I had missed this puzzle - a tough one.
Applying W12+OR5W12 leaves it in T&E(2).
The solution I'll propose here will use both eleven replacement and anti-tridagon rules.
it will rely on the following observation:
If an ORk-relation based on an anti-tridagon pattern with k guardians is found and eleven replacement is later applied in some of the 4 blocks to the 3 digits of the anti-tridagon, then the ORk-relation remains valid in the modified puzzle.
Proof: obvious. We only have to consider the 12 cells of the pattern. In any case, eleven replacement doesn't change anything to the digits not in the pattern and when applying it in some of the 4 blocks to the 3 digits of the anti-tridagon, it doesn't change anything to the anti-tridagon pattern (It may even re-add digits of the pattern that were eliminated before replacement, thus restoring a non-degenerated anti-tridagon - except of course the 3 that are fixed by replacement)
We have two OR3-anti-tridagons in the same blocks at the start:
- Code: Select all
Resolution state after Singles and whips[1]:
+----------------------+----------------------+----------------------+
! 23567 23479 23589 ! 2378 2389 236 ! 4569 45789 1 !
! 1567 1479 1589 ! 178 189 16 ! 4569 2 3 !
! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 !
+----------------------+----------------------+----------------------+
! 1235 1239 4 ! 6 7 123 ! 12359 1359 8 !
! 1237 6 1239 ! 123 5 8 ! 12349 13479 2479 !
! 8 1237 1235 ! 9 123 4 ! 1235 6 257 !
+----------------------+----------------------+----------------------+
! 123 8 6 ! 12345 123 9 ! 7 1345 245 !
! 4 5 123 ! 123 6 7 ! 8 139 29 !
! 9 123 7 ! 123458 1238 123 ! 123456 1345 2456 !
+----------------------+----------------------+----------------------+
209 candidates.
hidden-pairs-in-a-column: c4{n4 n5}{r7 r9} ==> r9c4≠8, r9c4≠3, r9c4≠2, r9c4≠1, r7c4≠3, r7c4≠2, r7c4≠1
hidden-single-in-a-block ==> r9c5=8
+----------------------+----------------------+----------------------+
! 23567 23479 23589 ! 2378 239 236 ! 4569 45789 1 !
! 1567 1479 1589 ! 178 19 16 ! 4569 2 3 !
! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 !
+----------------------+----------------------+----------------------+
! 1235 1239 4 ! 6 7 123 ! 12359 1359 8 !
! 1237 6 1239 ! 123 5 8 ! 12349 13479 2479 !
! 8 1237 1235 ! 9 123 4 ! 1235 6 257 !
+----------------------+----------------------+----------------------+
! 123 8 6 ! 45 123 9 ! 7 1345 245 !
! 4 5 123 ! 123 6 7 ! 8 139 29 !
! 9 123 7 ! 45 8 123 ! 123456 1345 2456 !
+----------------------+----------------------+----------------------+
OR3-anti-tridagon[12] for digits 1, 2 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: r9c6, r7c5, r8c4
with 3 guardians: n9r4c2 n7r5c1 n5r6c3
OR3-anti-tridagon[12] for digits 1, 2 and 3 in blocks:
b4, with cells: r4c1, r5c3, r6c2
b5, with cells: r4c6, r5c4, r6c5
b7, with cells: r7c1, r8c3, r9c2
b8, with cells: r7c5, r8c4, r9c6
with 3 guardians: n5r4c1 n9r5c3 n7r6c2
Based on the first anti-tridagon:
Trid-OR3-whip[7]: b3n5{r2c7 r1c8} - c8n8{r1 r3} - c8n7{r3 r5} - OR3{{n7r5c1 n5r6c3 | n9r4c2}} - c8n9{r4 r8} - r8c9{n9 n2} - r6c9{n2 .} ==> r6c7≠5
Trid-OR3-ctr-whip[8]: c7n2{r6 r9} - r9n6{c7 c9} - c9n4{r9 r7} - c9n5{r7 r6} - b6n7{r6c9 r5c8} - r5n4{c8 c7} - r5n9{c7 c3} - OR3{{n5r6c3 n7r5c1 n9r4c2 | .}} ==> r5c9≠2
- Code: Select all
***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE FOR GENERAL TRIDAGON in resolution state: *****
RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.
+----------------------+----------------------+----------------------+
! 23567 23479 23589 ! 2378 239 236 ! 4569 45789 1 !
! 1567 1479 1589 ! 178 19 16 ! 4569 2 3 !
! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 !
+----------------------+----------------------+----------------------+
! 1235 1239 4 ! 6 7 123 ! 12359 1359 8 !
! 1237 6 1239 ! 123 5 8 ! 12349 13479 479 !
! 8 1237 1235 ! 9 123 4 ! 123 6 257 !
+----------------------+----------------------+----------------------+
! 123 8 6 ! 45 123 9 ! 7 1345 245 !
! 4 5 123 ! 123 6 7 ! 8 139 29 !
! 9 123 7 ! 45 8 123 ! 123456 1345 2456 !
+----------------------+----------------------+----------------------+
Trying in block 8
+----------------------+----------------------+----------------------+
! 123567 123479 123589 ! 12378 1239 1236 ! 4569 45789 123 !
! 123567 123479 123589 ! 12378 1239 1236 ! 4569 123 123 !
! 12367 12379 12389 ! 12378 4 5 ! 69 789 679 !
+----------------------+----------------------+----------------------+
! 1235 1239 4 ! 6 7 123 ! 12359 12359 8 !
! 1237 6 1239 ! 123 5 8 ! 12349 123479 479 !
! 8 1237 1235 ! 9 123 4 ! 123 6 12357 !
+----------------------+----------------------+----------------------+
! 123 8 6 ! 45 3 9 ! 7 12345 12345 !
! 4 5 123 ! 2 6 7 ! 8 1239 1239 !
! 9 123 7 ! 45 8 1 ! 123456 12345 123456 !
+----------------------+----------------------+----------------------+
whip[1]: r3n2{c1 .} ==> r2c3≠2, r2c2≠2, r2c1≠2, r1c3≠2, r1c2≠2, r1c1≠2
whip[1]: c7n1{r4 .} ==> r4c8≠1, r5c8≠1, r6c9≠1
whip[7]: r9c2{n3 n2} - r7c1{n2 n1} - r4n1{c1 c7} - r4n9{c7 c8} - b4n9{r4c2 r5c3} - r5n1{c3 c4} - b5n3{r5c4 .} ==> r4c2≠3
whip[6]: b4n5{r6c3 r4c1} - b6n5{r4c7 r6c9} - r6n7{c9 c2} - r6n3{c2 c7} - r4n3{c8 c6} - b5n2{r4c6 .} ==> r6c3≠2
This is where the fun begins:
Based on the second anti-tridagon:
Trid-OR3-ctr-whip[6]: c3n2{r3 r5} - b4n9{r5c3 r4c2} - c2n1{r4 r6} - r6n7{c2 c9} - r6n5{c9 c3} - OR3{{n5r4c1 n9r5c3 n7r6c2 | .}} ==> r3c3≠1
Trid-OR3-whip[6]: r6n7{c2 c9} - r6n5{c9 c3} - OR3{{n5r4c1 n7r6c2 | n9r5c3}} - r4c2{n9 n2} - r4c6{n2 n3} - r4c1{n3 .} ==> r6c2≠1
whip[6]: r5c4{n3 n1} - r6c5{n1 n2} - r6c2{n2 n7} - r5c1{n7 n2} - b7n2{r7c1 r9c2} - b7n3{r9c2 .} ==> r5c3≠3
whip[6]: r4c6{n2 n3} - r5c4{n3 n1} - r5c3{n1 n9} - r4c2{n9 n1} - r3n1{c2 c1} - r7c1{n1 .} ==> r4c1≠2
z-chain[5]: r6n7{c2 c9} - r6n5{c9 c3} - r4c1{n5 n1} - b7n1{r7c1 r8c3} - b7n3{r8c3 .} ==> r6c2≠3
z-chain[3]: r6n3{c9 c3} - b7n3{r8c3 r9c2} - c7n3{r9 .} ==> r5c8≠3, r4c8≠3
Based on the fist anti-tridagon:
Trid-OR3-whip[5]: c9n5{r7 r6} - r6n7{c9 c2} - OR3{{n7r5c1 n5r6c3 | n9r4c2}} - r4c8{n9 n2} - c7n2{r4 .} ==> r9c7≠5
Trid-OR3-whip[6]: r6n3{c9 c3} - b4n5{r6c3 r4c1} - r4n1{c1 c2} - OR3{{n9r4c2 n5r6c3 | n7r5c1}} - b6n7{r5c8 r6c9} - r6n5{c9 .} ==> r4c7≠3
The rest is almost trivial:
- Code: Select all
whip[6]: b4n3{r4c1 r6c3} - b6n3{r6c9 r5c7} - r5c4{n3 n1} - r3n1{c4 c2} - b4n1{r4c2 r4c1} - b4n5{r4c1 .} ==> r3c1≠3
whip[5]: r6c2{n7 n2} - r6c5{n2 n1} - r5c4{n1 n3} - r3n3{c4 c3} - c3n2{r3 .} ==> r3c2≠7
whip[6]: c3n2{r5 r3} - c1n2{r3 r7} - r9c2{n2 n3} - r3n3{c2 c4} - b5n3{r5c4 r4c6} - b5n2{r4c6 .} ==> r6c2≠2
naked-single ==> r6c2=7
Note that the two anti-tridagons come again into play:
- 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: n5r6c3 n9r4c2
+-------------------+-------------------+-------------------+
! 13567 13479 13589 ! 1378 129 236 ! 4569 45789 123 !
! 13567 13479 13589 ! 1378 129 236 ! 4569 123 123 !
! 1267 1239 2389 ! 1378 4 5 ! 69 789 679 !
+-------------------+-------------------+-------------------+
! 135 129 4 ! 6 7 23 ! 1259 259 8 !
! 123 6 129 ! 13 5 8 ! 12349 2479 479 !
! 8 7 135 ! 9 12 4 ! 123 6 235 !
+-------------------+-------------------+-------------------+
! 12 8 6 ! 45 3 9 ! 7 1245 1245 !
! 4 5 13 ! 2 6 7 ! 8 139 139 !
! 9 23 7 ! 45 8 1 ! 2346 2345 23456 !
+-------------------+-------------------+-------------------+
At least one candidate of a previous Trid-OR3-relation has just been eliminated.
There remains a Trid-OR2-relation between candidates: n9r4c2 n5r6c3
+-------------------+-------------------+-------------------+
! 13567 13479 13589 ! 1378 129 236 ! 4569 45789 123 !
! 13567 13479 13589 ! 1378 129 236 ! 4569 123 123 !
! 1267 1239 2389 ! 1378 4 5 ! 69 789 679 !
+-------------------+-------------------+-------------------+
! 135 129 4 ! 6 7 23 ! 1259 259 8 !
! 123 6 129 ! 13 5 8 ! 12349 2479 479 !
! 8 7 135 ! 9 12 4 ! 123 6 235 !
+-------------------+-------------------+-------------------+
! 12 8 6 ! 45 3 9 ! 7 1245 1245 !
! 4 5 13 ! 2 6 7 ! 8 139 139 !
! 9 23 7 ! 45 8 1 ! 2346 2345 23456 !
+-------------------+-------------------+-------------------+
- Code: Select all
z-chain[4]: r3n1{c1 c4} - r5c4{n1 n3} - r5c1{n3 n2} - r7c1{n2 .} ==> r1c1≠1, r2c1≠1
t-whip[4]: c2n1{r1 r4} - b4n9{r4c2 r5c3} - b4n2{r5c3 r5c1} - r7c1{n2 .} ==> r3c1≠1
z-chain[4]: r3n1{c2 c4} - r3n3{c4 c3} - c3n2{r3 r5} - c3n9{r5 .} ==> r3c2≠9
z-chain[5]: b5n2{r4c6 r6c5} - b5n1{r6c5 r5c4} - r3n1{c4 c2} - c2n2{r3 r9} - c7n2{r9 .} ==> r4c8≠2
Trid-OR2-whip[2]: OR2{{n5r6c3 | n9r4c2}} - r4c8{n9 .} ==> r6c9≠5 (possibly not necessary)
- Code: Select all
hidden-single-in-a-row ==> r6c3=5
whip[1]: r6n3{c9 .} ==> r5c7≠3
whip[1]: b4n3{r4c1 .} ==> r1c1≠3, r2c1≠3
whip[1]: c9n5{r9 .} ==> r9c8≠5, r7c8≠5
naked-triplets-in-a-column: c1{r4 r5 r7}{n1 n3 n2} ==> r3c1≠2
naked-triplets-in-a-row: r3{c1 c7 c9}{n7 n6 n9} ==> r3c8≠9, r3c8≠7, r3c4≠7, r3c3≠9
naked-single ==> r3c8=8
whip[1]: r3n9{c9 .} ==> r2c7≠9, r1c8≠9, r1c7≠9
naked-pairs-in-a-column: c4{r3 r5}{n1 n3} ==> r2c4≠3, r2c4≠1, r1c4≠3, r1c4≠1
naked-triplets-in-a-column: c9{r1 r2 r6}{n2 n1 n3} ==> r9c9≠3, r9c9≠2, r8c9≠3, r8c9≠1, r7c9≠2, r7c9≠1
singles ==> r8c9=9, r3c7=9
whip[1]: b9n1{r8c8 .} ==> r2c8≠1
naked-pairs-in-a-row: r7{c4 c9}{n4 n5} ==> r7c8≠4
naked-triplets-in-a-column: c8{r2 r7 r8}{n3 n2 n1} ==> r9c8≠3, r9c8≠2, r5c8≠2
singles ==> r9c8=4, r7c9=5, r7c4=4, r9c9=6, r3c9=7, r1c8=5, r4c8=9, r5c8=7, r3c1=6, r1c1=7, r1c4=8, r2c4=7, r2c1=5, r5c9=4, r9c4=5, r2c3=8, r5c3=9, r3c3=2, r4c7=5
whip[1]: r4n1{c1 .} ==> r5c1≠1
naked-pairs-in-a-block: b1{r1c3 r3c2}{n1 n3} ==> r2c2≠3, r2c2≠1, r1c2≠3, r1c2≠1
finned-x-wing-in-columns: n3{c3 c8}{r8 r1} ==> r1c9≠3
whip[1]: b3n3{r2c9 .} ==> r2c6≠3
finned-x-wing-in-rows: n2{r5 r7}{c1 c7} ==> r9c7≠2
stte
