.
Thanks for your solutions.
Yes, this is a very hard puzzle. I proposed it as a case where identifying the anti-tridagon pattern doesn't help much. But it's also a case where eleven replacement does wonders - not general eleven replacement, but the very restrictive form of it that tries only the 3 cells of a tridagon block.
I'm currently running computations showing (until now) that when the anti-tridagon chain rules are not enough, they can always be supplemented with such tridagon-restricted replacement.
Totuan: it seems other (more complex) impossible patterns appear when an anti-tridagon is present. How frequent this is remains to be explored. Is your E2 "eleven pattern" isomorphic to the only one that I identified (
http://forum.enjoysudoku.com/chromatic-patterns-t39885-50.html) as requiring T&E(3) to be proven contradictory in his list here:
http://forum.enjoysudoku.com/chromatic-patterns-t39885-41.html?
I haven't followed this approach of finding other contradictory patterns, because they are hard to code.
Marek, good idea to use intermediary relations, though your cryptic notation is illegible to me: 7r4c8 = 7r4c1 – 7# = [(3|5)&249#, (3|5|8)r9c3, 249b7# \ r7c16] – (249=7)r7c6 => 7# = 7r7c6 => –7r7c8
After singles and whips[1], there is indeed a Trid-OR5-whip elimination, with a very long whip:
z-chain[3]: c8n6{r9 r2} - c8n3{r2 r7} - r8c7{n3 .} ==> r9c8≠9
z-chain[5]: c4n8{r4 r8} - c4n5{r8 r9} - c3n5{r9 r6} - r4n5{c1 c6} - r4n8{c6 .} ==> r4c4≠2, r4c4≠9, r4c4≠4
Trid-OR5-whip[10]: r8c7{n9 n3} - r1n3{c7 c3} - r2n3{c3 c8} - r7c8{n3 n7} - r7c9{n7 n8} - r9n8{c9 c3} - c3n5{r9 r6} - c3n7{r6 r2} - OR5{{n7r2c1 n3r2c1 n5r6c1 n5r6c6 | n7r6c1}} - b6n7{r6c7 .} ==> r9c9≠9After that there's no elimination available with chains of length ≤ 12.
My solution will completely discard the Tridagon elimination and rely on replacement:
- Code: Select all
z-chain[3]: c8n6{r9 r2} - c8n3{r2 r7} - r8c7{n3 .} ==> r9c8≠9
z-chain[5]: c4n8{r4 r8} - c4n5{r8 r9} - c3n5{r9 r6} - r4n5{c1 c6} - r4n8{c6 .} ==> r4c4≠2, r4c4≠9, r4c4≠4
+----------------------+----------------------+----------------------+
! 1 249 23479 ! 249 5 6 ! 2379 8 479 !
! 23479 5 23479 ! 1 8 249 ! 2379 234679 4679 !
! 8 6 249 ! 7 249 3 ! 129 5 149 !
+----------------------+----------------------+----------------------+
! 24579 249 1 ! 58 249 24589 ! 6 2479 3 !
! 6 3 249 ! 249 7 1 ! 8 249 5 !
! 24579 8 24579 ! 3 6 2459 ! 1279 2479 1479 !
+----------------------+----------------------+----------------------+
! 2349 249 23489 ! 6 1 2479 ! 5 379 789 !
! 3459 7 6 ! 4589 349 4589 ! 39 1 2 !
! 2359 1 23589 ! 259 239 2579 ! 4 367 6789 !
+----------------------+----------------------+----------------------+
- Code: Select all
***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE FOR GENERAL TRIDAGON *****
RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.
Trying in block 2
+----------------------+----------------------+----------------------+
! 1 249 23479 ! 9 5 6 ! 23479 8 2479 !
! 23479 5 23479 ! 1 8 4 ! 23479 234679 24679 !
! 8 6 249 ! 7 2 3 ! 1249 5 1249 !
+----------------------+----------------------+----------------------+
! 24579 249 1 ! 58 249 24589 ! 6 2479 3 !
! 6 3 249 ! 249 7 1 ! 8 249 5 !
! 24579 8 24579 ! 3 6 2459 ! 12479 2479 12479 !
+----------------------+----------------------+----------------------+
! 2349 249 23489 ! 6 1 2479 ! 5 23479 24789 !
! 23459 7 6 ! 24589 2349 24589 ! 2349 1 249 !
! 23459 1 234589 ! 2459 2349 24579 ! 249 367 246789 !
+----------------------+----------------------+----------------------+
The rest is routine solving in W6:
- Code: Select all
finned-x-wing-in-columns: n9{c2 c5}{r4 r7} ==> r7c6≠9
finned-x-wing-in-rows: n9{r5 r3}{c3 c8} ==> r2c8≠9
biv-chain[2]: c2n9{r7 r4} - r5n9{c3 c8} ==> r7c8≠9
whip[1]: c8n9{r4 .} ==> r6c7≠9, r6c9≠9
t-whip[4]: c9n8{r7 r9} - r9n6{c9 c8} - r9n7{c8 c6} - r7c6{n7 .} ==> r7c9≠2
z-chain[5]: r3c3{n9 n4} - r5c3{n4 n2} - r5c4{n2 n4} - r4c5{n4 n9} - c2n9{r4 .} ==> r9c3≠9, r7c3≠9
z-chain[5]: r4c5{n9 n4} - r4c2{n4 n2} - r1c2{n2 n4} - r3c3{n4 n9} - r5n9{c3 .} ==> r4c8≠9
t-whip[5]: c2n9{r7 r4} - r4c5{n9 n4} - r5c4{n4 n2} - r5c3{n2 n4} - b1n4{r3c3 .} ==> r7c2≠4
biv-chain[4]: r5n9{c8 c3} - r3c3{n9 n4} - c2n4{r1 r4} - b5n4{r4c5 r5c4} ==> r5c8≠4
z-chain[5]: c2n4{r4 r1} - r3c3{n4 n9} - r5c3{n9 n2} - r4c2{n2 n9} - r4c5{n9 .} ==> r4c1≠4
z-chain[5]: b6n4{r6c9 r4c8} - r4c5{n4 n9} - r4c2{n9 n2} - r5c3{n2 n9} - r3c3{n9 .} ==> r6c3≠4
t-whip[5]: c9n8{r7 r9} - r9n6{c9 c8} - r9n7{c8 c6} - r7c6{n7 n2} - r7c2{n2 .} ==> r7c9≠9
whip[1]: r7n9{c1 .} ==> r8c1≠9, r9c1≠9
z-chain[6]: r5n4{c3 c4} - r4c5{n4 n9} - r4c2{n9 n2} - r5c3{n2 n9} - r3c3{n9 n4} - c2n4{r1 .} ==> r6c1≠4
whip[1]: r6n4{c9 .} ==> r4c8≠4
whip[1]: c1n4{r9 .} ==> r7c3≠4, r9c3≠4
t-whip[6]: r7c6{n7 n2} - b5n2{r4c6 r5c4} - b5n4{r5c4 r4c5} - c2n4{r4 r1} - c2n2{r1 r4} - r4c8{n2 .} ==> r7c8≠7
hidden-triplets-in-a-block: b9{n6 n7 n8}{r9c9 r9c8 r7c9} ==> r9c9≠9, r9c9≠4, r9c9≠2, r9c8≠3, r7c9≠4
finned-x-wing-in-columns: n3{c8 c1}{r2 r7} ==> r7c3≠3
hidden-pairs-in-a-row: r7{n3 n4}{c1 c8} ==> r7c8≠2, r7c1≠9, r7c1≠2
hidden-single-in-a-block ==> r7c2=9
t-whip[2]: r7n2{c3 c6} - b5n2{r4c6 .} ==> r5c3≠2
naked-pairs-in-a-column: c3{r3 r5}{n4 n9} ==> r6c3≠9, r2c3≠9, r1c3≠4
biv-chain[3]: r4c2{n2 n4} - r5c3{n4 n9} - r5c8{n9 n2} ==> r4c8≠2
singles ==> r4c8=7, r9c8=6, r2c9=6
whip[1]: c7n7{r1 .} ==> r1c9≠7
naked-pairs-in-a-row: r1{c2 c9}{n2 n4} ==> r1c7≠4, r1c7≠2, r1c3≠2
finned-x-wing-in-rows: n2{r7 r4}{c6 c3} ==> r6c3≠2
biv-chain[3]: r1n3{c3 c7} - b3n7{r1c7 r2c7} - r2n9{c7 c1} ==> r2c1≠3
whip[1]: c1n3{r9 .} ==> r9c3≠3
biv-chain[3]: c9n9{r8 r3} - r3c3{n9 n4} - r1n4{c2 c9} ==> r8c9≠4
biv-chain[4]: r4c5{n9 n4} - b4n4{r4c2 r5c3} - c3n9{r5 r3} - c9n9{r3 r8} ==> r8c5≠9
biv-chain[3]: r8c5{n4 n3} - r9n3{c5 c1} - r7c1{n3 n4} ==> r8c1≠4
biv-chain[4]: r5c8{n2 n9} - c3n9{r5 r3} - b1n4{r3c3 r1c2} - r1n2{c2 c9} ==> r6c9≠2, r2c8≠2
stte
After replacement, a puzzle that was in T&E(3) is now solved in T&E(1). This is consistent with my analysis of replacement as being in T&E(2) complexity-wise.
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