   ! 359   1     3459  !
   ! 12369 1256  12359 ! 3589  12359 4     ! 3589  359   7     !
   +-------------------+-------------------+-------------------+
196 candidates.

[DEFISE]

A monster!
Using Tridagon, solvable in T&E(1,S4) but ending with two UR.

[DEFISE]

Simple T&E(1) is enough except one time when I have to use T&E(1,S2) to exploit an UR with 3 guardians.

[denis_berthier]

.
Yes, a monster indeed.

What I wanted to show with this puzzle is, finding and applying anti-tridagon rules may leave us with a very hard puzzle.
I also wanted to illustrate the necessity of using degenerated Trid-ORk-relations.

1) Let's first evacuate the use of eleven replacement at the start: it leads to an easy solution in W10 (I mean easy for a puzzle in T&E(3)):

Code: Select all

(solve-sukaku-grid-by-eleven-replacement3 3 5 9
1 5
2 6
3 4
   +-------------------+-------------------+-------------------+
   ! 1239  125   12359 ! 4     359   6     ! 7     8     359   !
   ! 349   45    7     ! 1     8     359   ! 2     3569  3569  !
   ! 369   568   3589  ! 359   7     2     ! 359   4     1     !
   +-------------------+-------------------+-------------------+
   ! 127   9     128   ! 35678 356   3578  ! 4     3567  3568  !
   ! 5     3     48    ! 6789  469   789   ! 1     679   2     !
   ! 47    48    6     ! 2     3459  1     ! 3589  3579  3589  !
   +-------------------+-------------------+-------------------+
   ! 1349  145   13459 ! 35789 1359  35789 ! 6     2     34589 !
   ! 8     7     23459 ! 3569  23569 359   ! 359   1     3459  !
   ! 12369 1256  12359 ! 3589  12359 4     ! 3589  359   7     !
   +-------------------+-------------------+-------------------+
)

Code: Select all

AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to 3 digits 3, 5 and 9 in 3 cells r1c5, r2c6 and r3c4,
the resolution state is:
   +----------------------+----------------------+----------------------+
   ! 12359  12359  12359  ! 4      3      6      ! 7      8      359    !
   ! 3594   4359   7      ! 1      8      5      ! 2      3596   3596   !
   ! 3596   35968  3598   ! 9      7      2      ! 359    4      1      !
   +----------------------+----------------------+----------------------+
   ! 127    359    128    ! 359678 3596   35978  ! 4      35967  35968  !
   ! 359    359    48     ! 678359 46359  78359  ! 1      67359  2      !
   ! 47     48     6      ! 2      3594   1      ! 3598   3597   3598   !
   +----------------------+----------------------+----------------------+
   ! 13594  14359  13594  ! 35978  1359   35978  ! 6      2      35948  !
   ! 8      7      23594  ! 3596   23596  359    ! 359    1      3594   !
   ! 123596 123596 12359  ! 3598   12359  4      ! 3598   359    7      !
   +----------------------+----------------------+----------------------+

THIS IS THE PUZZLE THAT WILL NOW BE SOLVED.
RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.

Hidden Text: Show

whip[1]: r6n3{c9 .} ==> r5c8≠3, r4c8≠3, r4c9≠3
naked-pairs-in-a-block: b4{r5c3 r6c2}{n4 n8} ==> r6c1≠4, r4c3≠8
naked-single ==> r6c1=7
z-chain[3]: r9n6{c1 c2} - c2n2{r9 r1} - c2n1{r1 .} ==> r9c1≠1
z-chain[3]: c2n6{r9 r3} - r3n8{c2 c3} - c3n3{r3 .} ==> r9c2≠3
t-whip[5]: c3n3{r9 r3} - r3n8{c3 c2} - c2n6{r3 r9} - c2n2{r9 r1} - c2n1{r1 .} ==> r7c2≠3
whip[5]: c7n8{r6 r9} - c7n9{r9 r8} - r8c6{n9 n3} - r9c4{n3 n5} - c8n5{r9 .} ==> r6c7≠5
whip[5]: r1c9{n9 n5} - r3c7{n5 n3} - r6c7{n3 n8} - r4c9{n8 n6} - r2c9{n6 .} ==> r6c9≠9
whip[7]: r1c9{n9 n5} - r3c7{n5 n3} - r2c9{n3 n6} - r4c9{n6 n8} - r6n8{c9 c2} - r5c3{n8 n4} - r8n4{c3 .} ==> r8c9≠9
t-whip[9]: r6n9{c8 c5} - r6n4{c5 c2} - c2n8{r6 r3} - c2n6{r3 r9} - c2n2{r9 r1} - c2n1{r1 r7} - r7c5{n1 n5} - r4c5{n5 n6} - r5n6{c5 .} ==> r5c8≠9
t-whip[9]: r6n5{c9 c5} - r6n4{c5 c2} - c2n8{r6 r3} - c2n6{r3 r9} - c2n2{r9 r1} - c2n1{r1 r7} - r7c5{n1 n9} - r4c5{n9 n6} - r5n6{c5 .} ==> r5c8≠5
t-whip[10]: r6n9{c8 c5} - r6n4{c5 c2} - c2n8{r6 r3} - c2n6{r3 r9} - c2n2{r9 r1} - c2n1{r1 r7} - r7c5{n1 n5} - r4c5{n5 n6} - r5n6{c5 c8} - c8n7{r5 .} ==> r4c8≠9
whip[8]: c9n6{r2 r4} - r5c8{n6 n7} - r4c8{n7 n5} - r4c5{n5 n9} - r4c2{n9 n3} - r2c2{n3 n4} - r6n4{c2 c5} - r6n5{c5 .} ==> r2c9≠9
whip[5]: r8c6{n3 n9} - r8c7{n9 n5} - b3n5{r3c7 r1c9} - b3n9{r1c9 r2c8} - r9c8{n9 .} ==> r8c9≠3
whip[7]: c3n9{r9 r1} - c9n9{r1 r4} - b4n9{r4c2 r5c1} - b5n9{r5c5 r6c5} - r6n4{c5 c2} - r2c2{n4 n3} - b4n3{r4c2 .} ==> r7c2≠9
whip[9]: c8n5{r6 r9} - c7n5{r9 r3} - r1c9{n5 n9} - c8n9{r2 r6} - r6n3{c8 c7} - r8c7{n3 n9} - r8c6{n9 n3} - r9c4{n3 n8} - r9c7{n8 .} ==> r6c9≠5
whip[9]: r5n4{c3 c5} - r6n4{c5 c2} - r7n4{c2 c9} - b9n8{r7c9 r9c7} - r6n8{c7 c9} - c9n3{r6 r2} - c9n6{r2 r4} - c5n6{r4 r8} - r8n2{c5 .} ==> r8c3≠4
hidden-single-in-a-row ==> r8c9=4
whip[9]: b9n8{r9c7 r7c9} - b6n8{r6c9 r6c7} - c7n9{r6 r8} - r8c6{n9 n3} - b9n3{r8c7 r9c8} - c7n3{r9 r3} - c3n3{r3 r7} - c3n4{r7 r5} - b4n8{r5c3 .} ==> r9c7≠5
whip[9]: r2c9{n6 n3} - r3c7{n3 n5} - c9n5{r1 r7} - c9n8{r7 r6} - r6c2{n8 n4} - r7c2{n4 n1} - r7c5{n1 n9} - r6c5{n9 n5} - r4c5{n5 .} ==> r4c9≠6
hidden-single-in-a-column ==> r2c9=6
hidden-pairs-in-a-column: c8{n6 n7}{r4 r5} ==> r4c8≠5
biv-chain[3]: r2c8{n3 n9} - r1c9{n9 n5} - b6n5{r4c9 r6c8} ==> r6c8≠3
t-whip[6]: r6n3{c9 c7} - c7n8{r6 r9} - c7n9{r9 r8} - r8c6{n9 n3} - r9c4{n3 n5} - r9c8{n5 .} ==> r7c9≠3
hidden-single-in-a-column ==> r6c9=3
finned-x-wing-in-columns: n8{c9 c6}{r7 r4} ==> r4c4≠8
whip[9]: c3n4{r7 r5} - b4n8{r5c3 r6c2} - b6n8{r6c7 r4c9} - r7c9{n8 n5} - r7c5{n5 n1} - r7c2{n1 n4} - r7c1{n4 n3} - c3n3{r7 r3} - r3n8{c3 .} ==> r7c3≠9
z-chain[5]: b9n3{r9c8 r8c7} - r8c6{n3 n9} - c3n9{r8 r1} - b3n9{r1c9 r2c8} - c8n3{r2 .} ==> r9c3≠3
whip[5]: b3n9{r2c8 r1c9} - c3n9{r1 r8} - r8c6{n9 n3} - r8c7{n3 n5} - b3n5{r3c7 .} ==> r9c8≠9
biv-chain[3]: r9c8{n5 n3} - r2c8{n3 n9} - r1c9{n9 n5} ==> r7c9≠5
biv-chain[2]: r6n5{c5 c8} - b9n5{r9c8 r8c7} ==> r8c5≠5
biv-chain[4]: r9n8{c4 c7} - r6c7{n8 n9} - c8n9{r6 r2} - c8n3{r2 r9} ==> r9c4≠3
biv-chain[4]: r9c4{n5 n8} - c7n8{r9 r6} - c2n8{r6 r3} - c2n6{r3 r9} ==> r9c2≠5
biv-chain[4]: c2n8{r3 r6} - r6c7{n8 n9} - c8n9{r6 r2} - b3n3{r2c8 r3c7} ==> r3c2≠3
biv-chain[4]: c8n9{r2 r6} - r6n5{c8 c5} - r6n4{c5 c2} - b1n4{r2c2 r2c1} ==> r2c1≠9
whip[4]: r1c9{n5 n9} - r2n9{c8 c2} - r5c2{n9 n3} - r4c2{n3 .} ==> r1c2≠5
biv-chain[5]: r3n6{c1 c2} - c2n8{r3 r6} - r6c7{n8 n9} - c8n9{r6 r2} - b3n3{r2c8 r3c7} ==> r3c1≠3
t-whip[5]: b4n5{r5c2 r5c1} - r3c1{n5 n6} - r9n6{c1 c2} - c2n2{r9 r1} - c2n1{r1 .} ==> r7c2≠5
biv-chain[5]: r9c4{n5 n8} - c7n8{r9 r6} - r6c2{n8 n4} - r7c2{n4 n1} - b8n1{r7c5 r9c5} ==> r9c5≠5
whip[5]: r3c7{n3 n5} - r1c9{n5 n9} - b9n9{r7c9 r9c7} - c3n9{r9 r8} - r8c6{n9 .} ==> r8c7≠3
whip[1]: b9n3{r9c8 .} ==> r9c1≠3
whip[5]: r3n5{c3 c7} - r3n3{c7 c3} - b7n3{r8c3 r7c1} - b7n5{r7c1 r9c1} - b9n5{r9c8 .} ==> r1c3≠5
z-chain[6]: r9c4{n5 n8} - c7n8{r9 r6} - c2n8{r6 r3} - r3c3{n8 n3} - c7n3{r3 r9} - r9c8{n3 .} ==> r9c3≠5
naked-triplets-in-a-column: c3{r1 r4 r9}{n9 n2 n1} ==> r8c3≠9, r8c3≠2, r7c3≠1
hidden-single-in-a-row ==> r8c5=2
hidden-single-in-a-block ==> r8c4=6
biv-chain[3]: r8n9{c6 c7} - r7c9{n9 n8} - r4n8{c9 c6} ==> r4c6≠9
biv-chain[3]: r8c3{n3 n5} - c7n5{r8 r3} - r3n3{c7 c3} ==> r7c3≠3
z-chain[3]: b7n9{r9c3 r7c1} - b7n3{r7c1 r8c3} - r8c6{n3 .} ==> r9c5≠9
naked-single ==> r9c5=1
biv-chain[3]: r9c4{n8 n5} - r7c5{n5 n9} - r7c9{n9 n8} ==> r7c4≠8, r7c6≠8, r9c7≠8
hidden-single-in-a-block ==> r7c9=8
hidden-single-in-a-block ==> r6c7=8
naked-single ==> r6c2=4
naked-single ==> r5c3=8
naked-single ==> r7c2=1
hidden-single-in-a-block ==> r3c2=8
hidden-single-in-a-block ==> r3c1=6
hidden-single-in-a-block ==> r9c2=6
hidden-single-in-a-column ==> r1c2=2
hidden-single-in-a-block ==> r4c6=8
hidden-single-in-a-column ==> r9c4=8
hidden-single-in-a-block ==> r2c1=4
hidden-single-in-a-block ==> r7c3=4
hidden-single-in-a-row ==> r5c5=4
hidden-single-in-a-block ==> r4c5=6
naked-single ==> r4c8=7
naked-single ==> r5c8=6
whip[1]: b8n5{r7c5 .} ==> r7c1≠5
whip[1]: c2n5{r5 .} ==> r5c1≠5
naked-pairs-in-a-column: c1{r5 r7}{n3 n9} ==> r9c1≠9, r1c1≠9
finned-x-wing-in-columns: n9{c1 c6}{r5 r7} ==> r7c5≠9
stte

2) Starting from the resolution state after Singles and whips[1], we quickly find two anti-tridagon patterns on different cells in b8 and b9:

Code: Select all

naked-pairs-in-a-block: b4{r5c3 r6c2}{n4 n8} ==> r6c1≠4, r4c3≠8
naked-single ==> r6c1=7
179 g-candidates, 922 csp-glinks and 547 non-csp glinks
   +-------------------+-------------------+-------------------+
   ! 1239  125   12359 ! 4     359   6     ! 7     8     359   !
   ! 349   45    7     ! 1     8     359   ! 2     3569  3569  !
   ! 369   568   3589  ! 359   7     2     ! 359   4     1     !
   +-------------------+-------------------+-------------------+
   ! 12    9     12    ! 35678 356   3578  ! 4     3567  3568  !
   ! 5     3     48    ! 6789  469   789   ! 1     679   2     !
   ! 7     48    6     ! 2     3459  1     ! 3589  359   3589  !
   +-------------------+-------------------+-------------------+
   ! 1349  145   13459 ! 35789 1359  35789 ! 6     2     34589 !
   ! 8     7     23459 ! 3569  23569 359   ! 359   1     3459  !
   ! 12369 1256  12359 ! 3589  12359 4     ! 3589  359   7     !
   +-------------------+-------------------+-------------------+

OR5-anti-tridagon[12] for digits 3, 5 and 9 in blocks:
        b2, with cells: r1c5, r2c6, r3c4
        b3, with cells: r1c9, r2c8, r3c7
        b8, with cells: r7c5, r8c6, r9c4
        b9, with cells: r7c9, r8c7, r9c8
with 5 guardians: n6r2c8 n1r7c5 n4r7c9 n8r7c9 n8r9c4

OR8-anti-tridagon[12] for digits 3, 5 and 9 in blocks:
        b2, with cells: r1c5, r2c6, r3c4
        b3, with cells: r1c9, r2c8, r3c7
        b8, with cells: r9c5, r7c6, r8c4
        b9, with cells: r9c8, r7c9, r8c7
with 8 guardians: n6r2c8 n7r7c6 n8r7c6 n4r7c9 n8r7c9 n6r8c4 n1r9c5 n2r9c5

Code: Select all

biv-chain[3]: r2c2{n5 n4} - b4n4{r6c2 r5c3} - c3n8{r5 r3} ==> r3c3≠5
z-chain[3]: r9n6{c1 c2} - c2n2{r9 r1} - c2n1{r1 .} ==> r9c1≠1
t-whip[4]: r2c2{n5 n4} - r6c2{n4 n8} - b6n8{r6c9 r4c9} - c9n6{r4 .} ==> r2c9≠5
t-whip[4]: r6n5{c9 c5} - r6n4{c5 c2} - r2c2{n4 n5} - r1n5{c3 .} ==> r4c9≠5
whip[7]: r7n7{c4 c6} - b8n8{r7c6 r9c4} - c7n8{r9 r6} - r6c2{n8 n4} - r2c2{n4 n5} - c6n5{r2 r4} - r4n8{c6 .} ==> r7c4≠5
g-whip[7]: r8n4{c9 c3} - r5n4{c3 c5} - r6n4{c5 c2} - r6n8{c2 c789} - r4c9{n8 n6} - c5n6{r4 r8} - r8n2{c5 .} ==> r8c9≠3
g-whip[8]: b6n8{r4c9 r6c789} - r6c2{n8 n4} - r5n4{c3 c5} - c5n6{r5 r8} - c5n2{r8 r9} - c5n1{r9 r7} - r7c2{n1 n5} - r2c2{n5 .} ==> r4c9≠6
hidden-single-in-a-column ==> r2c9=6

At least one candidate of a previous Trid-OR5-relation has just been eliminated.
There remains a Trid-OR4-relation between candidates: n1r7c5 n4r7c9 n8r7c9 n8r9c4

Code: Select all

   +-------------------+-------------------+-------------------+
   ! 1239  125   12359 ! 4     359   6     ! 7     8     359   !
   ! 349   45    7     ! 1     8     359   ! 2     359   6     !
   ! 369   568   389   ! 359   7     2     ! 359   4     1     !
   +-------------------+-------------------+-------------------+
   ! 12    9     12    ! 35678 356   3578  ! 4     3567  38    !
   ! 5     3     48    ! 6789  469   789   ! 1     679   2     !
   ! 7     48    6     ! 2     3459  1     ! 3589  359   3589  !
   +-------------------+-------------------+-------------------+
   ! 1349  145   13459 ! 3789  1359  35789 ! 6     2     34589 !
   ! 8     7     23459 ! 3569  23569 359   ! 359   1     459   !
   ! 2369  1256  12359 ! 3589  12359 4     ! 3589  359   7     !
   +-------------------+-------------------+-------------------+

At this point, notice that the Trid-OR4-relation is degenerated and CSP-Rules wouldn't have found if it was not inherited from the previous Trid-OR5-relation.

hidden-pairs-in-a-column: c8{n6 n7}{r4 r5} ==> r5c8≠9, r4c8≠5, r4c8≠3
whip[1]: b6n5{r6c9 .} ==> r6c5≠5
whip[1]: b6n9{r6c9 .} ==> r6c5≠9
whip[5]: c3n8{r3 r5} - r5n4{c3 c5} - r6c5{n4 n3} - r1n3{c5 c9} - r4n3{c9 .} ==> r3c3≠3
Trid-OR4-whip[6]: r6c5{n3 n4} - r6c2{n4 n8} - c7n8{r6 r9} - OR4{{n8r9c4 n8r7c9 n1r7c5 | n4r7c9}} - c1n4{r7 r2} - c2n4{r2 .} ==> r7c5≠3
Trid-OR4-whip[7]: c1n4{r7 r2} - r2c2{n4 n5} - r7c2{n5 n4} - r6c2{n4 n8} - c7n8{r6 r9} - OR4{{n8r9c4 n8r7c9 n4r7c9 | n1r7c5}} - r7c5{n5 .} ==> r7c1≠1
Trid-OR4-whip[7]: r2c2{n5 n4} - c1n4{r2 r7} - c3n4{r8 r5} - r6c2{n4 n8} - c7n8{r6 r9} - OR4{{n8r9c4 n8r7c9 n4r7c9 | n1r7c5}} - r7c2{n1 .} ==> r9c2≠5
Trid-OR4-whip[7]: r2c2{n5 n4} - c1n4{r2 r7} - c3n4{r8 r5} - r6c2{n4 n8} - c7n8{r6 r9} - OR4{{n8r9c4 n8r7c9 n4r7c9 | n1r7c5}} - r7c2{n1 .} ==> r1c2≠5
biv-chain[3]: r1c2{n2 n1} - c1n1{r1 r4} - b4n2{r4c1 r4c3} ==> r1c3≠2
Trid-OR4-whip[7]: r2c2{n5 n4} - c1n4{r2 r7} - c3n4{r8 r5} - r6c2{n4 n8} - c7n8{r6 r9} - OR4{{n8r9c4 n8r7c9 n4r7c9 | n1r7c5}} - r7c2{n1 .} ==> r3c2≠5
z-chain[5]: c1n1{r1 r4} - c1n2{r4 r9} - c1n6{r9 r3} - r3c2{n6 n8} - r3c3{n8 .} ==> r1c1≠9
Trid-OR4-whip[8]: c2n1{r9 r1} - c2n2{r1 r9} - c2n6{r9 r3} - c2n8{r3 r6} - c7n8{r6 r9} - OR4{{n8r9c4 n8r7c9 n1r7c5 | n4r7c9}} - c1n4{r7 r2} - c2n4{r2 .} ==> r7c3≠1
whip[9]: r7n7{c6 c4} - r7n8{c4 c9} - c7n8{r9 r6} - r6c2{n8 n4} - r2c2{n4 n5} - r7c2{n5 n1} - r7c5{n1 n5} - r8c6{n5 n3} - r2c6{n3 .} ==> r7c6≠9
whip[9]: r7n7{c4 c6} - b8n8{r7c6 r9c4} - c7n8{r9 r6} - r6c2{n8 n4} - r2c2{n4 n5} - r7c2{n5 n1} - r7c5{n1 n5} - c6n5{r8 r4} - r4n8{c6 .} ==> r7c4≠9
whip[10]: r8n4{c9 c3} - b4n4{r5c3 r6c2} - r6c5{n4 n3} - b6n3{r6c7 r4c9} - r1c9{n3 n5} - r3n5{c7 c4} - b2n3{r3c4 r2c6} - b3n3{r2c8 r3c7} - r8c7{n3 n5} - r8c6{n5 .} ==> r8c9≠9
whip[10]: r7n7{c6 c4} - r7n8{c4 c9} - c7n8{r9 r6} - r6c2{n8 n4} - r2c2{n4 n5} - r2c6{n5 n9} - r8c6{n9 n5} - r8c9{n5 n4} - c3n4{r8 r7} - r7n5{c3 .} ==> r7c6≠3
whip[7]: c1n6{r9 r3} - r3c2{n6 n8} - r6c2{n8 n4} - r6c5{n4 n3} - c8n3{r6 r2} - c6n3{r2 r8} - c7n3{r8 .} ==> r9c1≠3
t-whip[11]: r8n4{c9 c3} - c1n4{r7 r2} - c2n4{r2 r6} - r7n4{c2 c9} - b9n8{r7c9 r9c7} - r6n8{c7 c9} - c9n9{r6 r1} - r2n9{c8 c6} - r2n3{c6 c8} - b9n3{r9c8 r8c7} - r8c6{n3 .} ==> r8c9≠5
naked-single ==> r8c9=4

At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n1r7c5 n8r7c9 n8r9c4

Code: Select all

   +-------------------+-------------------+-------------------+
   ! 123   12    1359  ! 4     359   6     ! 7     8     359   !
   ! 349   45    7     ! 1     8     359   ! 2     359   6     !
   ! 369   68    89    ! 359   7     2     ! 359   4     1     !
   +-------------------+-------------------+-------------------+
   ! 12    9     12    ! 35678 356   3578  ! 4     67    38    !
   ! 5     3     48    ! 6789  469   789   ! 1     67    2     !
   ! 7     48    6     ! 2     34    1     ! 3589  359   3589  !
   +-------------------+-------------------+-------------------+
   ! 349   145   3459  ! 378   159   578   ! 6     2     3589  !
   ! 8     7     2359  ! 3569  23569 359   ! 359   1     4     !
   ! 269   126   12359 ! 3589  12359 4     ! 3589  359   7     !
   +-------------------+-------------------+-------------------+

Trid-OR3-whip[5]: r6c2{n8 n4} - r2c2{n4 n5} - r7c2{n5 n1} - OR3{{n1r7c5 n8r7c9 | n8r9c4}} - c7n8{r9 .} ==> r6c9≠8
finned-x-wing-in-columns: n8{c9 c6}{r7 r4} ==> r4c4≠8
Trid-OR3-whip[5]: c2n5{r7 r2} - c6n5{r2 r4} - r4n8{c6 c9} - OR3{{n8r7c9 n1r7c5 | n8r9c4}} - c7n8{r9 .} ==> r7c5≠5
whip[7]: r7n7{c4 c6} - r7n8{c6 c9} - c7n8{r9 r6} - r6c2{n8 n4} - r2c2{n4 n5} - r7n5{c2 c3} - c3n4{r7 .} ==> r7c4≠3
whip[7]: b9n8{r9c7 r7c9} - r4n8{c9 c6} - r5n8{c4 c3} - r3c3{n8 n9} - b7n9{r7c3 r7c1} - r7n3{c1 c3} - c3n4{r7 .} ==> r9c7≠9
whip[7]: r9n6{c1 c2} - r3c2{n6 n8} - r3c3{n8 n9} - c1n9{r3 r7} - r7c5{n9 n1} - c2n1{r7 r1} - r1n2{c2 .} ==> r9c1≠2
hidden-pairs-in-a-column: c1{n1 n2}{r1 r4} ==> r1c1≠3
naked-pairs-in-a-block: b1{r1c1 r1c2}{n1 n2} ==> r1c3≠1
t-whip[9]: b9n8{r9c7 r7c9} - r4c9{n8 n3} - b5n3{r4c4 r6c5} - r6n4{c5 c2} - r2c2{n4 n5} - r7c2{n5 n1} - r7c5{n1 n9} - r1c5{n9 n5} - b3n5{r1c9 .} ==> r9c7≠5
PUZZLE 0 IS NOT SOLVED. 50 VALUES MISSING.
Final resolution state:

Code: Select all

   +-------------------+-------------------+-------------------+
   ! 12    12    359   ! 4     359   6     ! 7     8     359   !
   ! 349   45    7     ! 1     8     359   ! 2     359   6     !
   ! 369   68    89    ! 359   7     2     ! 359   4     1     !
   +-------------------+-------------------+-------------------+
   ! 12    9     12    ! 3567  356   3578  ! 4     67    38    !
   ! 5     3     48    ! 6789  469   789   ! 1     67    2     !
   ! 7     48    6     ! 2     34    1     ! 3589  359   359   !
   +-------------------+-------------------+-------------------+
   ! 349   145   3459  ! 78    19    578   ! 6     2     3589  !
   ! 8     7     2359  ! 3569  23569 359   ! 359   1     4     !
   ! 69    126   12359 ! 3589  12359 4     ! 38    359   7     !
   +-------------------+-------------------+-------------------+

After all the sweating, we're left with a puzzle in T&E(W2, 1) - the initial puzzle is in T&E(W2, 2).

Of course, we can still use eleven replacement to finish the work:

Code: Select all

(solve-sukaku-grid-by-eleven-replacement3 3 5 9
1 5
2 6
3 4
    +-------------------+-------------------+-------------------+
    ! 12    12    359   ! 4     359   6     ! 7     8     359   !
    ! 349   45    7     ! 1     8     359   ! 2     359   6     !
    ! 369   68    89    ! 359   7     2     ! 359   4     1     !
    +-------------------+-------------------+-------------------+
    ! 12    9     12    ! 3567  356   3578  ! 4     67    38    !
    ! 5     3     48    ! 6789  469   789   ! 1     67    2     !
    ! 7     48    6     ! 2     34    1     ! 3589  359   359   !
    +-------------------+-------------------+-------------------+
    ! 349   145   3459  ! 78    19    578   ! 6     2     3589  !
    ! 8     7     2359  ! 3569  23569 359   ! 359   1     4     !
    ! 69    126   12359 ! 3589  12359 4     ! 38    359   7     !
    +-------------------+-------------------+-------------------+
)

Code: Select all

AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to 3 digits 3, 5 and 9 in 3 cells r1c5, r2c6 and r3c4,
the resolution state is:
   +----------------------+----------------------+----------------------+
   ! 12     12     359    ! 4      3      6      ! 7      8      359    !
   ! 3594   4359   7      ! 1      8      5      ! 2      359    6      !
   ! 3596   68     8359   ! 9      7      2      ! 359    4      1      !
   +----------------------+----------------------+----------------------+
   ! 12     359    12     ! 35967  3596   35978  ! 4      67     3598   !
   ! 359    359    48     ! 678359 46359  78359  ! 1      67     2      !
   ! 7      48     6      ! 2      3594   1      ! 3598   359    359    !
   +----------------------+----------------------+----------------------+
   ! 3594   14359  3594   ! 78     1359   35978  ! 6      2      3598   !
   ! 8      7      2359   ! 3596   23596  359    ! 359    1      4      !
   ! 6359   126    12359  ! 3598   12359  4      ! 3598   359    7      !
   +----------------------+----------------------+----------------------+

THIS IS THE PUZZLE THAT WILL NOW BE SOLVED.
RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.

and we find a solution in W6:

Hidden Text: Show

whip[1]: r6n3{c9 .} ==> r4c9≠3
z-chain[4]: r8c6{n9 n3} - r8c7{n3 n5} - b3n5{r3c7 r1c9} - r1n9{c9 .} ==> r8c3≠9
t-whip[5]: c3n3{r9 r3} - r3n8{c3 c2} - c2n6{r3 r9} - c2n2{r9 r1} - c2n1{r1 .} ==> r7c2≠3
whip[5]: c7n8{r6 r9} - c7n9{r9 r8} - r8c6{n9 n3} - r9c4{n3 n5} - c8n5{r9 .} ==> r6c7≠5
t-whip[3]: c8n5{r6 r9} - c7n5{r9 r3} - b3n3{r3c7 .} ==> r6c8≠3
biv-chain[3]: r6n3{c9 c7} - r3c7{n3 n5} - r1c9{n5 n9} ==> r6c9≠9
t-whip[5]: b1n5{r3c3 r3c1} - r3n6{c1 c2} - c2n8{r3 r6} - c7n8{r6 r9} - c7n5{r9 .} ==> r8c3≠5
biv-chain[3]: c5n2{r9 r8} - r8c3{n2 n3} - r8c6{n3 n9} ==> r9c5≠9
biv-chain[4]: c5n1{r7 r9} - b8n2{r9c5 r8c5} - r8c3{n2 n3} - r8c6{n3 n9} ==> r7c5≠9
whip[5]: c7n8{r6 r9} - c7n9{r9 r8} - b8n9{r8c6 r7c6} - r7n8{c6 c4} - r7n7{c4 .} ==> r6c7≠3
hidden-single-in-a-block ==> r6c9=3
z-chain[6]: r1n5{c9 c3} - b7n5{r9c3 r9c1} - r9n6{c1 c2} - r3c2{n6 n8} - r6n8{c2 c7} - b9n8{r9c7 .} ==> r7c9≠5
biv-chain[3]: b3n9{r2c8 r1c9} - c9n5{r1 r4} - r6c8{n5 n9} ==> r9c8≠9
biv-chain[4]: c8n3{r9 r2} - b3n9{r2c8 r1c9} - r7c9{n9 n8} - r9n8{c7 c4} ==> r9c4≠3
biv-chain[4]: c8n9{r2 r6} - r6n5{c8 c5} - r6n4{c5 c2} - b1n4{r2c2 r2c1} ==> r2c1≠9
biv-chain[4]: r7c9{n9 n8} - b6n8{r4c9 r6c7} - r6c2{n8 n4} - c3n4{r5 r7} ==> r7c3≠9
biv-chain[3]: c3n9{r9 r1} - r2n9{c2 c8} - c8n3{r2 r9} ==> r9c3≠3
t-whip[4]: r3c7{n3 n5} - r1c9{n5 n9} - c3n9{r1 r9} - b9n9{r9c7 .} ==> r8c7≠3
whip[1]: b9n3{r9c8 .} ==> r9c1≠3
biv-chain[4]: c7n3{r3 r9} - c7n8{r9 r6} - c2n8{r6 r3} - b1n6{r3c2 r3c1} ==> r3c1≠3
z-chain[5]: r7n5{c3 c5} - c5n1{r7 r9} - r9n2{c5 c2} - c3n2{r8 r4} - c3n1{r4 .} ==> r9c3≠5
z-chain[5]: c3n4{r7 r5} - c3n8{r5 r3} - c3n5{r3 r1} - b3n5{r1c9 r3c7} - r3n3{c7 .} ==> r7c3≠3
biv-chain[4]: r8c7{n9 n5} - r3c7{n5 n3} - c3n3{r3 r8} - r8n2{c3 c5} ==> r8c5≠9
whip[1]: b8n9{r8c6 .} ==> r4c6≠9, r5c6≠9
biv-chain[4]: r9n8{c4 c7} - c7n3{r9 r3} - c3n3{r3 r8} - r7n3{c1 c6} ==> r7c6≠8
whip[1]: b8n8{r9c4 .} ==> r5c4≠8
biv-chain[3]: r7n7{c6 c4} - r7n8{c4 c9} - r4n8{c9 c6} ==> r4c6≠7
biv-chain[4]: c6n8{r5 r4} - c9n8{r4 r7} - r7c4{n8 n7} - c6n7{r7 r5} ==> r5c6≠3
biv-chain[4]: r5c6{n7 n8} - c3n8{r5 r3} - c3n3{r3 r8} - r7n3{c1 c6} ==> r7c6≠7
stte

[yzfwsf]

No eleven replacement is used, but UR-like techniques are used, whip [15]

Hidden Text: Show

Locked Candidates 1 (Pointing): 6 in b3 => r2c1<>6,r2c2<>6
Locked Candidates 2 (Claiming): 6 in r8 => r9c4<>6,r9c5<>6
Naked Pair: in r5c3,r6c2 => r4c3<>8,r6c1<>4,
Naked Single: r6c1=7
Uniqueness Test 3: 12 in r14c13 => r1c2 <> 5
AIC Type 2: (5=4)r2c2 - r6c2 = (4-8)r5c3 = 8r3c3 => r3c3<>5
Grouped AIC Type 2: (5=4)r2c2 - (4=8)r6c2 - r6c79 = (8-6)r4c9 = 6r2c9 => r2c9<>5
Grouped AIC Type 2: 6r9c1 = (6-2)r9c2 = (2-1)r1c2 = 1r79c2 => r9c1<>1
Region Forcing Chain: Each 6 in r5 true in turn will all lead to: r2c8,r4c9<>6
6r5c4 - (6=34591)r14567c5 - (1=458)r267c2 - (8=35796)b6p25789
6r5c5 - (6=798)r5c468 - 8r4c46 = (8-6)r4c9 = 6r2c9
6r5c8
Hidden Single: 6 in r2 => r2c9=6
Hidden Pair: 67 in r4c8,r5c8 => r4c8<>35,r5c8<>9
Locked Candidates 2 (Claiming): 9 in r5 => r6c5<>9
Region Forcing Chain: Each 5 in r1 true in turn will all lead to: r4c9<>5
5r1c3 - (5=4)r2c2 - (4=3895)r6c2789
5r1c5 - 5r6c5 = 5r4c456
5r1c9
Locked Candidates 2 (Claiming): 5 in r4 => r6c5<>5
Grouped AIC Type 2: (3=8)r4c9 - r4c46 = r5c46 - (8=4)r5c3 - r8c3 = 4r8c9 => r8c9<>3
Region Forcing Chain: Each 3 in r1 true in turn will all lead to: r3c3<>3
3r1c1
3r1c3
3r1c5 - (3=4)r6c5 - (4=8)r6c2 - 8r3c2 = 8r3c3
3r1c9 - (3=8)r4c9 - 8r6c79 = 8r6c2 - 8r3c2 = 8r3c3
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r7c13<>1
1r7c5
8r9c4 - 8r9c7 = 8r6c7 - (8=451)r267c2
4r7c9 - 4r8c9 = 4r8c3 - 4r5c3 = 4r6c2 - (4=51)r27c2
8r7c9 - 8r9c7 = 8r6c7 - (8=451)r267c2
W-Wing: 12 in r1c2,r4c3 connected by 1c1 => r1c3<>2
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r1c1<>9
1r7c5 - (1=4586)r2367c2 - (6=349)r237c1
8r9c4 - 8r9c7 = 8r6c7 - 8r6c2 = 8r3c2 - (8=9)r3c3
4r7c9 - 4r8c9 = 4r8c3 - (4=8)r5c3 - (8=9)r3c3
8r7c9 - 8r9c7 = 8r6c7 - 8r6c2 = 8r3c2 - (8=9)r3c3
Region Forcing Chain: Each 5 in c6 true in turn will all lead to: r7c4<>5
5r2c6 - (5=4)r2c2 - (4=5893)r6c2789 - (3=8)r4c9 - (8=13495)r7c12359
(5-8)r4c6 = 8r4c9,r7c6 - (8=13495)r7c12359
5r7c6
5r8c6
UR Forcing Chain: Each true guardian of UR 79{r57c46} will all lead to: r7c4<>9
6r5c4 - (6=34591)r14567c5 - (1=458)r267c2 - 8r6c79 = 8r4c9 - (8=13459)r7c12359
8r5c4 - (8=1235697)b8p1245678
8r5c6 - 8r4c46 = 8r4c9 - (8=13459)r7c12359
3r7c4
8r7c4
(3-7)r7c6 = 7r7c4
(5-7)r7c6 = 7r7c4
(8-7)r7c6 = 7r7c4
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r7c5<>3
1r7c5
8r9c4 - 8r9c7 = 8r6c7 - (8=4)r6c2 - (4=3)r6c5
4r7c9 - 4r8c9 = 4r8c3 - 4r5c3 = 4r6c2 - (4=3)r6c5
8r7c9 - (8=3)r4c9 - 3r6c789 = 3r6c5
UR Forcing Chain: Each true guardian of UR 79{r57c46} will all lead to: r7c6<>9
6r5c4 - (6=34591)r14567c5 - (1=458)r267c2 - 8r6c79 = 8r4c9 - (8=13459)r7c12359
8r5c4 - 8r45c6 = 8r7c6
8r5c6 - 8r4c46 = 8r4c9 - (8=13459)r7c12359
(3-7)r7c4 = 7r7c6
(8-7)r7c4 = 7r7c6
3r7c6
5r7c6
8r7c6
UR Forcing Chain: Each true guardian of UR 25{r19c23} will all lead to: r9c2<>5
(1-2)r1c2 = 2r9c2
(1-5)r1c3 = 5r23c2
(3-5)r1c3 = 5r23c2
(9-5)r1c3 = 5r23c2
1r9c2
6r9c2
1r9c3 - (1=45682)r23679c2
3r9c3 - (3=895)r9c478
9r9c3 - (9=385)r9c478
Whip[11]: Supposing 3r7c4 would causes 4 to disappear in Box 1 => r7c4<>3
3r7c4 - 7r7(c4=c6) - 8r7(c6=c9) - 8r9(c7=c4) - 8r4(c4=c6) - 8r5(c6=c3) - r6c2(8=4) - r2c2(4=5) - 5c6(r2=r8) - 5r7(c5=c3) - 4r7(c3=c1) - 4b1(p4=.)
Whip[10]: Supposing 3r7c6 would causes 4 to disappear in Box 1 => r7c6<>3
3r7c6 - 7r7(c6=c4) - 8r7(c4=c9) - 8c7(r9=r6) - r6c2(8=4) - r2c2(4=5) - r2c6(5=9) - r8c6(9=5) - 5r7(c5=c3) - 4r7(c3=c1) - 4b1(p4=.)
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r9c1<>3
1r7c5 - (1=4586)r2367c2 - 6r9c2 = 6r9c1
8r9c4 - (8=145793)r7c123456
(4-3)r7c9 = 3r7c13
(8-3)r7c9 = 3r7c13
Whip[11]: Supposing 5r8c9 would causes 3 to disappear in Row 2 => r8c9<>5
5r8c9 - 4r8(c9=c3) - 4c1(r7=r2) - 4c2(r2=r6) - r6c5(4=3) - 3r4(c6=c9) - r1c9(3=9) - 9r2(c8=c6) - r8c6(9=3) - r8c7(3=9) - r9c8(9=3) - 3r2(c8=.)
Whip[14]: Supposing 4r7c9 would causes 2 to disappear in Row 8 => r7c9<>4
4r7c9 - 4r8(c9=c3) - r5c3(4=8) - 8r3(c3=c2) - r6c2(8=4) - r6c5(4=3) - 3r4(c6=c9) - 8c9(r4=r6) - 5c9(r6=r1) - 5r3(c7=c4) - 3b2(p7=p6) - 3c8(r2=r9) - 3r8(c7=c4) - 6r8(c4=c5) - 2r8(c5=.)
Hidden Single: 4 in c9 => r8c9=4
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r4c4<>8
1r7c5 - (1=458)r267c2 - 8r6c79 = 8r4c9
8r9c4
8r7c9 - 8r4c9,r7c6 = 8r4c6
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r6c9<>8
1r7c5 - (1=458)r267c2
8r9c4 - 8r9c7 = 8r6c7
8r7c9
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r7c5<>5
1r7c5
8r9c4 - (8=75)r7c46
8r7c9 - (8=75)r7c46
Region Forcing Chain: Each 9 in r8 true in turn will all lead to: r9c7<>9
9r8c3 - (9=8)r3c3 - 8r3c2 = 8r6c2 - 8r6c7 = 8r9c7
9r8c4 - (9=1)r7c5 - (1=458)r267c2 - 8r6c7 = 8r9c7
9r8c5 - (9=1)r7c5 - (1=458)r267c2 - 8r6c7 = 8r9c7
9r8c6 - (9=1)r7c5 - (1=458)r267c2 - 8r6c7 = 8r9c7
9r8c7
UR Forcing Chain: Each true guardian of UR 78{r57c46} will all lead to: r5c4<>7
6r5c4
9r5c4
(9-8)r5c6 = 8r7c69 - (8=7)r7c4
(5-7)r7c6 = 7r7c4
Whip[15]: Supposing 2r9c1 will result in all candidates in cell r9c8 being impossible => r9c1<>2
2r9c1 - 6c1(r9=r3) - 6c2(r3=r9) - 2c2(r9=r1) - 1c2(r1=r7) - r7c5(1=9) - 9c1(r7=r2) - 9b2(p6=p7) - 3r3(c4=c7) - r2c8(3=5) - r2c6(5=3) - r8c6(3=5) - r8c7(5=9) - r8c3(9=3) - 3r7(c1=c9) - r9c8(3=.)
Hidden Pair: 12 in r1c1,r4c1 => r1c1<>3
Locked Pair: in r1c1,r1c2 => r1c3<>1,r1c3<>1,
UL Type 3: {r1c12,r4c13,r9c23}(With Naked Quintuplet:35689) => r9c5<>359
W-Wing: 12 in r1c2,r9c5 connected by 1r7 => r9c2<>2
Hidden Single: 2 in c2 => r1c2=2
Hidden Single: 1 in r1 => r1c1=1
Hidden Single: 1 in r4 => r4c3=1
Hidden Single: 2 in r4 => r4c1=2
AIC Type 2: (9=6)r9c1 - (6=1)r9c2 - (1=2)r9c5 - r9c3 = 2r8c3 => r8c3<>9
Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r8c3<>3
1r7c5 - (1=2)r9c5 - 2r8c5 = 2r8c3
8r9c4 - (8=145793)r7c123456
(8-3)r7c9 = 3r7c13
Whip[14]: Supposing 5r3c2 would causes 1 to disappear in Box 8 => r3c2<>5
5r3c2 - r2c2(5=4) - 4c1(r2=r7) - r7c2(4=1) - r7c5(1=9) - 9r8(c6=c7) - r3c7(9=3) - 3c1(r3=r2) - r1c3(3=9) - r1c9(9=5) - 5r2(c8=c6) - 5r7(c6=c3) - r8c3(5=2) - 2r9(c3=c5) - 1b8(p8=.)
Region Forcing Chain: Each 5 in r9 true in turn will all lead to: r2c8<>5
5r9c3 - 5r7c2 = 5r2c2
5r9c4 - 5r3c4 = 5r3c7
(5-8)r9c7 = 8r7c9 - (8=75)r7c46 - 5r7c2 = 5r2c2
5r9c8
Cell Forcing Chain: Each candidate in r9c4 true in turn will all lead to: r8c7<>5
3r9c4 - 3r8c456 = 3r8c7
5r9c4 - 5r3c4 = 5r3c7
8r9c4 - 8r9c7 = 8r6c7 - (8=451)r267c2 - (1=345692)r145678c5 - (2=5)r8c3
9r9c4 - 9r8c456 = 9r8c7
Grouped AIC Type 2: 8r9c4 = r9c7 - r6c7 = r6c2 - (8=6)r3c2 - (6=1)r9c2 - (1=2)r9c5 - r9c3 = (2-5)r8c3 = 5r8c456 => r9c4<>5
Cell Forcing Chain: Each candidate in r1c5 true in turn will all lead to: r9c7<>5
3r1c5 - (3=4)r6c5 - (4=8)r6c2 - 8r6c7 = 8r9c7
5r1c5 - 5r3c4 = 5r3c7
9r1c5 - (9=1)r7c5 - (1=458)r267c2 - 8r6c7 = 8r9c7
Whip[9]: Supposing 3r2c1 would causes 3 to disappear in Box 8 => r2c1<>3
3r2c1 - r2c8(3=9) - r2c6(9=5) - 5c2(r2=r7) - 5c3(r9=r1) - 5c9(r1=r6) - 9c9(r6=r7) - r8c7(9=3) - 3r3(c7=c4) - 3b8(p7=.)
Cell Forcing Chain: Each candidate in r7c9 true in turn will all lead to: r3c7<>3
3r7c9 - 3r7c1 = 3r3c1
5r7c9 - 5r1c9 = 5r3c7
8r7c9 - (8=3)r9c7
9r7c9 - (9=3)r8c7
Region Forcing Chain: Each 3 in r9 true in turn will all lead to: r3c1<>9
3r9c3 - 3r7c1 = 3r3c1
3r9c4 - 3r3c4 = 3r3c1
(3-8)r9c7 = 8r6c7 - 8r6c2 = 8r3c2 - (8=9)r3c3
3r9c8 - 3r2c8 = 3r2c6 - 3r3c4 = 3r3c1
Cell Forcing Chain: Each candidate in r9c3 true in turn will all lead to: r1c9<>5
2r9c3 - (2=1)r9c5 - (1=6)r9c2 - (6=5893)r3c2347 - 5r3c4 = 5r3c7
3r9c3 - 3r7c13 = 3r7c9 - 3r89c7 = (3-5)r6c7 = 5r3c7
5r9c3 - 5r9c8 = 5r7c9
9r9c3 - (9=3685)b1p3789
Hidden Single: 5 in b3 => r3c7=5
W-Wing: 39 in r1c9,r3c4 connected by 3r2 => r1c5<>9
Grouped W-Wing: 39 in r3c4,r8c7 connected by 3b8 => r8c4<>9
XY-Wing: 369 in r3c1 r3c4 r9c1 => r9c4 <> 9
Naked Pair: in r9c4,r9c7 => r9c3<>3,r9c8<>3,
Locked Candidates 1 (Pointing): 3 in b7 => r7c9<>3
Locked Candidates 1 (Pointing): 3 in b9 => r6c7<>3
W-Wing: 38 in r4c9,r9c4 connected by 8c7 => r4c4<>3
XY-Wing: 389 in r4c9 r1c9 r6c7 => r6c9 <> 9
WXYZ-Wing: 3489 in r6c257,r8c7,Pivot Cell Is r6c7 => r8c5<>3
Sue de Coq: r13c3 - {3589} (r3c12 - {368}, r89c3 -{259}) => r7c3<>5 r7c3<>9
XY-Chain: (3=9)r8c7 - (9=8)r6c7 - (8=4)r6c2 - (4=8)r5c3 - (8=9)r3c3 - (9=3)r3c4 => r8c4<>3
Grouped Discontinuous Nice Loop: 5r1c5 = r1c3 - r9c3 = r9c8 - r6c8 = (5-3)r6c9 = r1c9,r6c5 - (3=5)r1c5 => r1c5=5
Hidden Single: 5 in r2 => r2c2=5
Hidden Single: 4 in r2 => r2c1=4
Locked Candidates 2 (Claiming): 9 in c1 => r9c3<>9
Locked Candidates 2 (Claiming): 3 in c5 => r4c6<>3
Hidden Triple: 578 in r7c4,r7c6,r7c9 => r7c9<>9
stte

[denis_berthier]

.
If I have to choose between eleven's technique and making the assumption of uniqueness, I have no hesitation.

The purpose f this puzzle was to show we don't get everything with the anti-tridagon techniques and what remains can be very hard.
Uniqueness always creates perturbations in assessing the difficulty of a puzzle.

[marek stefanik]

Code: Select all

.--------------------.---------------------.-------------------.
| 1239   12–5  12359 | 4     #359    6     | 7     8    #359   |
| 349   e45    7     | 1      8     #359   | 2   A#3569 B3569  |
| 369    68–5  3589  |#359    7      2     |#359   4     1     |
:--------------------+---------------------+-------------------:
| 12     9     12    | 35678  356    3578  | 4     3567 C3568  |
| 5      3    δ48    | 6789   469    789   | 1     679   2     |
| 7     d48    6     | 2      3459   1     |cD3589 359  D3589  |
:--------------------+---------------------+-------------------:
| 349–1 e145   3459–1| 35789Ω#1359   35789 | 6     2  aα#34589 |
| 8      7    γ23459 | 3569   23569 #359   |#359   1    β3459  |
| 12369  126–5 12359 |a#3589  12359  4     |b3589 #359   7     |
'--------------------'---------------------'-------------------'

TH kraken:

Code: Select all

8r7c9|8r9c4 – 8r9c7 = 8r6c7 – 8r6c2 ======\\
||                                         \\
6r2c8 – 6r2c9 = (6–8)r4c9 = 8r6c79 – 8r6c2 === (d)4r6c2 – (4=15)r27c2 – Loop
||                                         //
4r7c9 – 4r8c9 = 4r8c3 – 4r5c3 ============//
||
1r7c5
=> -5r139c2, -1r7c13, (1# –– 4|6|8#)

Code: Select all

.-------------------.---------------------.-------------------.
| 1239   12   12359 | 4    A#359    6     | 7     8   A#359   |
| 349    45   7     | 1      8     #359   | 2    #3569  3569  |
| 369    68   3589  |#359    7      2     |#359   4     1     |
:-------------------+---------------------+-------------------:
| 12     9    12    | 35678  356    3578  | 4     3567  3568  |
| 5      3    48    | 6789   469    789   | 1     679   2     |
| 7      48   6     | 2      3459   1     | 3589  359   3589  |
:-------------------+---------------------+-------------------:
| 349    145  3459  | 35789 #1359   35789 | 6     2   A#34589 |
| 8      7    23459 | 3569   23569 #359   |#359   1     3459  |
| 12369  126  12359 |#3589   12359  4     | 3589 #359   7     |
'-------------------'---------------------'-------------------'
4|6|8# = (TH[11], RT359A)
     \\= 1# – 1r7c2 = (12–6)r19c2 = (126–3|5|9)r9c12 = 359b7p12369
=> 4|6|8# == [359A && 359b7p12369]

1# –– 4|6|8# == [359b7A \ r17c3] – (3|5|9=12)r14c3 – (1|2)r789c3 = 126b7p278 – 1r7c5 => –1r7c5

5r14\c59b5 => –5r7c5

Code: Select all

.-------------.------------------.-----------------.
| 13   2  13  | 4     #59  6     | 7     8   #59   |
| 4    5  7   | 1      8  #39    | 2    #39   6    |
| 6    8  9   |#35     7   2     |#35    4    1    |
:-------------+------------------+-----------------:
| 12   9  12  | 58     6   58    | 4     7    3    |
| 5    3  8   | 79     4   79    | 1     6    2    |
| 7    4  6   | 2      3   1     | 589   59   589  |
:-------------+------------------+-----------------:
| 39   1  345 | 35789 #59  35789 | 6     2   #4589 |
| 8    7  345 | 6      2  #359   |#359   1    459  |
| 239  6  235 |#3589   1   4     | 3589 #359  7    |
'-------------'------------------'-----------------'

Notice that r1c59 r7c5 can no longer be a remote triple, therefore a TH guardian outside r7c9 is needed. -359r9c4, stte

Marek