The New Sudoku Players' Forum

Website · by **denis_berthier** » Fri Mar 10, 2023 6:32 am

.
One more illustration of eleven's 630 impossible patterns.

Code: Select all: +-------+-------+-------+ ! 1 2 3 ! . . 6 ! . . . ! ! . . . ! . 8 . ! 2 . . ! ! . . . ! 7 . 2 ! . . . ! +-------+-------+-------+ ! 2 3 . ! . . . ! . . . ! ! . 1 6 ! 5 9 . ! 3 2 . ! ! 9 . 5 ! 3 2 . ! . 6 . ! +-------+-------+-------+ ! 3 . . ! . . . ! 6 5 . ! ! 5 9 . ! . . . ! 1 . 2 ! ! . 6 . ! . . 5 ! . 9 . ! +-------+-------+-------+ 123..6.......8.2.....7.2...23........1659.32.9.532..6.3.....65.59....1.2.6...5.9.;3169;166755 SER = 10.6

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 1 2 3 ! 49 45 6 ! 45789 478 45789 ! ! 467 457 479 ! 149 8 1349 ! 2 134 134569 ! ! 468 458 489 ! 7 1345 2 ! 459 134 134569 ! +----------------------+----------------------+----------------------+ ! 2 3 478 ! 1468 1467 1478 ! 45789 1478 145789 ! ! 478 1 6 ! 5 9 478 ! 3 2 478 ! ! 9 478 5 ! 3 2 1478 ! 478 6 1478 ! +----------------------+----------------------+----------------------+ ! 3 478 12478 ! 12489 147 14789 ! 6 5 478 ! ! 5 9 478 ! 468 3467 3478 ! 1 3478 2 ! ! 478 6 12478 ! 1248 1347 5 ! 478 9 3478 ! +----------------------+----------------------+----------------------+ 185 candidates.

by **totuan** » Tue Mar 14, 2023 8:24 am

Code: Select all: *-----------------------------------------------------------------------------* | 1 2 3 | 49 45 6 | 45789 478 45789 | | 467 457 479 | 149 8 1349 | 2 134 134569 | | 468 458 489 | 7 1345 2 | 459 134 134569 | |-------------------------+-------------------------+-------------------------| | 2 3 *478# | 1468 1467 1478 | 59 *1478 59 | |*478# 1 6 | 5 9 478# | 3 2 *478# | | 9 *478# 5 | 3 2 1478# |*478# 6 1478 | |-------------------------+-------------------------+-------------------------| | 3 *478# 12 | 12489 147 14789 | 6 5 *478# | | 5 9 *478# | 468 3467 3478# | 1 *478-3 2 | |*478# 6 12 | 1248 1347 5 |*478# 9 3478# | *-----------------------------------------------------------------------------*

My path for this one:
Tridagon (478) * marked cells => (1)r4c8=(3)r8c8
Impossible pattern (478) # marked cells => (1)r6c6=(3)r8c6/r9c9
01: (3)r8c6/r9c9==(1)r6c6-r6c9=r4c8-(1=34)r23c8 => r8c8<>3, r9c9=3
02: Tridagon (478) => r4c8=1, some singles
Impossible pattern (478):

Hidden Text: Show

Code: Select all: *--------------------------------------------------------------------* | 1 2 3 | 49 45 6 | 78 78 59 | | 467 5 479 | 19 8 39 | 2 34 16 | | 468 48 489 | 7 135 2 | 59 34 16 | |----------------------+----------------------+----------------------| | 2 3 *478 | 468 467 478 | 59 1 59 | |*478 1 6 | 5 9 *478 | 3 2 *478 | | 9 *478 5 | 3 2 1 |*478 6 *478 | |----------------------+----------------------+----------------------| | 3 *478 12 | 12489 147 4789 | 6 5 *478 | | 5 9 *478 | 468 3467 *478+3 | 1 *78 2 | |*478 6 12 | 1248 147 5 |*478 9 3 | *--------------------------------------------------------------------*

03: Impossible pattern (478) * marked cells => r8c6=3, some singles
Impossible pattern (478):

Hidden Text: Show

Add: or eleven’s impossible pattern - form abc rectangle

Hidden Text: Show

Code: Select all: *--------------------------------------------------* | 1 2 3 | 4 5 6 | 78 78 9 | | 47 5 47 | 1 8 9 | 2 3 6 | | 6 8 9 | 7 3 2 | 5 4 1 | |----------------+----------------+----------------| | 2 3 478 | 68 467 478 | 9 1 5 | | 478 1 6 | 5 9 478 | 3 2 47 | | 9 a47 5 | 3 2 1 | 78-4 6 478 | |----------------+----------------+----------------| | 3 b47 2 | 9 1 478 | 6 5 c478 | | 5 9 478 | 68 467 3 | 1 78 2 | | 478 6 1 | 2 47 5 |d478 9 3 | *--------------------------------------------------*

04: 4’s abcd => r6c7<>4, stte

Thanks for the puzzle!
totuan

Website · by **denis_berthier** » Tue Mar 14, 2023 4:07 pm

;
It seems we find the same patterns, in different order.

Code: Select all: hidden-pairs-in-a-column: c3{n1 n2}{r7 r9} ==> r9c3≠8, r9c3≠7, r9c3≠4, r7c3≠8, r7c3≠7, r7c3≠4 hidden-pairs-in-a-row: r4{n5 n9}{c7 c9} ==> r4c9≠8, r4c9≠7, r4c9≠4, r4c9≠1, r4c7≠8, r4c7≠7, r4c7≠4 finned-x-wing-in-rows: n3{r9 r3}{c5 c9} ==> r2c9≠3

We now find the 3 ORk-relations that will be useful (among more useless ones):

Code: Select all: OR2-anti-tridagon[12] for digits 7, 8 and 4 in blocks: b4, with cells (marked #): r4c3, r5c1, r6c2 b6, with cells (marked #): r4c8, r5c9, r6c7 b7, with cells (marked #): r8c3, r9c1, r7c2 b9, with cells (marked #): r8c8, r9c7, r7c9 with 2 guardians (in cells marked @): n1r4c8 n3r8c8 +----------------------+----------------------+----------------------+ ! 1 2 3 ! 49 45 6 ! 45789 478 45789 ! ! 467 457 479 ! 149 8 1349 ! 2 134 14569 ! ! 468 458 489 ! 7 1345 2 ! 459 134 134569 ! +----------------------+----------------------+----------------------+ ! 2 3 478# ! 1468 1467 1478 ! 59 1478#@ 59 ! ! 478# 1 6 ! 5 9 478 ! 3 2 478# ! ! 9 478# 5 ! 3 2 1478 ! 478# 6 1478 ! +----------------------+----------------------+----------------------+ ! 3 478# 12 ! 12489 147 14789 ! 6 5 478# ! ! 5 9 478# ! 468 3467 3478 ! 1 3478#@ 2 ! ! 478# 6 12 ! 1248 1347 5 ! 478# 9 3478 ! +----------------------+----------------------+----------------------+ EL13c290-OR2-relation for digits: 4, 7 and 8 in cells (marked #): (r4c3 r6c6 r6c7 r6c2 r5c6 r5c9 r5c1 r9c7 r9c1 r7c9 r7c2 r8c6 r8c3) with 2 guardians (in cells marked @) : n1r6c6 n3r8c6 +----------------------+----------------------+----------------------+ ! 1 2 3 ! 49 45 6 ! 45789 478 45789 ! ! 467 457 479 ! 149 8 1349 ! 2 134 14569 ! ! 468 458 489 ! 7 1345 2 ! 459 134 134569 ! +----------------------+----------------------+----------------------+ ! 2 3 478# ! 1468 1467 1478 ! 59 1478 59 ! ! 478# 1 6 ! 5 9 478# ! 3 2 478# ! ! 9 478# 5 ! 3 2 1478#@ ! 478# 6 1478 ! +----------------------+----------------------+----------------------+ ! 3 478# 12 ! 12489 147 14789 ! 6 5 478# ! ! 5 9 478# ! 468 3467 3478#@ ! 1 3478 2 ! ! 478# 6 12 ! 1248 1347 5 ! 478# 9 3478 ! +----------------------+----------------------+----------------------+ EL14c13-OR3-relation for digits: 4, 7 and 8 in cells (marked #): (r4c3 r6c7 r6c9 r6c2 r5c6 r5c9 r5c1 r7c9 r7c2 r9c7 r9c1 r8c6 r8c8 r8c3) with 3 guardians (in cells marked @) : n1r6c9 n3r8c6 n3r8c8 +----------------------+----------------------+----------------------+ ! 1 2 3 ! 49 45 6 ! 45789 478 45789 ! ! 467 457 479 ! 149 8 1349 ! 2 134 14569 ! ! 468 458 489 ! 7 1345 2 ! 459 134 134569 ! +----------------------+----------------------+----------------------+ ! 2 3 478# ! 1468 1467 1478 ! 59 1478 59 ! ! 478# 1 6 ! 5 9 478# ! 3 2 478# ! ! 9 478# 5 ! 3 2 1478 ! 478# 6 1478#@ ! +----------------------+----------------------+----------------------+ ! 3 478# 12 ! 12489 147 14789 ! 6 5 478# ! ! 5 9 478# ! 468 3467 3478#@ ! 1 3478#@ 2 ! ! 478# 6 12 ! 1248 1347 5 ! 478# 9 3478 ! +----------------------+----------------------+----------------------+

Trid-OR2-whip[3]: OR2{{n1r4c8 | n3r8c8}} - c8n7{r8 r1} - c8n8{r1 .} ==> r4c8≠4
Trid-OR2-whip[3]: OR2{{n3r8c8 | n1r4c8}} - c8n7{r4 r1} - c8n8{r1 .} ==> r8c8≠4
whip[1]: c8n4{r3 .} ==> r1c7≠4, r1c9≠4, r2c9≠4, r3c7≠4, r3c9≠4
naked-pairs-in-a-column: c7{r3 r4}{n5 n9} ==> r1c7≠9, r1c7≠5
EL13c290-OR2-whip[3]: r2n3{c8 c6} - OR2{{n3r8c6 | n1r6c6}} - c9n1{r6 .} ==> r2c8≠1
z-chain[3]: b2n3{r3c5 r2c6} - r2c8{n3 n4} - r1n4{c8 .} ==> r3c5≠4
Trid-OR2-ctr-whip[3]: r2c8{n4 n3} - r3c8{n3 n1} - OR2{{n1r4c8 n3r8c8 | .}} ==> r1c8≠4
whip[1]: r1n4{c5 .} ==> r2c4≠4, r2c6≠4
naked-pairs-in-a-block: b3{r1c7 r1c8}{n7 n8} ==> r1c9≠8, r1c9≠7
naked-pairs-in-a-block: b3{r1c9 r3c7}{n5 n9} ==> r3c9≠9, r3c9≠5, r2c9≠9, r2c9≠5
hidden-single-in-a-row ==> r2c2=5
Trid-OR2-whip[3]: c6n3{r2 r8} - OR2{{n3r8c8 | n1r4c8}} - b5n1{r4c4 .} ==> r2c6≠1
finned-x-wing-in-rows: n1{r2 r6}{c9 c4} ==> r4c4≠1
EL14c13-OR3-whip[3]: c8n1{r3 r4} - OR3{{n1r6c9 n3r8c8 | n3r8c6}} - r2n3{c6 .} ==> r3c8≠3
x-wing-in-columns: n3{c6 c8}{r2 r8} ==> r8c5≠3
biv-chain[3]: r3c8{n1 n4} - r2c8{n4 n3} - b2n3{r2c6 r3c5} ==> r3c5≠1
singles ==> r2c4=1, r2c9=6, r3c1=6
EL13c290-OR2-whip[3]: r9n3{c9 c5} - OR2{{n3r8c6 | n1r6c6}} - c9n1{r6 .} ==> r3c9≠3

Code: Select all: singles ==> r3c9=1, r3c8=4, r2c8=3, r2c6=9, r1c4=4, r1c5=5, r1c9=9, r3c7=5, r4c7=9, r4c9=5, r3c5=3, r3c2=8, r3c3=9, r8c6=3, r9c9=3, r7c4=9, r9c4=2, r9c3=1, r7c3=2, r4c8=1, r6c6=1, r7c5=1 whip[1]: r6n8{c9 .} ==> r5c9≠8 x-wing-in-columns: n8{c3 c4}{r4 r8} ==> r8c8≠8, r4c6≠8 singles ==> r8c8=7, r1c8=8, r1c7=7 finned-x-wing-in-columns: n4{c2 c9}{r7 r6} ==> r6c7≠4 stte

by **Cenoman** » Tue Mar 14, 2023 9:34 pm

Of course, the most interesting steps in such a puzzle are the first moves.
My first step was not yet at the right level. I propose just a variation for the end.

Code: Select all: +--------------------+-------------------------+--------------------------+ | 1 2 3 | 49 45 6 | 45789 478 45789 | | 467 457 479 | 149 8 1349 | 2 134 134569 | | 468 458 489 | 7 1345 2 | 459 134 134569 | +--------------------+-------------------------+--------------------------+ | 2 3 478# | 1468 1467 1478 | 59 1478# 59 | | 478# 1 6 | 5 9 478# | 3 2 478# | | 9 478# 5 | 3 2 1478# | 478# 6 1478 | +--------------------+-------------------------+--------------------------+ | 3 478# 12 | 12489 147 14789 | 6 5 478# | | 5 9 478# | 468 3467 3478# | 1 3478 2 | | 478# 6 12 | 1248 1347 5 | 478# 9 3478 | +--------------------+-------------------------+--------------------------+

TH(478)b4679 having two guardians (1r4c8, 3r8c8) and Impossible Pattern at cells marked (#) having three guardians (1r4c8, 1r6c6, 3r8c6) 1r4c8 and 1c6c6 are True or False together (both conjugates of 1r6c9) =>one chain on any one is enough. Note that this IP is not minimal, it has cell r4c8 in excess wrt EL13c290 used by Denis.
1. (3)r8c6 == (1)r6c6 - r6c9 = r4c8 - (1=43)r23c8 => -3 r8c8; 1 placement

2. TH(478)b4679 has now a single guardian => +1 r4c8; lcls, 3 placements

Code: Select all: +--------------------+------------------------+-------------------+ | 1 2 3 | 49 45 6 | 78 78 59 | | 467 5 479 | 19 8 39 | 2 34 16 | | 468 48 489 | 7 135 2 | 59 34 16 | +--------------------+------------------------+-------------------+ | 2 3 478* | 468 467 478 | 59 1 59 | | 478 1 6 | 5 9 478* | 3 2 478 | | 9 478 5 | 3 2 1 | 478 6 478 | +--------------------+------------------------+-------------------+ | 3 478 12 | 12489 147 4789 | 6 5 478 | | 5 9 478# | 468 3467 3-478 | 1 78# 2 | | 478 6 12 | 1248 147 5 | 478 9 3 | +--------------------+------------------------+-------------------+

3. TH(478)b4679 => RT(478)r4c3, r8c38. Look at ER(4,7,8)b5 => r4c3 and r5c6 have the same digit => RT(478)r5c6, r8c38 => -478 r8c6; 22 placements

4. End as above: Finned X-Wing (4)c29\r67, fr5c9 => -4 r6c7; ste

Website · by **denis_berthier** » Wed Mar 15, 2023 3:36 am

.
In my above solution, I used two different patterns: EL13c290 and EL14c13 (in addition to tridagon). The max length of chains was 3.
(Note that those two "useful" patterns were found among the full set of 630.)

The two patterns are very close to tridagon:
EL13c290: 2 cells missing, 3 added:

Code: Select all: +-------+-------+-------+ ! . . . ! . . - ! . . X ! ! . . Z ! . X . ! . X . ! ! . . Z ! X . . ! X . . ! +-------+-------+-------+ ! . . Z ! . . - ! . . X ! ! . . . ! X . . ! . X . ! ! . . . ! . X . ! X . . ! +-------+-------+-------+ ! o o . ! . . o ! . . . ! ! o o . ! . . o ! . . . ! ! o o . ! . . o ! . . . ! +-------+-------+-------+

EL14c13: 1 cell missing, 3 added

Code: Select all: +-------+-------+-------+ ! . . Z ! . . X ! . . X ! ! . . . ! . X . ! . X . ! ! . . . ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! . . - ! . . X ! ! . . . ! X Z . ! . X . ! ! . . Z ! . X . ! X . . ! +-------+-------+-------+ ! o o . ! . . . ! . . . ! ! o o . ! . . . ! . . . ! ! o o . ! . . . ! . . . ! +-------+-------+-------+

One can also solve the puzzle using only one of the two patterns (still in addition to tridagon):
- using only EL13c290, as Cenoman, but then max length is 6,
- using only EL14c13, but then max length is 7.

Starting from the point where the ORk-relations are found:

using only EL13c290: Show

Code: Select all: Trid-OR2-whip[3]: OR2{{n1r4c8 | n3r8c8}} - c8n7{r8 r1} - c8n8{r1 .} ==> r4c8≠4 Trid-OR2-whip[3]: OR2{{n3r8c8 | n1r4c8}} - c8n7{r4 r1} - c8n8{r1 .} ==> r8c8≠4 whip[1]: c8n4{r3 .} ==> r1c7≠4, r1c9≠4, r2c9≠4, r3c7≠4, r3c9≠4 naked-pairs-in-a-column: c7{r3 r4}{n5 n9} ==> r1c7≠9, r1c7≠5 EL13c290-OR2-whip[3]: r2n3{c8 c6} - OR2{{n3r8c6 | n1r6c6}} - c9n1{r6 .} ==> r2c8≠1 z-chain[3]: b2n3{r3c5 r2c6} - r2c8{n3 n4} - r1n4{c8 .} ==> r3c5≠4 Trid-OR2-ctr-whip[3]: r2c8{n4 n3} - r3c8{n3 n1} - OR2{{n1r4c8 n3r8c8 | .}} ==> r1c8≠4 whip[1]: r1n4{c5 .} ==> r2c4≠4, r2c6≠4 naked-pairs-in-a-block: b3{r1c7 r1c8}{n7 n8} ==> r1c9≠8, r1c9≠7 naked-pairs-in-a-block: b3{r1c9 r3c7}{n5 n9} ==> r3c9≠9, r3c9≠5, r2c9≠9, r2c9≠5 hidden-single-in-a-row ==> r2c2=5 Trid-OR2-whip[3]: c6n3{r2 r8} - OR2{{n3r8c8 | n1r4c8}} - b5n1{r4c4 .} ==> r2c6≠1 finned-x-wing-in-rows: n1{r2 r6}{c9 c4} ==> r4c4≠1 Trid-OR2-whip[4]: r6n1{c6 c9} - OR2{{n1r4c8 | n3r8c8}} - c6n3{r8 r2} - c6n9{r2 .} ==> r7c6≠1 whip[1]: c6n1{r6 .} ==> r4c5≠1 EL13c290-OR2-whip[4]: r4n1{c8 c6} - OR2{{n1r6c6 | n3r8c6}} - r8c8{n3 n8} - r1c8{n8 .} ==> r4c8≠7 z-chain[2]: c2n7{r7 r6} - b6n7{r6c7 .} ==> r7c9≠7 EL13c290-OR2-whip[4]: r4n1{c8 c6} - OR2{{n1r6c6 | n3r8c6}} - r8c8{n3 n7} - r1c8{n7 .} ==> r4c8≠8 singles ==> r4c8=1, r6c6=1 naked-pairs-in-a-block: b3{r2c8 r3c8}{n3 n4} ==> r3c9≠3 hidden-single-in-a-column ==> r9c9=3 whip[1]: c9n7{r6 .} ==> r6c7≠7 z-chain[3]: r8n3{c6 c5} - c5n6{r8 r4} - c5n7{r4 .} ==> r8c6≠7 biv-chain[4]: r1c4{n4 n9} - r2c6{n9 n3} - r8n3{c6 c5} - b8n6{r8c5 r8c4} ==> r8c4≠4 t-whip[6]: r7c9{n8 n4} - b6n4{r6c9 r6c7} - c2n4{r6 r3} - r2n4{c3 c8} - r2n3{c8 c6} - c6n9{r2 .} ==> r7c6≠8 z-chain[5]: r8n3{c5 c6} - r2c6{n3 n9} - r7c6{n9 n4} - b9n4{r7c9 r9c7} - b9n7{r9c7 .} ==> r8c5≠7 whip[5]: c5n3{r8 r3} - r2c6{n3 n9} - r7c6{n9 n7} - r9c5{n7 n1} - r7c5{n1 .} ==> r8c5≠4 finned-x-wing-in-rows: n4{r8 r4}{c3 c6} ==> r5c6≠4 whip[1]: b5n4{r4c6 .} ==> r4c3≠4 biv-chain[2]: c7n4{r9 r6} - r5n4{c9 c1} ==> r9c1≠4 biv-chain[3]: r8n4{c3 c6} - c6n3{r8 r2} - r2c8{n3 n4} ==> r2c3≠4 biv-chain[3]: r4c3{n8 n7} - c2n7{r6 r7} - b7n4{r7c2 r8c3} ==> r8c3≠8 biv-chain[3]: r8c3{n4 n7} - r8c8{n7 n8} - r7c9{n8 n4} ==> r7c2≠4 singles ==> r8c3=4, r8c8=7, r1c8=8, r1c7=7 whip[1]: r8n8{c6 .} ==> r7c4≠8, r9c4≠8 hidden-pairs-in-a-column: c4{n6 n8}{r4 r8} ==> r4c4≠4 finned-x-wing-in-rows: n8{r7 r6}{c2 c9} ==> r5c9≠8 whip[1]: b6n8{r6c9 .} ==> r6c2≠8 biv-chain[3]: r9n8{c1 c7} - r7c9{n8 n4} - r5n4{c9 c1} ==> r5c1≠8 stte

using only EL14c13: Show

Code: Select all: Trid-OR2-whip[3]: OR2{{n1r4c8 | n3r8c8}} - c8n7{r8 r1} - c8n8{r1 .} ==> r4c8≠4 Trid-OR2-whip[3]: OR2{{n3r8c8 | n1r4c8}} - c8n7{r4 r1} - c8n8{r1 .} ==> r8c8≠4 whip[1]: c8n4{r3 .} ==> r1c7≠4, r1c9≠4, r2c9≠4, r3c7≠4, r3c9≠4 naked-pairs-in-a-column: c7{r3 r4}{n5 n9} ==> r1c7≠9, r1c7≠5 Trid-OR2-whip[3]: c6n3{r2 r8} - OR2{{n3r8c8 | n1r4c8}} - b5n1{r4c4 .} ==> r2c6≠1 z-chain[4]: b2n1{r2c4 r3c5} - b2n3{r3c5 r2c6} - c6n9{r2 r7} - c6n1{r7 .} ==> r4c4≠1 Trid-OR2-whip[4]: b5n1{r6c6 r4c5} - OR2{{n1r4c8 | n3r8c8}} - c6n3{r8 r2} - c6n9{r2 .} ==> r7c6≠1 +-------------------+-------------------+-------------------+ ! 1 2 3 ! 49 45 6 ! 78 478 5789 ! ! 467 457 479 ! 149 8 349 ! 2 134 1569 ! ! 468 458 489 ! 7 1345 2 ! 59 134 13569 ! +-------------------+-------------------+-------------------+ ! 2 3 478 ! 468 1467 1478 ! 59 178 59 ! ! 478 1 6 ! 5 9 478 ! 3 2 478 ! ! 9 478 5 ! 3 2 1478 ! 478 6 1478 ! +-------------------+-------------------+-------------------+ ! 3 478 12 ! 12489 147 4789 ! 6 5 478 ! ! 5 9 478 ! 468 3467 3478 ! 1 378 2 ! ! 478 6 12 ! 1248 1347 5 ! 478 9 3478 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL14c13-OR5-relation between candidates n3r9c9 n1r7c6 n9r7c6 n1r4c6 n1r4c8 has just been eliminated. There remains an EL14c13-OR4-relation between candidates: n3r9c9 n9r7c6 n1r4c6 n1r4c8 whip[1]: c6n1{r6 .} ==> r4c5≠1 Trid-OR2-whip[4]: c9n3{r3 r9} - OR2{{n3r8c8 | n1r4c8}} - b3n1{r2c8 r2c9} - c9n6{r2 .} ==> r3c9≠9 Trid-OR2-whip[4]: c9n3{r3 r9} - OR2{{n3r8c8 | n1r4c8}} - b3n1{r2c8 r2c9} - c9n6{r2 .} ==> r3c9≠5 Trid-OR2-whip[4]: r2n3{c6 c8} - OR2{{n3r8c8 | n1r4c8}} - r3c8{n1 n4} - b1n4{r3c1 .} ==> r2c6≠4 Trid-OR2-whip[4]: c9n6{r2 r3} - c9n3{r3 r9} - OR2{{n3r8c8 | n1r4c8}} - b3n1{r2c8 .} ==> r2c9≠9 Trid-OR2-whip[4]: c9n6{r2 r3} - c9n3{r3 r9} - OR2{{n3r8c8 | n1r4c8}} - b3n1{r2c8 .} ==> r2c9≠5 hidden-single-in-a-row ==> r2c2=5 hidden-pairs-in-a-block: b3{n5 n9}{r1c9 r3c7} ==> r1c9≠8, r1c9≠7 hidden-pairs-in-a-block: b3{n7 n8}{r1c7 r1c8} ==> r1c8≠4 whip[1]: r1n4{c5 .} ==> r2c4≠4, r3c5≠4 EL14c13-OR3-whip[4]: OR3{{n3r8c8 n3r8c6 | n1r6c9}} - r2c9{n1 n6} - r3c9{n6 n3} - r9n3{c9 .} ==> r8c5≠3 x-wing-in-columns: n3{c5 c9}{r3 r9} ==> r3c8≠3 biv-chain[3]: r3c8{n4 n1} - r2c9{n1 n6} - b1n6{r2c1 r3c1} ==> r3c1≠4 biv-chain[3]: b2n1{r2c4 r3c5} - r3n3{c5 c9} - b3n6{r3c9 r2c9} ==> r2c9≠1 singles ==> r2c9=6, r3c1=6 +-------------------+-------------------+-------------------+ ! 1 2 3 ! 49 45 6 ! 78 78 59 ! ! 47 5 479 ! 19 8 39 ! 2 134 6 ! ! 6 48 489 ! 7 135 2 ! 59 14 13 ! +-------------------+-------------------+-------------------+ ! 2 3 478 ! 468 467 1478 ! 59 178 59 ! ! 478 1 6 ! 5 9 478 ! 3 2 478 ! ! 9 478 5 ! 3 2 1478 ! 478 6 1478 ! +-------------------+-------------------+-------------------+ ! 3 478 12 ! 12489 147 4789 ! 6 5 478 ! ! 5 9 478 ! 468 467 3478 ! 1 378 2 ! ! 478 6 12 ! 1248 1347 5 ! 478 9 3478 ! +-------------------+-------------------+-------------------+ EL14c13-OR3-relation between candidates n1r6c9, n3r8c6 and n3r8c8 + same valence for candidates n1r6c9 and n3r8c8 via c-chain[4]: n1r6c9,n1r3c9,n3r3c9,n3r2c8,n3r8c8 ==> EL14c13-OR3-relation can be split into two EL14c13-OR2-relations with respective lists of guardians: n3r8c6 n3r8c8 and n1r6c9 n3r8c6 . +-------------------+-------------------+-------------------+ ! 1 2 3 ! 49 45 6 ! 78 78 59 ! ! 47 5 479 ! 19 8 39 ! 2 134 6 ! ! 6 48 489 ! 7 135 2 ! 59 14 13 ! +-------------------+-------------------+-------------------+ ! 2 3 478 ! 468 467 1478 ! 59 178 59 ! ! 478 1 6 ! 5 9 478 ! 3 2 478 ! ! 9 478 5 ! 3 2 1478 ! 478 6 1478 ! +-------------------+-------------------+-------------------+ ! 3 478 12 ! 12489 147 4789 ! 6 5 478 ! ! 5 9 478 ! 468 467 3478 ! 1 378 2 ! ! 478 6 12 ! 1248 1347 5 ! 478 9 3478 ! +-------------------+-------------------+-------------------+ EL14c13-OR4-relation between candidates n3r9c9, n9r7c6, n1r4c6 and n1r4c8 + same valence for candidates n1r4c8 and n3r9c9 via c-chain[4]: n1r4c8,n1r6c9,n1r3c9,n3r3c9,n3r9c9 ==> EL14c13-OR4-relation can be split into two EL14c13-OR3-relations with respective lists of guardians: n3r9c9 n9r7c6 n1r4c6 and n9r7c6 n1r4c6 n1r4c8 . biv-chain[5]: r9c3{n1 n2} - r7n2{c3 c4} - r7n9{c4 c6} - r2c6{n9 n3} - b8n3{r8c6 r9c5} ==> r9c5≠1 hidden-pairs-in-a-row: r9{n1 n2}{c3 c4} ==> r9c4≠8, r9c4≠4 t-whip[6]: b9n4{r9c9 r7c9} - b6n4{r6c9 r6c7} - c2n4{r6 r3} - c8n4{r3 r2} - r2n3{c8 c6} - c5n3{r3 .} ==> r9c5≠4 biv-chain[3]: r9c5{n7 n3} - c6n3{r8 r2} - c6n9{r2 r7} ==> r7c6≠7 whip[6]: r8n3{c8 c6} - r2c6{n3 n9} - r7n9{c6 c4} - c4n8{r7 r4} - c3n8{r4 r3} - c3n9{r3 .} ==> r8c8≠8 z-chain[2]: c1n8{r5 r9} - b9n8{r9c7 .} ==> r5c9≠8 whip[6]: r5c9{n4 n7} - r6c7{n7 n8} - r6c2{n8 n7} - r7n7{c2 c5} - c5n1{r7 r3} - c9n1{r3 .} ==> r6c9≠4 whip[6]: b2n4{r1c5 r1c4} - r8n4{c4 c3} - r4n4{c3 c6} - c6n1{r4 r6} - c9n1{r6 r3} - c5n1{r3 .} ==> r7c5≠4 t-whip[6]: c5n7{r9 r4} - c5n6{r4 r8} - c5n4{r8 r1} - r1c4{n4 n9} - r2n9{c4 c3} - c3n7{r2 .} ==> r8c6≠7 whip[1]: b8n7{r9c5 .} ==> r4c5≠7 biv-chain[3]: r1n4{c4 c5} - r4c5{n4 n6} - b8n6{r8c5 r8c4} ==> r8c4≠4 t-whip[7]: b9n8{r9c9 r9c7} - c7n4{r9 r6} - r5c9{n4 n7} - r7c9{n7 n4} - c2n4{r7 r3} - r3c8{n4 n1} - c9n1{r3 .} ==> r6c9≠8 whip[1]: c9n8{r9 .} ==> r9c7≠8 biv-chain[4]: c9n8{r7 r9} - c9n3{r9 r3} - r2n3{c8 c6} - c6n9{r2 r7} ==> r7c6≠8 z-chain[4]: r4c5{n4 n6} - r4c4{n6 n8} - b6n8{r4c8 r6c7} - b6n4{r6c7 .} ==> r5c6≠4 finned-x-wing-in-rows: n4{r5 r9}{c1 c9} ==> r7c9≠4 whip[1]: b9n4{r9c9 .} ==> r9c1≠4 biv-chain[3]: r9c1{n8 n7} - r2c1{n7 n4} - r3c2{n4 n8} ==> r7c2≠8 biv-chain[2]: r5n8{c6 c1} - b7n8{r9c1 r8c3} ==> r8c6≠8 whip[1]: c6n8{r6 .} ==> r4c4≠8 naked-pairs-in-a-block: b5{r4c4 r4c5}{n4 n6} ==> r6c6≠4, r4c6≠4 whip[1]: c6n4{r8 .} ==> r7c4≠4, r8c5≠4 whip[1]: b5n4{r4c5 .} ==> r4c3≠4 biv-chain[3]: b1n7{r2c1 r2c3} - r4c3{n7 n8} - b7n8{r8c3 r9c1} ==> r9c1≠7 stte

by **marek stefanik** » Sat Mar 18, 2023 10:59 am

We can use a remote pentuple.

Code: Select all: .---------------.--------------------.---------------------. | 1 2 3 | 49 45 6 | 45789 478 45789 | | 467 457 479 | 149 8 1349 | 2 134 134569 | | 468 458 489 | 7 1345 2 | 459 134 134569 | :---------------+--------------------+---------------------: | 2 3 #478 | 1468 1467 1478 | 59 #1478 59 | |*478 1 6 | 5 9 478 | 3 2 *478 | | 9 *478 5 | 3 2 1478 |*478 6 1478 | :---------------+--------------------+---------------------: | 3 *478 12 | 12489 147 14789 | 6 5 *478 | | 5 9 #478 | 468 3467 #3478 | 1 #3478 2 | |*478 6 12 | 1248 1347 5 |*478 9 3478 | '---------------'--------------------'---------------------'

TH 8-loop 478* => each of 478 must appear in b49* and in b67*
remote pentuple 13478#
Can only contain one 1 (in r4c8) and one 3 (in r8c68).
For 478:
r8c6 sees r4c38 via b5,
r4c3 sees r8c8 via b49*,
r4c8 sees r8c3 via b67*,
remaining pairs of cells see each other directly.
Therefore each digit can only appear in #-marked cells once and all five of them are needed.

We can place 1r4c8 (only cell in #).
After basics, we get 3r8c6 (only cell in #).
After basics, 8c3 is forced into # => we can eliminate other 8 candidates in # (–8r8c8).
Also 4# is forced into c3 => we can eliminate other 4 candidates in c3 (–4r2c3).

Finally, we arrive here:

Code: Select all: .------------.--------------.------------. | 1 2 3 | 4 5 6 | 7 8 9 | | 4 5 7 | 1 8 9 | 2 3 6 | | 6 8 9 | 7 3 2 | 5 4 1 | :------------+--------------+------------: | 2 3 48 | 68 467 478 | 9 1 5 | | 78 1 6 | 5 9 48 | 3 2 47 | | 9 47 5 | 3 2 1 | 48 6 478 | :------------+--------------+------------: | 3 a47 2 | 9 1 478 | 6 5 8–4 | | 5 9 48 | 68 46 3 | 1 7 2 | |b78 6 1 | 2 47 5 |c48 9 3 | '------------'--------------'------------'

(4=7)r7c2 – (7=8)r9c1 – (8=4)r9c7 => –4r7c9, stte

Marek

The New Sudoku Players' Forum

#13975 in 158,276 T&E(3) min-expands

#13975 in 158,276 T&E(3) min-expands

Re: #13975 in 158,276 T&E(3) min-expands

Re: #13975 in 158,276 T&E(3) min-expands

Re: #13975 in 158,276 T&E(3) min-expands

Re: #13975 in 158,276 T&E(3) min-expands

Re: #13975 in 158,276 T&E(3) min-expands