.
That's a real monster. I don't know if there is another T&E(3) pattern that'd simplify it, but if not, it requires gW15+OR4gW15 (maybe slightly less if we also allow Forcing-g-whips).
As of now, it's the hardest puzzle (in mith's 63137 collection) I've solved without uniqueness or replacement. Thanks for submitting it.
Notice that we start with an OR4 relation, which degenerates to an OR3 one and then to an OR2 one (using ORk ultra-persistency rules, see https://www.researchgate.net/publication/365186265_Augmented_User_Manual_for_CSP-Rules-V21). The 3 of them lead to ORk eliminations.
It's the first time I see all this in a puzzle.- Code: Select all
Resolution state after Singles and whips[1]:
+-------------------------+-------------------------+-------------------------+
! 15 1259 2359 ! 4 1259 6 ! 7 8 1239 !
! 14567 12569 2345679 ! 1259 8 1279 ! 129 1239 123469 !
! 8 1269 24679 ! 129 3 1279 ! 5 129 12469 !
+-------------------------+-------------------------+-------------------------+
! 2 56 1 ! 35689 579 389 ! 4 3579 379 !
! 3 7 568 ! 12569 1259 4 ! 1289 1259 129 !
! 9 4 58 ! 1235 1257 123 ! 128 6 1237 !
+-------------------------+-------------------------+-------------------------+
! 145 1259 2459 ! 12389 129 12389 ! 6 1279 1279 !
! 16 8 269 ! 7 4 129 ! 3 129 5 !
! 17 3 279 ! 129 6 5 ! 129 4 8 !
+-------------------------+-------------------------+-------------------------+
197 candidates.
- Code: Select all
hidden-pairs-in-a-row: r7{n3 n8}{c4 c6} ==> r7c6≠9, r7c6≠2, r7c6≠1, r7c4≠9, r7c4≠2, r7c4≠1
hidden-pairs-in-a-column: c9{n4 n6}{r2 r3} ==> r3c9≠9, r3c9≠2, r3c9≠1, r2c9≠9, r2c9≠3, r2c9≠2, r2c9≠1
152 g-candidates, 834 csp-glinks and 471 non-csp glinks
+-------------------------+-------------------------+-------------------------+
! 15 1259 2359 ! 4 1259 6 ! 7 8 1239 !
! 14567 12569 2345679 ! 1259 8 1279 ! 129 1239 46 !
! 8 1269 24679 ! 129 3 1279 ! 5 129 46 !
+-------------------------+-------------------------+-------------------------+
! 2 56 1 ! 35689 579 389 ! 4 3579 379 !
! 3 7 568 ! 12569 1259 4 ! 1289 1259 129 !
! 9 4 58 ! 1235 1257 123 ! 128 6 1237 !
+-------------------------+-------------------------+-------------------------+
! 145 1259 2459 ! 38 129 38 ! 6 1279 1279 !
! 16 8 269 ! 7 4 129 ! 3 129 5 !
! 17 3 279 ! 129 6 5 ! 129 4 8 !
+-------------------------+-------------------------+-------------------------+
OR4-anti-tridagon[12] for digits 1, 2 and 9 in blocks:
b2, with cells: r1c5, r2c6, r3c4
b3, with cells: r1c9, r2c7, r3c8
b8, with cells: r7c5, r8c6, r9c4
b9, with cells: r7c9, r8c8, r9c7
with 4 guardians: n5r1c5 n3r1c9 n7r2c6 n7r7c9
biv-chain[3]: r5n6{c4 c3} - r4c2{n6 n5} - b6n5{r4c8 r5c8} ==> r5c4≠5
t-whip[5]: b4n5{r6c3 r4c2} - r4n6{c2 c4} - c4n8{r4 r7} - c4n3{r7 r6} - c4n5{r6 .} ==> r2c3≠5
whip[7]: r1c1{n5 n1} - c2n1{r3 r7} - c2n5{r7 r4} - r4n6{c2 c4} - c4n5{r4 r6} - c4n3{r6 r7} - c4n8{r7 .} ==> r2c1≠5
whip[8]: r1c1{n5 n1} - c2n1{r3 r7} - c2n5{r7 r4} - c8n5{r4 r5} - c5n5{r5 r6} - c5n1{r6 r5} - c9n1{r5 r6} - r6n7{c9 .} ==> r1c3≠5
Nothing unusual until now. Not even a large number of guardians.
This is where we take off (hardest steps in blue):
g-whip[9]: c2n2{r3 r7} - c2n1{r7 r123} - r1c1{n1 n5} - c2n5{r2 r4} - c8n5{r4 r5} - c5n5{r5 r6} - r6n7{c5 c9} - c9n2{r6 r5} - c5n2{r5 .} ==> r1c3≠2
whip[13]: c6n7{r3 r2} - c1n7{r2 r9} - c3n7{r9 r3} - r3n4{c3 c9} - r3n6{c9 c2} - b4n6{r4c2 r5c3} - c4n6{r5 r4} - r4n8{c4 c6} - c6n9{r4 r8} - r8c3{n9 n2} - r8c8{n2 n1} - r3n1{c8 c4} - r9n1{c4 .} ==> r3c6≠2
Trid-OR4-whip[13]: r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 n5} - r2n5{c2 c4} - r6n5{c4 c5} - c3n5{r6 r7} - c3n4{r7 r2} - b1n7{r2c3 r2c1} - c1n6{r2 r8} - r8c3{n6 n9} - r1c3{n9 n3} - OR4{{n3r1c9 n5r1c5 n7r2c6 | n7r7c9}} - r6n7{c9 .} ==> r3c3≠2
Trid-OR4-whip[13]: r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 n5} - c3n5{r6 r7} - c3n4{r7 r2} - b1n7{r2c3 r2c1} - c1n4{r2 r7} - c1n5{r7 r1} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - OR4{{n7r7c9 n5r1c5 n7r2c6 | n3r1c9}} - r1c3{n3 .} ==> r3c3≠9
g-whip[13]: b3n3{r2c8 r1c9} - r1c3{n3 n9} - b7n9{r7c3 r7c2} - c2n1{r7 r123} - r1c1{n1 n5} - c2n5{r1 r4} - b4n6{r4c2 r5c3} - r8c3{n6 n2} - r8c8{n2 n9} - r9n9{c7 c4} - r3n9{c4 c6} - r3n7{c6 c3} - r9c3{n7 .} ==> r2c8≠1Trid-OR4-gwhip[13]: b7n1{r7c2 r789c1} - r1c1{n1 n5} - r7n5{c1 c3} - c3n2{r7 r2} - c3n4{r2 r3} - b1n7{r3c3 r2c1} - r9n7{c1 c3} - b7n9{r9c3 r8c3} - r1c3{n9 n3} - OR4{{n3r1c9 n5r1c5 n7r2c6 | n7r7c9}} - r6n7{c9 c5} - r6n5{c5 c4} - r2n5{c4 .} ==> r7c2≠2whip[1]: b7n2{r9c3 .} ==> r2c3≠2
Trid-OR4-gwhip[13]: c2n9{r3 r7} - c2n1{r7 r123} - r1c1{n1 n5} - r7n5{c1 c3} - c3n4{r7 r3} - b1n7{r3c3 r2c1} - r2n4{c1 c9} - r2n6{c9 c2} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - OR4{{n7r7c9 n5r1c5 n7r2c6 | n3r1c9}} - r1c3{n3 .} ==> r2c3≠9- Code: Select all
+-------------------+-------------------+-------------------+
! 15 1259 39 ! 4 1259 6 ! 7 8 1239 !
! 1467 12569 3467 ! 1259 8 1279 ! 129 239 46 !
! 8 1269 467 ! 129 3 179 ! 5 129 46 !
+-------------------+-------------------+-------------------+
! 2 56 1 ! 35689 579 389 ! 4 3579 379 !
! 3 7 568 ! 1269 1259 4 ! 1289 1259 129 !
! 9 4 58 ! 1235 1257 123 ! 128 6 1237 !
+-------------------+-------------------+-------------------+
! 145 159 2459 ! 38 129 38 ! 6 1279 1279 !
! 16 8 269 ! 7 4 129 ! 3 129 5 !
! 17 3 279 ! 129 6 5 ! 129 4 8 !
+-------------------+-------------------+-------------------+
whip[14]: r5n6{c4 c3} - c3n8{r5 r6} - b4n5{r6c3 r4c2} - r4n6{c2 c4} - c4n8{r4 r7} - c4n3{r7 r6} - r6c6{n3 n1} - r6c7{n1 n2} - r9n2{c7 c3} - r8c3{n2 n9} - r8c6{n9 n2} - c5n2{r7 r1} - c5n1{r1 r7} - r7c2{n1 .} ==> r5c4≠2
whip[15]: c6n7{r3 r2} - c1n7{r2 r9} - c3n7{r9 r3} - r3n4{c3 c9} - r3n6{c9 c2} - b4n6{r4c2 r5c3} - r8n6{c3 c1} - r8n1{c1 c8} - r9n1{c7 c4} - r5c4{n1 n9} - c6n9{r4 r8} - r8n2{c6 c3} - r9n2{c3 c7} - c7n9{r9 r2} - r3n9{c8 .} ==> r3c6≠1whip[12]: r2n3{c8 c3} - r1c3{n3 n9} - r8n9{c3 c6} - b2n9{r2c6 r3c4} - r3c6{n9 n7} - c3n7{r3 r9} - r9c1{n7 n1} - b8n1{r9c4 r7c5} - b9n1{r7c8 r8c8} - r3c8{n1 n2} - r2c7{n2 n1} - b2n1{r2c4 .} ==> r2c8≠9
Trid-OR4-whip[14]: c4n6{r5 r4} - r4c2{n6 n5} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - r6n1{c9 c7} - r9n1{c7 c1} - c1n7{r9 r2} - OR4{{n7r2c6 n5r1c5 n7r7c9 | n3r1c9}} - r2c8{n3 n2} - r2c7{n2 n9} - r3c8{n9 n1} - r8n1{c8 c6} - r2c6{n1 .} ==> r5c4≠1g-whip[7]: c3n9{r9 r1} - c5n9{r1 r456} - r5c4{n9 n6} - b4n6{r5c3 r4c2} - c2n5{r4 r123} - r1c1{n5 n1} - b7n1{r7c1 .} ==> r7c2≠9
whip[1]: b7n9{r9c3 .} ==> r1c3≠9
naked-single ==> r1c3=3
- Code: Select all
At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n5r1c5 n7r2c6 n7r7c9
+-------------------+-------------------+-------------------+
! 15 1259 3 ! 4 1259 6 ! 7 8 129 !
! 1467 12569 467 ! 1259 8 1279 ! 129 23 46 !
! 8 1269 467 ! 129 3 79 ! 5 129 46 !
+-------------------+-------------------+-------------------+
! 2 56 1 ! 35689 579 389 ! 4 3579 379 !
! 3 7 568 ! 69 1259 4 ! 1289 1259 129 !
! 9 4 58 ! 1235 1257 123 ! 128 6 1237 !
+-------------------+-------------------+-------------------+
! 145 15 2459 ! 38 129 38 ! 6 1279 1279 !
! 16 8 269 ! 7 4 129 ! 3 129 5 !
! 17 3 279 ! 129 6 5 ! 129 4 8 !
+-------------------+-------------------+-------------------+
hidden-single-in-a-block ==> r2c8=3
Trid-OR3-whip[7]: r4c2{n6 n5} - r2n5{c2 c4} - r6n5{c4 c5} - r6n7{c5 c9} - OR3{{n7r7c9 n5r1c5 | n7r2c6}} - r2n2{c6 c7} - r2n9{c7 .} ==> r2c2≠6
Trid-OR3-whip[7]: c8n5{r5 r4} - c8n7{r4 r7} - OR3{{n7r7c9 n5r1c5 | n7r2c6}} - b1n7{r2c1 r3c3} - r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 .} ==> r5c5≠5whip[7]: c5n7{r6 r4} - c5n5{r4 r1} - r1c1{n5 n1} - b7n1{r7c1 r7c2} - c5n1{r7 r5} - c9n1{r5 r6} - r6n7{c9 .} ==> r6c5≠2
Trid-OR3-whip[7]: r7c2{n1 n5} - b1n5{r1c2 r1c1} - OR3{{n5r1c5 n7r7c9 | n7r2c6}} - b1n7{r2c1 r3c3} - r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 .} ==> r7c9≠1
Trid-OR3-whip[9]: r5n5{c8 c3} - c3n8{r5 r6} - r6c7{n8 n1} - c9n1{r6 r1} - c9n2{r1 r7} - c5n2{r7 r1} - OR3{{n5r1c5 n7r7c9 | n7r2c6}} - r3c6{n7 n9} - r3c8{n9 .} ==> r5c8≠2
Trid-OR3-whip[10]: c8n5{r5 r4} - c8n7{r4 r7} - b9n1{r7c8 r9c7} - b3n1{r2c7 r1c9} - r1c1{n1 n5} - OR3{{n5r1c5 n7r7c9 | n7r2c6}} - b1n7{r2c1 r3c3} - r3n4{c3 c9} - r3n6{c9 c2} - r4c2{n6 .} ==> r5c8≠1
Trid-OR3-ctr-whip[12]: r6n8{c7 c3} - r5n8{c3 c7} - r5n1{c7 c5} - c9n1{r5 r1} - r1c1{n1 n5} - b2n5{r1c5 r2c4} - r6n5{c4 c5} - c5n7{r6 r4} - c8n7{r4 r7} - c8n1{r7 r8} - c6n1{r8 r2} - OR3{{n5r1c5 n7r2c6 n7r7c9 | .}} ==> r6c7≠1whip[11]: r1c1{n1 n5} - r7c1{n5 n4} - b7n1{r7c1 r7c2} - r7n5{c2 c3} - r6c3{n5 n8} - r6c7{n8 n2} - r2c7{n2 n9} - r9c7{n9 n1} - b8n1{r9c4 r8c6} - c6n2{r8 r2} - r2c2{n2 .} ==> r2c1≠1
naked-triplets-in-a-block: b1{r2c1 r2c3 r3c3}{n4 n6 n7} ==> r3c2≠6
singles ==> r4c2=6, r5c4=6
whip[1]: b4n5{r6c3 .} ==> r7c3≠5
hidden-pairs-in-a-row: r3{n4 n6}{c3 c9} ==> r3c3≠7
hidden-single-in-a-row ==> r3c6=7
- Code: Select all
At least one candidate of a previous Trid-OR3-relation has just been eliminated.
There remains a Trid-OR2-relation between candidates: n5r1c5 n7r7c9
+----------------+----------------+----------------+
! 15 1259 3 ! 4 1259 6 ! 7 8 129 !
! 467 1259 467 ! 1259 8 129 ! 129 3 46 !
! 8 129 46 ! 129 3 7 ! 5 129 46 !
+----------------+----------------+----------------+
! 2 6 1 ! 3589 579 389 ! 4 579 379 !
! 3 7 58 ! 6 129 4 ! 1289 59 129 !
! 9 4 58 ! 1235 157 123 ! 28 6 1237 !
+----------------+----------------+----------------+
! 145 15 249 ! 38 129 38 ! 6 1279 279 !
! 16 8 269 ! 7 4 129 ! 3 129 5 !
! 17 3 279 ! 129 6 5 ! 129 4 8 !
+----------------+----------------+----------------+
Trid-OR2-whip[2]: OR2{{n5r1c5 | n7r7c9}} - r6n7{c9 .} ==> r6c5≠5
Trid-OR2-whip[9]: OR2{{n7r7c9 | n5r1c5}} - r1c1{n5 n1} - c2n1{r3 r7} - r7c5{n1 n9} - r7c8{n9 n7} - c8n2{r7 r3} - b3n1{r3c8 r2c7} - b2n1{r2c4 r3c4} - r9n1{c4 .} ==> r7c9≠2
Trid-OR2-whip[9]: r5c3{n8 n5} - c8n5{r5 r4} - c8n7{r4 r7} - OR2{{n7r7c9 | n5r1c5}} - c1n5{r1 r7} - b7n4{r7c1 r7c3} - r7n2{c3 c5} - r5n2{c5 c9} - r6c7{n2 .} ==> r5c7≠8singles ==> r6c7=8, r6c3=5, r5c3=8, r5c8=5
Trid-OR2-whip[3]: OR2{{n7r7c9 | n5r1c5}} - r4c5{n5 n9} - r4c8{n9 .} ==> r4c9≠7
Trid-OR2-whip[9]: b6n7{r4c8 r6c9} - r7c9{n7 n9} - OR2{{n7r7c9 | n5r1c5}} - r1c1{n5 n1} - r1c9{n1 n2} - b3n9{r1c9 r2c7} - r3c8{n9 n1} - r8n1{c8 c6} - c6n9{r8 .} ==> r4c8≠9
singles ==> r4c8=7, r6c5=7, r7c9=7
whip[8]: b2n5{r2c4 r1c5} - r1c1{n5 n1} - c2n1{r3 r7} - c5n1{r7 r5} - c5n2{r5 r7} - r9c4{n2 n1} - r3n1{c4 c8} - b9n1{r7c8 .} ==> r2c4≠9
Resolution state RS3:
- Code: Select all
+----------------+----------------+----------------+
! 15 1259 3 ! 4 1259 6 ! 7 8 129 !
! 467 1259 467 ! 125 8 129 ! 129 3 46 !
! 8 129 46 ! 129 3 7 ! 5 129 46 !
+----------------+----------------+----------------+
! 2 6 1 ! 3589 59 389 ! 4 7 39 !
! 3 7 8 ! 6 129 4 ! 129 5 129 !
! 9 4 5 ! 123 7 123 ! 8 6 123 !
+----------------+----------------+----------------+
! 145 15 249 ! 38 129 38 ! 6 129 7 !
! 16 8 269 ! 7 4 129 ! 3 129 5 !
! 17 3 279 ! 129 6 5 ! 129 4 8 !
+----------------+----------------+----------------+
Still in T&E(2)
[Edit: simplified printing for splitting rule]
[Edit 2: corrected the end]