.
Here's my solution, using tridagons + eleven's pattern #97-15c
We shall find ten short tridagon ORk-whips + one EL97 ORk-whip, for a solution in
W6+OR3W4.
- Code: Select all
hidden-pairs-in-a-row: r2{n4 n5}{c1 c2} ==> r2c2≠9, r2c2≠6, r2c2≠1, r2c1≠9, r2c1≠7, r2c1≠6, r2c1≠1
singles ==> r3c1=7, r3c3=8
+----------------------+----------------------+----------------------+
! 169 2 3 ! 4 5 169 ! 7 8 169 !
! 45 45 169 ! 1789 1678 1679 ! 169 2 3 !
! 7 169 8 ! 2 3 169 ! 5 169 4 !
+----------------------+----------------------+----------------------+
! 1269 169 5 ! 179 12467 3 ! 8 14679 1679 !
! 3 8 169 ! 1579 1467 15679 ! 1469 14679 2 !
! 1269 7 4 ! 189 1268 1269 ! 3 169 5 !
+----------------------+----------------------+----------------------+
! 14569 14569 1269 ! 3 127 8 ! 12469 145679 1679 !
! 1458 145 12 ! 6 9 1257 ! 124 3 178 !
! 15689 3 7 ! 15 12 4 ! 1269 1569 1689 !
+----------------------+----------------------+----------------------+
OR3-anti-tridagon[12] for digits 6, 9 and 1 in blocks:
b1, with cells: r1c1, r2c3, r3c2
b3, with cells: r1c9, r2c7, r3c8
b4, with cells: r6c1, r5c3, r4c2
b6, with cells: r6c8, r5c7, r4c9
with 3 guardians: n7r4c9 n4r5c7 n2r6c1
At this point there are several ORk-anti-eleven#97[15] relations, including one with 10 guardians, but only the following one will be useful:
- Code: Select all
OR7-anti-eleven#97[15] for digits 6, 9 and 1
free cells: r8c4 r8c5 r8c6
3-cell blocks:
b3, with cells: r3c8, r2c7, r1c9
b4, with cells: r4c2, r5c3, r6c1
b1, with cells: r3c2, r2c3, r1c1
2-cell blocks:
b9, with cells: r9c7, r7c9
b6, with cells: r6c8, r4c9
b7, with cells: r7c3, r9c1
with 7 guardians: n7r4c9 n2r6c1 n2r7c3 n7r7c9 n5r9c1 n8r9c1 n2r9c7
A quite standard mix of easy eliminations and reductions of the numbers of guardians:
- Code: Select all
biv-chain[3]: r8n7{c9 c6} - b8n5{r8c6 r9c4} - b9n5{r9c8 r7c8} ==> r7c8≠7
whip[1]: b9n7{r8c9 .} ==> r4c9≠7
+-------------------+-------------------+-------------------+
! 169 2 3 ! 4 5 169 ! 7 8 169 !
! 45 45 169 ! 1789 1678 1679 ! 169 2 3 !
! 7 169 8 ! 2 3 169 ! 5 169 4 !
+-------------------+-------------------+-------------------+
! 1269 169 5 ! 179 12467 3 ! 8 14679 169 !
! 3 8 169 ! 1579 1467 15679 ! 1469 14679 2 !
! 1269 7 4 ! 189 1268 1269 ! 3 169 5 !
+-------------------+-------------------+-------------------+
! 14569 14569 1269 ! 3 127 8 ! 12469 14569 1679 !
! 1458 145 12 ! 6 9 1257 ! 124 3 178 !
! 15689 3 7 ! 15 12 4 ! 1269 1569 1689 !
+-------------------+-------------------+-------------------+
At least one candidate of a previous Trid-OR3-relation between candidates n7r4c9 n4r5c7 n2r6c1 has just been eliminated.
There remains a Trid-OR2-relation between candidates: n4r5c7 n2r6c1
At least one candidate of a previous El97-OR7-relation between candidates n7r4c9 n2r6c1 n2r7c3 n7r7c9 n5r9c1 n8r9c1 n2r9c7 has just been eliminated.
There remains an El97-OR6-relation between candidates: n2r6c1 n2r7c3 n7r7c9 n5r9c1 n8r9c1 n2r9c7
Trid-OR2-whip[3]: OR2{{n4r5c7 | n2r6c1}} - b5n2{r6c5 r4c5} - r4n4{c5 .} ==> r5c8≠4
Trid-OR2-whip[3]: r4n2{c5 c1} - OR2{{n2r6c1 | n4r5c7}} - c5n4{r5 .} ==> r4c5≠1
Trid-OR2-whip[3]: r4n2{c5 c1} - OR2{{n2r6c1 | n4r5c7}} - c5n4{r5 .} ==> r4c5≠6
Trid-OR2-whip[3]: r4n2{c5 c1} - OR2{{n2r6c1 | n4r5c7}} - c5n4{r5 .} ==> r4c5≠7biv-chain[4]: c8n7{r5 r4} - c8n4{r4 r7} - b9n5{r7c8 r9c8} - c4n5{r9 r5} ==> r5c4≠7
biv-chain[4]: r9c4{n1 n5} - c8n5{r9 r7} - c8n4{r7 r4} - r4n7{c8 c4} ==> r4c4≠1
biv-chain[4]: r9c5{n1 n2} - r4c5{n2 n4} - c8n4{r4 r7} - b9n5{r7c8 r9c8} ==> r9c8≠1
Trid-OR2-whip[4]: c6n2{r8 r6} - OR2{{n2r6c1 | n4r5c7}} - r8c7{n4 n2} - r8c3{n2 .} ==> r8c6≠1
Trid-OR2-whip[4]: OR2{{n4r5c7 | n2r6c1}} - c6n2{r6 r8} - r8c3{n2 n1} - r8c7{n1 .} ==> r7c7≠4
Trid-OR2-whip[4]: r8c3{n1 n2} - r8c7{n2 n4} - OR2{{n4r5c7 | n2r6c1}} - c6n2{r6 .} ==> r8c2≠1- Code: Select all
naked-pairs-in-a-column: c2{r2 r8}{n4 n5} ==> r7c2≠5, r7c2≠4
hidden-pairs-in-a-row: r7{n4 n5}{c1 c8} ==> r7c8≠9, r7c8≠6, r7c8≠1, r7c1≠9, r7c1≠6, r7c1≠1
naked-pairs-in-a-block: b7{r7c1 r8c2}{n4 n5} ==> r9c1≠5, r8c1≠5, r8c1≠4
+-------------------+-------------------+-------------------+
! 169 2 3 ! 4 5 169 ! 7 8 169 !
! 45 45 169 ! 1789 1678 1679 ! 169 2 3 !
! 7 169 8 ! 2 3 169 ! 5 169 4 !
+-------------------+-------------------+-------------------+
! 1269 169 5 ! 79 24 3 ! 8 14679 169 !
! 3 8 169 ! 159 1467 15679 ! 1469 1679 2 !
! 1269 7 4 ! 189 1268 1269 ! 3 169 5 !
+-------------------+-------------------+-------------------+
! 45 169 1269 ! 3 127 8 ! 1269 45 1679 !
! 18 45 12 ! 6 9 257 ! 124 3 178 !
! 1689 3 7 ! 15 12 4 ! 1269 569 1689 !
+-------------------+-------------------+-------------------+
At least one candidate of a previous El97-OR6-relation between candidates n2r6c1 n2r7c3 n7r7c9 n5r9c1 n8r9c1 n2r9c7 has just been eliminated.
There remains an El97-OR5-relation between candidates: n2r6c1 n2r7c3 n7r7c9 n8r9c1 n2r9c7
biv-chain[4]: b8n7{r7c5 r8c6} - r8n5{c6 c2} - r8n4{c2 c7} - r5n4{c7 c5} ==> r5c5≠7
Trid-OR2-whip[4]: OR2{{n2r6c1 | n4r5c7}} - r8n4{c7 c2} - r8n5{c2 c6} - c6n2{r8 .} ==> r6c5≠2
Trid-OR2-whip[4]: c6n2{r8 r6} - OR2{{n2r6c1 | n4r5c7}} - r8n4{c7 c2} - r8n5{c2 .} ==> r8c6≠7- Code: Select all
singles ==> r7c5=7, r8c9=7, r9c9=8, r8c1=8
+-------------------+-------------------+-------------------+
! 169 2 3 ! 4 5 169 ! 7 8 169 !
! 45 45 169 ! 1789 168 1679 ! 169 2 3 !
! 7 169 8 ! 2 3 169 ! 5 169 4 !
+-------------------+-------------------+-------------------+
! 1269 169 5 ! 79 24 3 ! 8 14679 169 !
! 3 8 169 ! 159 146 15679 ! 1469 1679 2 !
! 1269 7 4 ! 189 168 1269 ! 3 169 5 !
+-------------------+-------------------+-------------------+
! 45 169 1269 ! 3 7 8 ! 1269 45 169 !
! 8 45 12 ! 6 9 25 ! 124 3 7 !
! 169 3 7 ! 15 12 4 ! 1269 569 8 !
+-------------------+-------------------+-------------------+
At least one candidate of a previous El97-OR5-relation between candidates n2r6c1 n2r7c3 n7r7c9 n8r9c1 n2r9c7 has just been eliminated.
There remains an El97-OR3-relation between candidates: n2r6c1 n2r7c3 n2r9c7
whip[1]: b8n1{r9c5 .} ==> r9c1≠1, r9c7≠1
+-------------------+-------------------+-------------------+
! 169 2 3 ! 4 5 169 ! 7 8 169 !
! 45 45 169 ! 1789 168 1679 ! 169 2 3 !
! 7 169 8 ! 2 3 169 ! 5 169 4 !
+-------------------+-------------------+-------------------+
! 1269 169 5 ! 79 24 3 ! 8 14679 169 !
! 3 8 169 ! 159 146 15679 ! 1469 1679 2 !
! 1269 7 4 ! 189 168 1269 ! 3 169 5 !
+-------------------+-------------------+-------------------+
! 45 169 1269 ! 3 7 8 ! 1269 45 169 !
! 8 45 12 ! 6 9 25 ! 124 3 7 !
! 69 3 7 ! 15 12 4 ! 269 569 8 !
+-------------------+-------------------+-------------------+
This resolution state is in T&E(1) but it requires long chains. Using the following pattern will make it simpler.
Eleven's #97-15c pattern has not yet been used, but at this point it has already degenerated. Remember the cells in the pattern: two 123-candidates are missing: n1r9c7 n1r9c1
- Code: Select all
3-cell blocks:
b3, with cells: r3c8, r2c7, r1c9
b4, with cells: r4c2, r5c3, r6c1
b1, with cells: r3c2, r2c3, r1c1
2-cell blocks:
b9, with cells: r9c7, r7c9
b6, with cells: r6c8, r4c9
b7, with cells: r7c3, r9c1
This pattern wouldn't be found if it hadn't been detected earlier in the resolution path.
What this example shows is also very unfortunate if one wished to delay the detection of the pattern.
Here, it is destroyed by a whip[1]. Which means that, like any resolution rule that doesn't belong to a family with the conflence property, there's no natural place to put it in the hierarchy of rules complexity that can ensure it will not be destroyed by simpler rules.
El97-OR3-whip[3]: OR3{{n2r9c7 n2r7c3 | n2r6c1}} - c6n2{r6 r8} - c3n2{r8 .} ==> r7c7≠2- Code: Select all
singles ==> r7c3=2, r8c3=1
finned-x-wing-in-columns: n1{c2 c8}{r3 r4} ==> r4c9≠1
finned-x-wing-in-columns: n6{c3 c5}{r2 r5} ==> r5c6≠6
finned-x-wing-in-columns: n9{c3 c4}{r2 r5} ==> r5c6≠9
biv-chain[3]: r4c9{n6 n9} - r4c4{n9 n7} - c8n7{r4 r5} ==> r5c8≠6
t-whip[6]: c6n7{r2 r5} - r4c4{n7 n9} - r4c9{n9 n6} - r4c2{n6 n1} - c1n1{r6 r1} - r1n6{c1 .} ==> r2c6≠6
whip[6]: c4n8{r6 r2} - r2n7{c4 c6} - r5n7{c6 c8} - r5n1{c8 c7} - r2n1{c7 c5} - r9n1{c5 .} ==> r6c4≠1
whip[6]: b7n9{r9c1 r7c2} - c9n9{r7 r1} - r4c9{n9 n6} - c2n6{r4 r3} - b3n6{r3c8 r2c7} - r7n6{c7 .} ==> r4c1≠9
whip[6]: c2n1{r3 r4} - b6n1{r4c8 r5c7} - r5n4{c7 c5} - r4c5{n4 n2} - r4c1{n2 n6} - r5n6{c3 .} ==> r3c8≠1
whip[1]: c8n1{r6 .} ==> r5c7≠1
z-chain[4]: r5n1{c6 c8} - r5n7{c8 c6} - c6n5{r5 r8} - c6n2{r8 .} ==> r6c6≠1
z-chain[3]: c6n1{r3 r5} - c6n5{r5 r8} - r9c4{n5 .} ==> r2c4≠1
hidden-pairs-in-a-column: c4{n1 n5}{r5 r9} ==> r5c4≠9
t-whip[5]: r2n7{c6 c4} - r4c4{n7 n9} - r4c9{n9 n6} - r4c2{n6 n1} - r3n1{c2 .} ==> r2c6≠1
biv-chain[3]: r5n7{c8 c6} - r2c6{n7 n9} - c3n9{r2 r5} ==> r5c8≠9
Probably not necessary to finish the puzzle, but it's at level L2 so it's applied now:
Trid-OR2-whip[2]: OR2{{n2r6c1 | n4r5c7}} - r5n9{c7 .} ==> r6c1≠9Easy end in Z4
- Code: Select all
naked-triplets-in-a-row: r5{c4 c6 c8}{n1 n5 n7} ==> r5c5≠1
biv-chain[3]: b1n1{r3c2 r1c1} - c1n9{r1 r9} - b7n6{r9c1 r7c2} ==> r3c2≠6
z-chain[4]: r2n1{c7 c5} - c5n8{r2 r6} - c5n6{r6 r5} - c3n6{r5 .} ==> r2c7≠6
z-chain[4]: r2c7{n9 n1} - r7c7{n1 n6} - c2n6{r7 r4} - b4n9{r4c2 .} ==> r5c7≠9
singles ==> r5c3=9, r2c3=6
whip[1]: c5n6{r6 .} ==> r6c6≠6
biv-chain[3]: c6n6{r1 r3} - r3c8{n6 n9} - b1n9{r3c2 r1c1} ==> r1c6≠9
finned-x-wing-in-rows: n9{r1 r9}{c1 c9} ==> r7c9≠9
biv-chain[3]: r9c1{n6 n9} - r1n9{c1 c9} - b3n6{r1c9 r3c8} ==> r9c8≠6
biv-chain[3]: r9n6{c1 c7} - r9n2{c7 c5} - r4n2{c5 c1} ==> r4c1≠6
biv-chain[3]: r4c2{n6 n1} - r3n1{c2 c6} - r3n6{c6 c8} ==> r4c8≠6
z-chain[4]: c7n9{r9 r2} - r2n1{c7 c5} - r9n1{c5 c4} - r9n5{c4 .} ==> r9c8≠9
stte
