A puzzle with lots of guardians: #1182 in mith's list of 63,137 min-expands.
- Code: Select all
+-------+-------+-------+
! . 2 . ! 4 . . ! . . . !
! . . 7 ! . . . ! . . 6 !
! 6 . 8 ! . . . ! . 1 5 !
+-------+-------+-------+
! . . . ! 5 . 4 ! . 6 1 !
! . . . ! . 9 . ! . . 2 !
! . 6 . ! . 1 2 ! . 9 . !
+-------+-------+-------+
! . . . ! . . 5 ! 1 . . !
! 5 . . ! . . . ! . 2 4 !
! 9 . 1 ! 2 4 . ! . . . !
+-------+-------+-------+
.2.4.......7.....66.8....15...5.4.61....9...2.6..12.9......51..5......249.124....;236;29694
SER = 11.7
If we try to find an anti-tridagon pattern just after Singles, we find two, one with 9 guardians and one with 12:
- Code: Select all
+----------------------+----------------------+----------------------+
! 13 2 359 ! 4 35678 136789 ! 3789 378 3789 !
! 134 13459 7 ! 1389 2358 1389 ! 23489 348 6 !
! 6 349 8 ! 379 237 379 ! 23479 1 5 !
+----------------------+----------------------+----------------------+
! 2378 3789 239 ! 5 378 4 ! 378 6 1 !
! 13478 134578 345 ! 3678 9 3678 ! 34578 34578 2 !
! 3478 6 345 ! 378 1 2 ! 34578 9 378 !
+----------------------+----------------------+----------------------+
! 23478 3478 2346 ! 36789 3678 5 ! 1 378 3789 !
! 5 378 36 ! 136789 3678 136789 ! 36789 2 4 !
! 9 378 1 ! 2 4 3678 ! 35678 3578 378 !
+----------------------+----------------------+----------------------+
OR9-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:
b5, with cells: r4c5, r5c6, r6c4
b6, with cells: r4c7, r5c8, r6c9
b8, with cells: r8c5, r9c6, r7c4
b9, with cells: r8c7, r9c9, r7c8
with 9 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c4 n9r7c4 n6r8c5 n6r8c7 n9r8c7 n6r9c6
OR12-anti-tridagon[12] (type antidiag) for digits 7, 8 and 3 in blocks:
b5, with cells: r4c5, r5c6, r6c4
b6, with cells: r4c7, r5c8, r6c9
b8, with cells: r7c5, r9c6, r8c4
b9, with cells: r7c9, r9c8, r8c7
with 12 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c5 n9r7c9 n1r8c4 n6r8c4 n9r8c4 n6r8c7 n9r8c7 n6r9c6 n5r9c8
In instead of Singles, we allow Subsets + Finned Fish (IMO, the minimal reasonable thing to do before looking for any anti-tridagon), the situation is hardly better:
- Code: Select all
hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3
+----------------------+----------------------+----------------------+
! 13 2 359 ! 4 35678 136789 ! 3789 378 3789 !
! 134 15 7 ! 1389 2358 1389 ! 23489 348 6 !
! 6 349 8 ! 379 237 379 ! 23479 1 5 !
+----------------------+----------------------+----------------------+
! 2378 3789 239 ! 5 378 4 ! 378 6 1 !
! 13478 15 345 ! 3678 9 3678 ! 34578 34578 2 !
! 3478 6 345 ! 378 1 2 ! 34578 9 378 !
+----------------------+----------------------+----------------------+
! 23478 3478 2346 ! 36789 3678 5 ! 1 378 3789 !
! 5 378 36 ! 136789 3678 136789 ! 36789 2 4 !
! 9 378 1 ! 2 4 3678 ! 35678 3578 378 !
+----------------------+----------------------+----------------------+
OR9-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:
b5, with cells: r4c5, r5c6, r6c4
b6, with cells: r4c7, r5c8, r6c9
b8, with cells: r8c5, r9c6, r7c4
b9, with cells: r8c7, r9c9, r7c8
with 9 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c4 n9r7c4 n6r8c5 n6r8c7 n9r8c7 n6r9c6
OR12-anti-tridagon[12] (type antidiag) for digits 7, 8 and 3 in blocks:
b5, with cells: r4c5, r5c6, r6c4
b6, with cells: r4c7, r5c8, r6c9
b8, with cells: r7c5, r9c6, r8c4
b9, with cells: r7c9, r9c8, r8c7
with 12 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c5 n9r7c9 n1r8c4 n6r8c4 n9r8c4 n6r8c7 n9r8c7 n6r9c6 n5r9c8
The situation is quite better is one starts with Subsets + Finned Fish + Whips[≤4]: there remains only one anti-tridagon-pattern and it has only 5 guardians:
- Code: Select all
hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3
whip[3]: r1c1{n3 n1} - r2c2{n1 n5} - c5n5{r2 .} ==> r1c5≠3
whip[4]: c5n6{r8 r1} - r1n5{c5 c3} - r2c2{n5 n1} - c4n1{r2 .} ==> r8c4≠6
whip[4]: r3n4{c7 c2} - b1n9{r3c2 r1c3} - b3n9{r1c9 r3c7} - c7n2{r3 .} ==> r2c7≠4
whip[4]: r2c2{n1 n5} - r1n5{c3 c5} - r1n6{c5 c6} - r1n1{c6 .} ==> r2c1≠1
whip[3]: c7n2{r2 r3} - r3n4{c7 c2} - r2c1{n4 .} ==> r2c7≠3
whip[4]: r2n9{c6 c7} - r3n9{c7 c2} - r3n4{c2 c7} - c7n2{r3 .} ==> r1c6≠9
whip[4]: r3n7{c6 c7} - r3n4{c7 c2} - b1n9{r3c2 r1c3} - r1n5{c3 .} ==> r1c5≠7
whip[4]: r6n5{c7 c3} - r1n5{c3 c5} - r1n6{c5 c6} - r9n6{c6 .} ==> r9c7≠5
hidden-single-in-a-block ==> r9c8=5
whip[4]: r1c1{n3 n1} - r2c2{n1 n5} - r1n5{c3 c5} - r1n6{c5 .} ==> r1c6≠3
whip[3]: r3n4{c7 c2} - r2c1{n4 n3} - b2n3{r2c6 .} ==> r3c7≠3
whip[4]: r3n4{c7 c2} - b1n9{r3c2 r1c3} - b3n9{r1c9 r2c7} - c7n2{r2 .} ==> r3c7≠7
whip[1]: r3n7{c6 .} ==> r1c6≠7
whip[4]: r2n2{c7 c5} - c5n5{r2 r1} - r1n8{c5 c6} - r1n6{c6 .} ==> r2c7≠8
whip[4]: r1n6{c6 c5} - r1n5{c5 c3} - r2c2{n5 n1} - b2n1{r2c4 .} ==> r1c6≠8
whip[4]: r2c1{n3 n4} - r2c8{n4 n8} - r1n8{c9 c5} - c5n5{r1 .} ==> r2c5≠3
whip[4]: r1n6{c5 c6} - r1n1{c6 c1} - r2c2{n1 n5} - c5n5{r2 .} ==> r1c5≠8
whip[1]: r1n8{c9 .} ==> r2c8≠8
naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c6≠3, r2c4≠3
whip[1]: b2n3{r3c6 .} ==> r3c2≠3
whip[3]: c3n2{r7 r4} - r4n9{c3 c2} - c2n3{r4 .} ==> r7c3≠3
whip[3]: r4n2{c1 c3} - c3n9{r4 r1} - b1n3{r1c3 .} ==> r4c1≠3
whip[4]: c3n6{r8 r7} - c3n2{r7 r4} - r4n9{c3 c2} - c2n3{r4 .} ==> r8c3≠3
singles ==> r8c3=6, r9c7=6
+-------------------+-------------------+-------------------+
! 13 2 359 ! 4 56 16 ! 3789 378 3789 !
! 34 15 7 ! 189 258 189 ! 29 34 6 !
! 6 49 8 ! 379 237 379 ! 249 1 5 !
+-------------------+-------------------+-------------------+
! 278 3789 239 ! 5 378 4 ! 378 6 1 !
! 13478 15 345 ! 3678 9 3678 ! 34578 3478 2 !
! 3478 6 345 ! 378 1 2 ! 34578 9 378 !
+-------------------+-------------------+-------------------+
! 23478 3478 24 ! 36789 3678 5 ! 1 378 3789 !
! 5 378 6 ! 13789 378 13789 ! 3789 2 4 !
! 9 378 1 ! 2 4 378 ! 6 5 378 !
+-------------------+-------------------+-------------------+
OR5-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:
b5, with cells: r4c5, r5c6, r6c4
b6, with cells: r4c7, r5c8, r6c9
b8, with cells: r8c5, r9c6, r7c4
b9, with cells: r8c7, r9c9, r7c8
with 5 guardians: n6r5c6 n4r5c8 n6r7c4 n9r7c4 n9r8c7
Now, the first thing one may want to try is wether the puzzle can be solved in W4+OR5FW4. Unfortunately, the answer is negative.
From this point on, one can progressively increase the max sizes of allowed chains, independently the size of normal chains (bivalue, z-, whips...) and the size of ORk-Forcing-Whips.
For this puzzle, I found a good balance with all chains ≤ 6 and Forcing-Whips ≤ 9:
- Code: Select all
hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3
biv-chain[3]: r1c1{n3 n1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} ==> r1c5≠3
biv-chain[4]: r1c1{n3 n1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} - b2n6{r1c5 r1c6} ==> r1c6≠3
biv-chain[4]: r1n1{c1 c6} - b2n6{r1c6 r1c5} - b2n5{r1c5 r2c5} - r2c2{n5 n1} ==> r2c1≠1
biv-chain[3]: r2c1{n3 n4} - r3n4{c2 c7} - b3n2{r3c7 r2c7} ==> r2c7≠3
whip[3]: r3n4{c7 c2} - r2c1{n4 n3} - b2n3{r2c4 .} ==> r3c7≠3
biv-chain[4]: r1n1{c6 c1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} - b2n6{r1c5 r1c6} ==> r1c6≠7, r1c6≠8, r1c6≠9
biv-chain[3]: r8n1{c4 c6} - r1c6{n1 n6} - b5n6{r5c6 r5c4} ==> r8c4≠6
z-chain[3]: r1n8{c9 c5} - c5n5{r1 r2} - r2n2{c5 .} ==> r2c7≠8
biv-chain[4]: r1n5{c5 c3} - r2c2{n5 n1} - r1n1{c1 c6} - b2n6{r1c6 r1c5} ==> r1c5≠7, r1c5≠8
whip[1]: r1n8{c9 .} ==> r2c8≠8
whip[1]: r1n7{c9 .} ==> r3c7≠7
naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c7≠4, r2c6≠3, r2c5≠3, r2c4≠3
whip[1]: b2n3{r3c6 .} ==> r3c2≠3
z-chain[3]: c2n3{r9 r4} - r4n9{c2 c3} - c3n2{r4 .} ==> r7c3≠3
z-chain[3]: b1n3{r2c1 r1c3} - c3n9{r1 r4} - r4n2{c3 .} ==> r4c1≠3
biv-chain[4]: r6n5{c7 c3} - r1n5{c3 c5} - b2n6{r1c5 r1c6} - r9n6{c6 c7} ==> r9c7≠5
hidden-single-in-a-block ==> r9c8=5
z-chain[4]: c2n3{r9 r4} - r4n9{c2 c3} - c3n2{r4 r7} - c3n6{r7 .} ==> r8c3≠3
naked-single ==> r8c3=6
hidden-single-in-a-block ==> r9c7=6
z-chain[5]: b6n5{r6c7 r5c7} - b6n4{r5c7 r5c8} - r2n4{c8 c1} - r6n4{c1 c3} - r6n5{c3 .} ==> r6c7≠3, r6c7≠8, r6c7≠7
whip[6]: r5n6{c4 c6} - r1n6{c6 c5} - r1n5{c5 c3} - r5c3{n5 n4} - r6n4{c3 c7} - r6n5{c7 .} ==> r5c4≠3
+-------------------+-------------------+-------------------+
! 13 2 359 ! 4 56 16 ! 3789 378 3789 !
! 34 15 7 ! 189 258 189 ! 29 34 6 !
! 6 49 8 ! 379 237 379 ! 249 1 5 !
+-------------------+-------------------+-------------------+
! 278 3789 239 ! 5 378 4 ! 378 6 1 !
! 13478 15 345 ! 678 9 3678 ! 34578 3478 2 !
! 3478 6 345 ! 378 1 2 ! 45 9 378 !
+-------------------+-------------------+-------------------+
! 23478 3478 24 ! 36789 3678 5 ! 1 378 3789 !
! 5 378 6 ! 13789 378 13789 ! 3789 2 4 !
! 9 378 1 ! 2 4 378 ! 6 5 378 !
+-------------------+-------------------+-------------------+
OR5-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:
b5, with cells: r4c5, r5c6, r6c4
b6, with cells: r4c7, r5c8, r6c9
b8, with cells: r8c5, r9c6, r7c4
b9, with cells: r8c7, r9c9, r7c8
with 5 guardians: n6r5c6 n4r5c8 n6r7c4 n9r7c4 n9r8c7
OR5-forcing-whip-elim[9] based on OR5-anti-tridagon[12] for n9r8c7, n4r5c8, n9r7c4, n6r5c6 and n6r7c4:
|| n9r8c7 -
|| n4r5c8 - partial-whip[1]: c7n4{r6 r3} -
|| n9r7c4 - partial-whip[1]: c9n9{r7 r1} -
|| n6r5c6 - partial-whip[3]: r1n6{c6 c5} - r1n5{c5 c3} - b1n9{r1c3 r3c2} -
|| n6r7c4 - partial-whip[3]: c5n6{r7 r1} - r1n5{c5 c3} - b1n9{r1c3 r3c2} -
==> r3c7≠9
OR5-forcing-whip-elim[9] based on OR5-anti-tridagon[12] for n9r7c4, n9r8c7, n4r5c8, n6r5c6 and n6r7c4:
|| n9r7c4 - partial-whip[1]: c9n9{r7 r1} -
|| n9r8c7 - partial-whip[1]: c9n9{r7 r1} -
|| n4r5c8 - partial-whip[2]: r2n4{c8 c1} - r3c2{n4 n9} -
|| n6r5c6 - partial-whip[2]: r1n6{c6 c5} - r1n5{c5 c3} -
|| n6r7c4 - partial-whip[2]: c5n6{r7 r1} - r1n5{c5 c3} -
==> r1c3≠9
singles ==> r3c2=9, r2c1=4, r2c8=3, r3c7=4, r6c7=5, r2c7=2, r3c5=2, r5c8=4, r6c3=4, r7c3=2, r4c1=2, r7c2=4, r4c3=9
z-chain[5]: c5n6{r7 r1} - r1c6{n6 n1} - r1c1{n1 n3} - r7c1{n3 n8} - r7c8{n8 .} ==> r7c5≠7
z-chain[5]: c5n6{r7 r1} - r1c6{n6 n1} - r1c1{n1 n3} - r7c1{n3 n7} - r7c8{n7 .} ==> r7c5≠8
z-chain[4]: r7c5{n3 n6} - r1n6{c5 c6} - c6n1{r1 r2} - c6n9{r2 .} ==> r8c6≠3
z-chain[5]: r3c4{n7 n3} - r6c4{n3 n8} - r4c5{n8 n3} - r7c5{n3 n6} - c4n6{r7 .} ==> r5c4≠7
biv-chain[4]: r5c4{n8 n6} - r7n6{c4 c5} - r1c5{n6 n5} - r2c5{n5 n8} ==> r4c5≠8, r2c4≠8
biv-chain[3]: r8n1{c6 c4} - r2c4{n1 n9} - c6n9{r2 r8} ==> r8c6≠7, r8c6≠8
whip[3]: r4n8{c7 c2} - b7n8{r8c2 r7c1} - c8n8{r7 .} ==> r1c7≠8
biv-chain[3]: r7c8{n7 n8} - r1n8{c8 c9} - c9n9{r1 r7} ==> r7c9≠7
whip[6]: r4n8{c7 c2} - r9n8{c2 c6} - r5n8{c6 c4} - r5n6{c4 c6} - c6n3{r5 r3} - c6n7{r3 .} ==> r8c7≠8
whip[1]: c7n8{r5 .} ==> r6c9≠8
whip[4]: r4n8{c7 c2} - r6n8{c1 c4} - r8n8{c4 c5} - c5n7{r8 .} ==> r4c7≠7
z-chain[3]: b6n7{r5c7 r6c9} - r9n7{c9 c2} - r4n7{c2 .} ==> r5c6≠7
z-chain[3]: c5n3{r8 r4} - b5n7{r4c5 r6c4} - r3c4{n7 .} ==> r8c4≠3, r7c4≠3
whip[4]: r5n7{c1 c7} - r6c9{n7 n3} - b5n3{r6c4 r4c5} - r7n3{c5 .} ==> r5c1≠3
whip[4]: b5n7{r4c5 r6c4} - r6c9{n7 n3} - r7n3{c9 c1} - c2n3{r8 .} ==> r4c5≠3
naked-single ==> r4c5=7
whip[1]: c2n7{r9 .} ==> r7c1≠7
whip[1]: c5n3{r8 .} ==> r9c6≠3
finned-x-wing-in-rows: n3{r9 r4}{c2 c9} ==> r6c9≠3
singles ==> r6c9=7, r5c1=7, r5c2=1, r2c2=5, r1c3=3, r1c1=1, r1c6=6, r1c5=5, r5c3=5, r2c5=8, r8c5=3, r7c5=6, r5c4=6
finned-x-wing-in-rows: n8{r8 r6}{c4 c2} ==> r4c2≠8
stte