.
I have chosen this puzzle as an example where W + Trid-OR5W > gW + Trid-OR5gW.
(See
http://forum.enjoysudoku.com/the-tridagon-rule-t39859-114.html for general results about this.)
Without g-chains, one needs W8 + Trid-OR5W8 ; note that even with all of Imp630 activated, one still needs chains of length 8. (Totuan's solution above uses some degenerate impossible patterns and it nevertheless has much longer chains.)
Here's the solution in
gW6 + Trid-OR5gW6:
- Code: Select all
Trid-OR3-relation for digits 1, 3 and 4 in blocks:
b5, with cells (marked #): r4c6, r5c4, r6c5
b6, with cells (marked #): r4c9, r5c8, r6c7
b8, with cells (marked #): r9c6, r7c4, r8c5
b9, with cells (marked #): r9c7, r7c8, r8c9
with 3 guardians (in cells marked @): n9r5c8 n5r9c6 n7r9c7
+----------------------+----------------------+----------------------+
! 1 249 3 ! 24 456 456 ! 457 8 24679 !
! 24 5 6 ! 7 8 9 ! 134 134 1234 !
! 7 8 249 ! 1234 13456 13456 ! 45 49 2469 !
+----------------------+----------------------+----------------------+
! 2348 1234 1248 ! 5 9 134# ! 6 7 134# !
! 349 6 149 ! 134# 7 2 ! 8 1349#@ 5 !
! 3459 7 1459 ! 6 134# 8 ! 134# 2 1349 !
+----------------------+----------------------+----------------------+
! 34589 1349 145789 ! 134# 13456 13456 ! 2 134# 13478 !
! 6 1234 124 ! 8 134# 7 ! 9 5 134# !
! 3458 134 14578 ! 9 2 1345#@ ! 1347#@ 6 13478 !
+----------------------+----------------------+----------------------+
t-whip[3]: r1n9{c2 c9} - r3c8{n9 n4} - r2n4{c9 .} ==> r1c2≠4
biv-chain[3]: b1n4{r3c3 r2c1} - c1n2{r2 r4} - b4n8{r4c1 r4c3} ==> r4c3≠4
whip[5]: r2c1{n4 n2} - r1c2{n2 n9} - r7n9{c2 c3} - r7n7{c3 c9} - r7n8{c9 .} ==> r7c1≠4
g-whip[5]: c2n4{r9 r4} - c2n1{r4 r789} - r8c3{n1 n2} - c2n2{r8 r1} - r2c1{n2 .} ==> r9c1≠4
whip[6]: r7n7{c3 c9} - r7n8{c9 c1} - c1n5{r7 r6} - c1n9{r6 r5} - c8n9{r5 r3} - c3n9{r3 .} ==> r7c3≠5
whip[6]: r7n7{c9 c3} - r7n8{c3 c1} - r7n9{c1 c2} - r1n9{c2 c9} - c9n7{r1 r9} - r9n8{c9 .} ==> r7c9≠1
whip[6]: r7n7{c9 c3} - r7n8{c3 c1} - r7n9{c1 c2} - r1n9{c2 c9} - c9n7{r1 r9} - r9n8{c9 .} ==> r7c9≠3
whip[6]: r7n7{c9 c3} - r7n8{c3 c1} - r7n9{c1 c2} - r1n9{c2 c9} - c9n7{r1 r9} - r9n8{c9 .} ==> r7c9≠4
The interesting part:
Trid-OR3-whip[6]: c8n9{r5 r3} - c3n9{r3 r7} - r7n7{c3 c9} - OR3{{n7r9c7 n9r5c8 | n5r9c6}} - b7n5{r9c1 r7c1} - r7n8{c1 .} ==> r5c1≠9biv-chain[5]: c1n9{r7 r6} - r5n9{c3 c8} - r3c8{n9 n4} - b1n4{r3c3 r2c1} - r5c1{n4 n3} ==> r7c1≠3
whip[6]: r5c1{n3 n4} - b1n4{r2c1 r3c3} - b1n9{r3c3 r1c2} - c2n2{r1 r8} - r8c3{n2 n1} - b4n1{r4c3 .} ==> r4c2≠3
whip[1]: c2n3{r9 .} ==> r9c1≠3
Trid-OR3-gwhip[6]: r3c8{n4 n9} - r5n9{c8 c3} - c1n9{r6 r7} - r7n5{c1 c456} - OR3{{n5r9c6 n9r5c8 | n7r9c7}} - r1n7{c7 .} ==> r1c9≠4
Trid-OR3-gwhip[6]: c9n6{r1 r3} - c9n9{r3 r6} - c1n9{r6 r7} - r7n5{c1 c456} - OR3{{n5r9c6 n9r5c8 | n7r9c7}} - r1n7{c7 .} ==> r1c9≠2
Trid-OR3-gwhip[6]: c9n6{r3 r1} - c9n9{r1 r6} - c1n9{r6 r7} - r7n5{c1 c456} - OR3{{n5r9c6 n9r5c8 | n7r9c7}} - r1n7{c7 .} ==> r3c9≠2The end, in W6, has nothing noticeable:
- Code: Select all
singles ==> r2c9=2, r2c1=4, r5c1=3, r4c1=2, r4c3=8
z-chain[5]: c4n3{r7 r3} - c4n2{r3 r1} - c2n2{r1 r8} - r8n3{c2 c9} - r4n3{c9 .} ==> r9c6≠3, r7c6≠3
whip[6]: b5n3{r4c6 r6c5} - b5n4{r6c5 r5c4} - r1c4{n4 n2} - c2n2{r1 r8} - r8n3{c2 c9} - r4n3{c9 .} ==> r4c6≠1
whip[5]: r4n1{c2 c9} - c8n1{r5 r2} - c8n3{r2 r7} - b8n3{r7c4 r8c5} - r8n1{c5 .} ==> r7c2≠1
whip[6]: r8n2{c3 c2} - c2n4{r8 r4} - r4n1{c2 c9} - r8c9{n1 n3} - r9n3{c7 c2} - c2n1{r9 .} ==> r8c3≠4
biv-chain[4]: r8c3{n1 n2} - r3n2{c3 c4} - r1c4{n2 n4} - r5c4{n4 n1} ==> r5c3≠1
biv-chain[3]: r5c3{n4 n9} - r6c1{n9 n5} - c3n5{r6 r9} ==> r9c3≠4
biv-chain[4]: r8c3{n1 n2} - r3c3{n2 n9} - r5c3{n9 n4} - r4c2{n4 n1} ==> r8c2≠1, r9c2≠1, r6c3≠1
hidden-single-in-a-block ==> r4c2=1
whip[1]: c2n4{r9 .} ==> r7c3≠4
t-whip[4]: r4c9{n4 n3} - b5n3{r4c6 r6c5} - r8n3{c5 c2} - r9c2{n3 .} ==> r9c9≠4
z-chain[5]: r3c8{n4 n9} - r3c3{n9 n2} - r8c3{n2 n1} - r8c9{n1 n3} - r4c9{n3 .} ==> r3c9≠4
t-whip[5]: b8n3{r7c5 r8c5} - b5n3{r6c5 r4c6} - r4n4{c6 c9} - r8n4{c9 c2} - r9c2{n4 .} ==> r7c2≠3
biv-chain[3]: r7c2{n4 n9} - r1n9{c2 c9} - r3c8{n9 n4} ==> r7c8≠4
naked-pairs-in-a-column: c8{r2 r7}{n1 n3} ==> r5c8≠1
hidden-single-in-a-row ==> r5c4=1
biv-chain[2]: r4n4{c6 c9} - c8n4{r5 r3} ==> r3c6≠4
biv-chain[2]: b9n4{r9c7 r8c9} - r4n4{c9 c6} ==> r9c6≠4
biv-chain[3]: r4c9{n3 n4} - b9n4{r8c9 r9c7} - r9c2{n4 n3} ==> r9c9≠3
hidden-pairs-in-a-row: r9{n3 n4}{c2 c7} ==> r9c7≠7, r9c7≠1
singles ==> r1c7=7, r3c7=5, r3c8=4, r5c8=9, r5c3=4
hidden-pairs-in-a-column: c9{n7 n8}{r7 r9} ==> r9c9≠1
biv-chain[3]: r1n2{c2 c4} - c4n4{r1 r7} - r7c2{n4 n9} ==> r1c2≠9
stte
