.
This puzzle illustrates the use of
4 different impossible patterns in a single puzzle. It's also an example of using one more impossible pattern:
EL13c175.
As several previous examples, it illustrates the use of ultra-persistency of the ORk-relations and of the generic ORk-splitting rules. (Note that I've kept only those that were useful for resolution.)
Of course, it is NOT necessary to use all this arsenal. But I'm currently playing with the set of 1200+ rules automatically created by SudoRules from eleven's list of 630 impossible patterns.
The solution that follows doesn't use chains of length greater than 6. Using only tridagons, the puzzle is not solved with chains of length no greater than 8.
Starting as usual from the resolution state after Singles and whips[1]:
hidden-pairs-in-a-row: r1{n2 n3}{c2 c3} ==> r1c3≠9, r1c3≠5, r1c3≠1, r1c2≠9, r1c2≠5, r1c2≠1
In addition to tridagon, we now have a lot of ORk-relations. Here are the only 3 that will be used (EL13c290 and EL13c234 are the most frequent ones; EL13c290 was used in totuan's solution).
- Code: Select all
OR5-anti-tridagon[12] for digits 1, 5 and 9 in blocks:
b1, with cells (marked #): r1c1, r2c2, r3c3
b2, with cells (marked #): r1c5, r2c6, r3c4
b4, with cells (marked #): r6c1, r4c2, r5c3
b5, with cells (marked #): r6c5, r4c4, r5c6
with 5 guardians (in cells marked @): n4r1c5 n8r2c6 n8r5c6 n2r6c5 n8r6c5
+-------------------------+-------------------------+-------------------------+
! 159# 23 23 ! 1459 1459#@ 6 ! 1579 1578 789 !
! 4 159# 6 ! 7 1589 1589#@ ! 159 2 3 !
! 7 8 159# ! 159# 3 2 ! 159 6 4 !
+-------------------------+-------------------------+-------------------------+
! 2 159# 8 ! 159# 6 3 ! 4 157 79 !
! 3 4 159# ! 12589 7 1589#@ ! 6 158 289 !
! 159# 6 7 ! 12589 12589#@ 4 ! 1359 1358 289 !
+-------------------------+-------------------------+-------------------------+
! 16 1237 123 ! 123468 1248 18 ! 378 9 5 !
! 8 13579 4 ! 1359 159 159 ! 2 37 6 !
! 569 2359 2359 ! 235689 2589 7 ! 38 4 1 !
+-------------------------+-------------------------+-------------------------+
EL13c290-OR3-relation for digits: 1, 5 and 9
in cells (marked #): (r1c1 r3c7 r3c4 r3c3 r2c7 r2c6 r2c2 r4c4 r4c2 r5c6 r5c3 r6c7 r6c1)
with 3 guardians (in cells marked @) : n8r2c6 n8r5c6 n3r6c7
+----------------------+----------------------+----------------------+
! 159# 23 23 ! 1459 1459 6 ! 1579 1578 789 !
! 4 159# 6 ! 7 1589 1589#@ ! 159# 2 3 !
! 7 8 159# ! 159# 3 2 ! 159# 6 4 !
+----------------------+----------------------+----------------------+
! 2 159# 8 ! 159# 6 3 ! 4 157 79 !
! 3 4 159# ! 12589 7 1589#@ ! 6 158 289 !
! 159# 6 7 ! 12589 12589 4 ! 1359#@ 1358 289 !
+----------------------+----------------------+----------------------+
! 16 1237 123 ! 123468 1248 18 ! 378 9 5 !
! 8 13579 4 ! 1359 159 159 ! 2 37 6 !
! 569 2359 2359 ! 235689 2589 7 ! 38 4 1 !
+----------------------+----------------------+----------------------+
EL13c234-OR3-relation for digits: 1, 5 and 9
in cells (marked #): (r5c3 r4c4 r4c2 r6c7 r6c1 r1c4 r1c1 r3c7 r3c4 r3c3 r2c7 r2c5 r2c2)
with 3 guardians (in cells marked @) : n3r6c7 n4r1c4 n8r2c5
+----------------------+----------------------+----------------------+
! 159# 23 23 ! 1459#@ 1459 6 ! 1579 1578 789 !
! 4 159# 6 ! 7 1589#@ 1589 ! 159# 2 3 !
! 7 8 159# ! 159# 3 2 ! 159# 6 4 !
+----------------------+----------------------+----------------------+
! 2 159# 8 ! 159# 6 3 ! 4 157 79 !
! 3 4 159# ! 12589 7 1589 ! 6 158 289 !
! 159# 6 7 ! 12589 12589 4 ! 1359#@ 1358 289 !
+----------------------+----------------------+----------------------+
! 16 1237 123 ! 123468 1248 18 ! 378 9 5 !
! 8 13579 4 ! 1359 159 159 ! 2 37 6 !
! 569 2359 2359 ! 235689 2589 7 ! 38 4 1 !
+----------------------+----------------------+----------------------+
EL13c175-OR5-relation for digits: 1, 5 and 9
in cells (marked #): (r2c7 r2c2 r2c5 r3c7 r3c3 r3c4 r4c2 r4c4 r5c3 r5c6 r6c7 r6c1 r6c4)
with 5 guardians (in cells marked @) : n8r2c5 n8r5c6 n3r6c7 n2r6c4 n8r6c4
+-------------------------+-------------------------+-------------------------+
! 159 23 23 ! 1459 1459 6 ! 1579 1578 789 !
! 4 159# 6 ! 7 1589#@ 1589 ! 159# 2 3 !
! 7 8 159# ! 159# 3 2 ! 159# 6 4 !
+-------------------------+-------------------------+-------------------------+
! 2 159# 8 ! 159# 6 3 ! 4 157 79 !
! 3 4 159# ! 12589 7 1589#@ ! 6 158 289 !
! 159# 6 7 ! 12589#@ 12589 4 ! 1359#@ 1358 289 !
+-------------------------+-------------------------+-------------------------+
! 16 1237 123 ! 123468 1248 18 ! 378 9 5 !
! 8 13579 4 ! 1359 159 159 ! 2 37 6 !
! 569 2359 2359 ! 235689 2589 7 ! 38 4 1 !
+-------------------------+-------------------------+-------------------------+
Note that, among all the other ORk-relations available, there's one more EL13c234-OR3-relation, it differs by only one cell and one guardian; but it won't be used in resolution:
- Code: Select all
EL13c234-OR3-relation for digits: 1, 5 and 9
in cells (marked #): (r5c3 r4c4 r4c2 r6c7 r6c1 r1c4 r1c1 r3c7 r3c4 r3c3 r2c7 r2c6 r2c2)
with 3 guardians (in cells marked @) : n3r6c7 n4r1c4 n8r2c6
+----------------------+----------------------+----------------------+
! 159# 23 23 ! 1459#@ 1459 6 ! 1579 1578 789 !
! 4 159# 6 ! 7 1589 1589#@ ! 159# 2 3 !
! 7 8 159# ! 159# 3 2 ! 159# 6 4 !
+----------------------+----------------------+----------------------+
! 2 159# 8 ! 159# 6 3 ! 4 157 79 !
! 3 4 159# ! 12589 7 1589 ! 6 158 289 !
! 159# 6 7 ! 12589 12589 4 ! 1359#@ 1358 289 !
+----------------------+----------------------+----------------------+
! 16 1237 123 ! 123468 1248 18 ! 378 9 5 !
! 8 13579 4 ! 1359 159 159 ! 2 37 6 !
! 569 2359 2359 ! 235689 2589 7 ! 38 4 1 !
+----------------------+----------------------+----------------------+
biv-chain[3]: r7c6{n8 n1} - r7c1{n1 n6} - b8n6{r7c4 r9c4} ==> r9c4≠8
t-whip[4]: r8n9{c6 c2} - r8n7{c2 c8} - r4n7{c8 c9} - r4n9{c9 .} ==> r9c4≠9
EL13c290-OR3-ctr-whip[4]: c7n1{r3 r6} - c1n1{r6 r7} - r7c6{n1 n8} - OR3{{n3r6c7 n8r5c6 n8r2c6 | .}} ==> r1c8≠1whip[1]: c8n1{r6 .} ==> r6c7≠1
EL13c234-OR3-whip[5]: r7n4{c5 c4} - OR3{{n4r1c4 n8r2c5 | n3r6c7}} - c8n3{r6 r8} - c4n3{r8 r9} - c4n6{r9 .} ==> r7c5≠8
EL13c290-OR3-whip[5]: r7c6{n1 n8} - OR3{{n8r5c6 n8r2c6 | n3r6c7}} - c8n3{r6 r8} - r8n7{c8 c2} - r8n1{c2 .} ==> r7c4≠1
EL13c290-OR3-whip[5]: r7c6{n1 n8} - OR3{{n8r5c6 n8r2c6 | n3r6c7}} - c8n3{r6 r8} - r8n7{c8 c2} - r8n1{c2 .} ==> r7c5≠1
EL13c290-OR3-whip[6]: c4n6{r9 r7} - r7c1{n6 n1} - r7c6{n1 n8} - OR3{{n8r5c6 n8r2c6 | n3r6c7}} - b9n3{r7c7 r8c8} - c4n3{r8 .} ==> r9c4≠2
Trid-OR5-ctr-whip[5]: r7n4{c4 c5} - b8n2{r7c5 r9c5} - b8n8{r9c5 r7c6} - r2n8{c6 c5} - OR5{{n4r1c5 n8r2c6 n8r5c6 n2r6c5 n8r6c5 | .}} ==> r7c4≠6singles ==> r9c4=6, r7c1=6
biv-chain[3]: c4n3{r7 r8} - r8c8{n3 n7} - b7n7{r8c2 r7c2} ==> r7c2≠3
t-whip[4]: r9n3{c3 c7} - c7n8{r9 r7} - r7c6{n8 n1} - r8n1{c6 .} ==> r8c2≠3
EL13c290-OR3-ctr-whip[5]: c2n7{r8 r7} - b7n1{r7c2 r7c3} - r7c6{n1 n8} - r7c7{n8 n3} - OR3{{n8r2c6 n8r5c6 n3r6c7 | .}} ==> r8c2≠9whip[1]: r8n9{c6 .} ==> r9c5≠9
EL13c290-OR3-ctr-whip[5]: c2n7{r8 r7} - b7n1{r7c2 r7c3} - r7c6{n1 n8} - r7c7{n8 n3} - OR3{{n8r2c6 n8r5c6 n3r6c7 | .}} ==> r8c2≠5whip[1]: r8n5{c6 .} ==> r9c5≠5
biv-chain[4]: r9c5{n2 n8} - r9c7{n8 n3} - c2n3{r9 r1} - b1n2{r1c2 r1c3} ==> r9c3≠2
EL13c234-OR3-whip[4]: r7c5{n4 n2} - r9c5{n2 n8} - OR3{{n8r2c5 n4r1c4 | n3r6c7}} - r9c7{n3 .} ==> r7c4≠4singles ==> r7c5=4, r1c4=4
- Code: Select all
+-------------------+-------------------+-------------------+
! 159 23 23 ! 4 159 6 ! 1579 578 789 !
! 4 159 6 ! 7 1589 1589 ! 159 2 3 !
! 7 8 159 ! 159 3 2 ! 159 6 4 !
+-------------------+-------------------+-------------------+
! 2 159 8 ! 159 6 3 ! 4 157 79 !
! 3 4 159 ! 12589 7 1589 ! 6 158 289 !
! 159 6 7 ! 12589 12589 4 ! 359 1358 289 !
+-------------------+-------------------+-------------------+
! 6 127 123 ! 238 4 18 ! 378 9 5 !
! 8 17 4 ! 1359 159 159 ! 2 37 6 !
! 59 2359 359 ! 6 28 7 ! 38 4 1 !
+-------------------+-------------------+-------------------+
At least one candidate of a previous Trid-OR5-relation between candidates n4r1c5 n8r2c6 n8r5c6 n2r6c5 n8r6c5 has just been eliminated.
There remains a Trid-OR4-relation between candidates: n8r2c6 n8r5c6 n2r6c5 n8r6c5
At least one candidate of a previous EL13c179-OR6-relation between candidates n7r1c7 n4r1c5 n8r5c6 n3r6c7 n2r6c5 n8r6c5 has just been eliminated.
There remains an EL13c179-OR5-relation between candidates: n7r1c7 n8r5c6 n3r6c7 n2r6c5 n8r6c5
Trid-OR4-whip[3]: r2n8{c6 c5} - OR4{{n8r6c5 n8r2c6 n8r5c6 | n2r6c5}} - r9c5{n2 .} ==> r7c6≠8singles ==> r7c6=1, r8c2=1, r7c2=7, r8c8=7, r4c9=7, r1c7=7, r6c8=3, r8c4=3
- Code: Select all
+-------------------+-------------------+-------------------+
! 159 23 23 ! 4 159 6 ! 7 58 89 !
! 4 59 6 ! 7 1589 589 ! 159 2 3 !
! 7 8 159 ! 159 3 2 ! 159 6 4 !
+-------------------+-------------------+-------------------+
! 2 59 8 ! 159 6 3 ! 4 15 7 !
! 3 4 159 ! 12589 7 589 ! 6 158 289 !
! 159 6 7 ! 12589 12589 4 ! 59 3 289 !
+-------------------+-------------------+-------------------+
! 6 7 23 ! 28 4 1 ! 38 9 5 !
! 8 1 4 ! 3 59 59 ! 2 7 6 !
! 59 2359 359 ! 6 28 7 ! 38 4 1 !
+-------------------+-------------------+-------------------+
At least one candidate of a previous EL13c290-OR3-relation between candidates n8r2c6 n8r5c6 n3r6c7 has just been eliminated.
There remains an EL13c290-OR2-relation between candidates: n8r2c6 n8r5c6
At least one candidate of a previous EL13c175-OR5-relation between candidates n8r2c5 n8r5c6 n3r6c7 n2r6c4 n8r6c4 has just been eliminated.
There remains an EL13c175-OR4-relation between candidates: n8r2c5 n8r5c6 n2r6c4 n8r6c4
naked-pairs-in-a-column: c2{r2 r4}{n5 n9} ==> r9c2≠9, r9c2≠5
naked-pairs-in-a-block: b7{r7c3 r9c2}{n2 n3} ==> r9c3≠3
- Code: Select all
+-------------------+-------------------+-------------------+
! 159 23 23 ! 4 159 6 ! 7 58 89 !
! 4 59 6 ! 7 1589 589 ! 159 2 3 !
! 7 8 159 ! 159 3 2 ! 159 6 4 !
+-------------------+-------------------+-------------------+
! 2 59 8 ! 159 6 3 ! 4 15 7 !
! 3 4 159 ! 12589 7 589 ! 6 158 289 !
! 159 6 7 ! 12589 12589 4 ! 59 3 289 !
+-------------------+-------------------+-------------------+
! 6 7 23 ! 28 4 1 ! 38 9 5 !
! 8 1 4 ! 3 59 59 ! 2 7 6 !
! 59 23 59 ! 6 28 7 ! 38 4 1 !
+-------------------+-------------------+-------------------+
EL13c175-OR4-relation between candidates n8r2c5, n8r5c6, n2r6c4 and n8r6c4
+ same valence for candidates n8r5c6 and n8r2c5 via c-chain[2]: n8r5c6,n8r2c6,n8r2c5
==> EL13c175-OR4-relation can be split into two EL13c175-OR3-relations with respective lists of guardians:
n8r2c5 n2r6c4 n8r6c4 and n8r5c6 n2r6c4 n8r6c4 .
EL13c175-OR3-whip[2]: OR3{{n8r6c4 n8r5c6 | n2r6c4}} - r7c4{n2 .} ==> r5c4≠8EL13c175-OR3-whip[3]: OR3{{n8r6c4 n2r6c4 | n8r2c5}} - c5n1{r2 r1} - c1n1{r1 .} ==> r6c4≠1The end is easy:
- Code: Select all
x-wing-in-rows: n1{r1 r6}{c1 c5} ==> r2c5≠1
hidden-single-in-a-row ==> r2c7=1
finned-x-wing-in-columns: n5{c7 c4}{r3 r6} ==> r6c5≠5
finned-x-wing-in-columns: n9{c7 c4}{r3 r6} ==> r6c5≠9
finned-x-wing-in-rows: n9{r4 r2}{c2 c4} ==> r3c4≠9
whip[1]: c4n9{r6 .} ==> r5c6≠9
biv-chain[3]: r4c8{n1 n5} - c7n5{r6 r3} - r3c4{n5 n1} ==> r4c4≠1
hidden-single-in-a-row ==> r4c8=1
naked-pairs-in-a-row: r5{c6 c8}{n5 n8} ==> r5c9≠8, r5c4≠5, r5c3≠5
finned-x-wing-in-rows: n5{r4 r2}{c2 c4} ==> r3c4≠5
stte
