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by **yzfwsf** » Sat Mar 11, 2023 6:08 am

Code: Select all: ...4.6...4..18..6368..23...2.8......56..1....7913.......624..18...6..3.2..2.3.64.

18 Cells Impossible Pattern

Code: Select all: ....1.....11..1.....11...1....1..11....1....1..........1...11....1.1..1..........

Please ask the experts to identify if this puzzle cannot be solved using the Impossible Pattern forcing chain.

Website · by **denis_berthier** » Sat Mar 11, 2023 5:09 pm

.
It can be solved by using two impossible patterns (in addition to tridagon): EL14c1 and EL10c28.
Note that there are lots more patterns available, but none that gives shorter chains.

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 139 12357 3579 ! 4 579 6 ! 25789 25789 579 ! ! 4 257 579 ! 1 8 579 ! 2579 6 3 ! ! 6 8 579 ! 579 2 3 ! 14579 579 14579 ! +----------------------+----------------------+----------------------+ ! 2 34 8 ! 579 5679 4579 ! 14579 3579 145679 ! ! 5 6 34 ! 789 1 24789 ! 24789 23789 479 ! ! 7 9 1 ! 3 56 2458 ! 2458 258 456 ! +----------------------+----------------------+----------------------+ ! 39 357 6 ! 2 4 579 ! 579 1 8 ! ! 189 1457 4579 ! 6 579 15789 ! 3 579 2 ! ! 189 157 2 ! 5789 3 15789 ! 6 4 579 ! +----------------------+----------------------+----------------------+ 178 candidates.

hidden-pairs-in-a-row: r3{n1 n4}{c7 c9} ==> r3c9≠9, r3c9≠7, r3c9≠5, r3c7≠9, r3c7≠7, r3c7≠5

The 3 patterns are identified here:

Code: Select all: OR2-anti-tridagon[12] for digits 5, 7 and 9 in blocks: b2, with cells (marked #): r1c5, r2c6, r3c4 b3, with cells (marked #): r1c9, r2c7, r3c8 b8, with cells (marked #): r8c5, r7c6, r9c4 b9, with cells (marked #): r8c8, r7c7, r9c9 with 2 guardians (in cells marked @): n2r2c7 n8r9c4 +----------------------+----------------------+----------------------+ ! 139 12357 3579 ! 4 579# 6 ! 25789 25789 579# ! ! 4 257 579 ! 1 8 579# ! 2579#@ 6 3 ! ! 6 8 579 ! 579# 2 3 ! 14 579# 14 ! +----------------------+----------------------+----------------------+ ! 2 34 8 ! 579 5679 4579 ! 14579 3579 145679 ! ! 5 6 34 ! 789 1 24789 ! 24789 23789 479 ! ! 7 9 1 ! 3 56 2458 ! 2458 258 456 ! +----------------------+----------------------+----------------------+ ! 39 357 6 ! 2 4 579# ! 579# 1 8 ! ! 189 1457 4579 ! 6 579# 15789 ! 3 579# 2 ! ! 189 157 2 ! 5789#@ 3 15789 ! 6 4 579# ! +----------------------+----------------------+----------------------+ EL14c1s-OR3-relation for digits: 5, 7 and 9 in cells (marked #): (r4c8 r4c4 r4c6 r2c6 r3c8 r3c4 r1c9 r1c5 r8c8 r8c5 r9c9 r9c4 r7c7 r7c6) with 3 guardians (in cells marked @) : n3r4c8 n4r4c6 n8r9c4 +----------------------+----------------------+----------------------+ ! 139 12357 3579 ! 4 579# 6 ! 25789 25789 579# ! ! 4 257 579 ! 1 8 579# ! 2579 6 3 ! ! 6 8 579 ! 579# 2 3 ! 14 579# 14 ! +----------------------+----------------------+----------------------+ ! 2 34 8 ! 579# 5679 4579#@ ! 14579 3579#@ 145679 ! ! 5 6 34 ! 789 1 24789 ! 24789 23789 479 ! ! 7 9 1 ! 3 56 2458 ! 2458 258 456 ! +----------------------+----------------------+----------------------+ ! 39 357 6 ! 2 4 579# ! 579# 1 8 ! ! 189 1457 4579 ! 6 579# 15789 ! 3 579# 2 ! ! 189 157 2 ! 5789#@ 3 15789 ! 6 4 579# ! +----------------------+----------------------+----------------------+ EL10c28-OR4-relation for digits: 5, 7 and 9 in cells (marked #): (r3c3 r3c8 r2c6 r2c3 r2c7 r9c6 r7c6 r8c5 r8c3 r8c8) with 4 guardians (in cells marked @) : n2r2c7 n1r9c6 n8r9c6 n4r8c3 +-------------------------+-------------------------+-------------------------+ ! 139 12357 3579 ! 4 579 6 ! 25789 25789 579 ! ! 4 257 579# ! 1 8 579# ! 2579#@ 6 3 ! ! 6 8 579# ! 579 2 3 ! 14 579# 14 ! +-------------------------+-------------------------+-------------------------+ ! 2 34 8 ! 579 5679 4579 ! 14579 3579 145679 ! ! 5 6 34 ! 789 1 24789 ! 24789 23789 479 ! ! 7 9 1 ! 3 56 2458 ! 2458 258 456 ! +-------------------------+-------------------------+-------------------------+ ! 39 357 6 ! 2 4 579# ! 579 1 8 ! ! 189 1457 4579#@ ! 6 579# 15789 ! 3 579# 2 ! ! 189 157 2 ! 5789 3 15789#@ ! 6 4 579 ! +-------------------------+-------------------------+-------------------------+

Trid-OR2-whip[4]: r8n8{c1 c6} - OR2{{n8r9c4 | n2r2c7}} - c2n2{r2 r1} - r1n1{c2 .} ==> r8c1≠1
t-whip[5]: r6n8{c8 c6} - c6n2{r6 r5} - c6n4{r5 r4} - r4c2{n4 n3} - r5n3{c3 .} ==> r5c8≠8
EL14c1s-OR3-whip[5]: r6n4{c9 c6} - c6n2{r6 r5} - b5n8{r5c6 r5c4} - OR3{{n8r9c4 n4r4c6 | n3r4c8}} - r4c2{n3 .} ==> r4c9≠4, r4c7≠4
whip[7]: r1n8{c8 c7} - r1n2{c7 c2} - r1n1{c2 c1} - r1n3{c1 c3} - r5n3{c3 c8} - c8n2{r5 r6} - c8n8{r6 .} ==> r1c8≠5
whip[7]: r1n8{c8 c7} - r1n2{c7 c2} - r1n1{c2 c1} - r1n3{c1 c3} - r5n3{c3 c8} - c8n2{r5 r6} - c8n8{r6 .} ==> r1c8≠7
whip[7]: r1n8{c8 c7} - r1n2{c7 c2} - r1n1{c2 c1} - r1n3{c1 c3} - r5n3{c3 c8} - c8n2{r5 r6} - c8n8{r6 .} ==> r1c8≠9
whip[7]: c9n6{r4 r6} - r6c5{n6 n5} - r4c4{n5 n9} - r4c6{n9 n4} - b4n4{r4c2 r5c3} - c9n4{r5 r3} - c9n1{r3 .} ==> r4c9≠7
whip[7]: c9n6{r4 r6} - r6c5{n6 n5} - r4c4{n5 n7} - r4c6{n7 n4} - b4n4{r4c2 r5c3} - c9n4{r5 r3} - c9n1{r3 .} ==> r4c9≠9
Trid-OR2-ctr-whip[7]: c6n2{r5 r6} - c6n4{r6 r4} - b4n4{r4c2 r5c3} - r5n3{c3 c8} - r5n2{c8 c7} - r5n8{c7 c4} - OR2{{n8r9c4 n2r2c7 | .}} ==> r5c6≠7, r5c6≠9
Trid-OR2-whip[7]: c6n2{r5 r6} - b5n8{r6c6 r5c4} - OR2{{n8r9c4 | n2r2c7}} - c2n2{r2 r1} - b1n1{r1c2 r1c1} - r1n3{c1 c3} - r5c3{n3 .} ==> r5c6≠4
biv-chain[4]: r5n3{c8 c3} - b4n4{r5c3 r4c2} - c6n4{r4 r6} - b5n2{r6c6 r5c6} ==> r5c8≠2
hidden-pairs-in-a-column: c8{n2 n8}{r1 r6} ==> r6c8≠5
Trid-OR2-whip[5]: r6c8{n2 n8} - r1c8{n8 n2} - OR2{{n2r2c7 | n8r9c4}} - r5n8{c4 c6} - r5n2{c6 .} ==> r6c7≠2
whip[6]: r8n1{c6 c2} - c2n4{r8 r4} - c6n4{r4 r6} - c6n2{r6 r5} - c6n8{r5 r9} - r9n1{c6 .} ==> r8c6≠5
whip[6]: r8n1{c6 c2} - c2n4{r8 r4} - c6n4{r4 r6} - c6n2{r6 r5} - c6n8{r5 r9} - r9n1{c6 .} ==> r8c6≠7
whip[6]: r8n1{c6 c2} - c2n4{r8 r4} - c6n4{r4 r6} - c6n2{r6 r5} - c6n8{r5 r9} - r9n1{c6 .} ==> r8c6≠9
Trid-OR2-whip[6]: r4c2{n3 n4} - c6n4{r4 r6} - c6n2{r6 r5} - b5n8{r5c6 r5c4} - OR2{{n8r9c4 | n2r2c7}} - c2n2{r2 .} ==> r1c2≠3
biv-chain[4]: c1n1{r9 r1} - c1n3{r1 r7} - c2n3{r7 r4} - c2n4{r4 r8} ==> r8c2≠1
singles ==> r8c6=1, r8c1=8

Code: Select all: +-------------------+-------------------+-------------------+ ! 139 1257 3579 ! 4 579 6 ! 25789 28 579 ! ! 4 257 579 ! 1 8 579 ! 2579 6 3 ! ! 6 8 579 ! 579 2 3 ! 14 579 14 ! +-------------------+-------------------+-------------------+ ! 2 34 8 ! 579 5679 4579 ! 1579 3579 156 ! ! 5 6 34 ! 789 1 28 ! 24789 379 479 ! ! 7 9 1 ! 3 56 2458 ! 458 28 456 ! +-------------------+-------------------+-------------------+ ! 39 357 6 ! 2 4 579 ! 579 1 8 ! ! 8 457 4579 ! 6 579 1 ! 3 579 2 ! ! 19 157 2 ! 5789 3 5789 ! 6 4 579 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL10c28-OR4-relation between candidates n2r2c7 n1r9c6 n8r9c6 n4r8c3 has just been eliminated. There remains an EL10c28-OR3-relation between candidates: n2r2c7 n8r9c6 n4r8c3

Trid-OR2-whip[4]: OR2{{n8r9c4 | n2r2c7}} - r1n2{c8 c2} - r1n1{c2 c1} - r9c1{n1 .} ==> r9c4≠9
EL10c28-OR3-whip[5]: c6n2{r5 r6} - b6n2{r6c8 r5c7} - OR3{{n2r2c7 n8r9c6 | n4r8c3}} - r5n4{c3 c9} - r6n4{c7 .} ==> r5c6≠8
singles ==> r5c6=2, r6c8=2, r1c8=8
whip[6]: c1n9{r9 r1} - c5n9{r1 r4} - c4n9{r5 r3} - c8n9{r3 r5} - r5n3{c8 c3} - c3n4{r5 .} ==> r8c3≠9
whip[1]: b7n9{r9c1 .} ==> r1c1≠9
biv-chain[5]: c1n9{r9 r7} - r7n3{c1 c2} - r4c2{n3 n4} - c6n4{r4 r6} - c6n8{r6 r9} ==> r9c6≠9
whip[6]: r1n3{c3 c1} - r7c1{n3 n9} - b8n9{r7c6 r8c5} - b9n9{r8c8 r9c9} - r1c9{n9 n7} - r1c5{n7 .} ==> r1c3≠5
whip[6]: r1n3{c3 c1} - r7c1{n3 n9} - b8n9{r7c6 r8c5} - b9n9{r8c8 r9c9} - r1c9{n9 n5} - r1c5{n5 .} ==> r1c3≠7
Trid-OR2-whip[6]: r5n8{c7 c4} - OR2{{n8r9c4 | n2r2c7}} - c2n2{r2 r1} - b1n1{r1c2 r1c1} - r1n3{c1 c3} - r5c3{n3 .} ==> r5c7≠4
Trid-OR2-whip[7]: c2n3{r4 r7} - c1n3{r7 r1} - r1n1{c1 c2} - c2n2{r1 r2} - OR2{{n2r2c7 | n8r9c4}} - b5n8{r5c4 r6c6} - c6n4{r6 .} ==> r4c2≠4
singles ==> r4c2=3, r5c3=4, r8c2=4, r5c8=3, r1c3=3, r1c1=1, r9c1=9, r7c1=3, r9c2=1, r4c6=4

Code: Select all: +----------------+----------------+----------------+ ! 1 257 3 ! 4 579 6 ! 2579 8 579 ! ! 4 257 579 ! 1 8 579 ! 2579 6 3 ! ! 6 8 579 ! 579 2 3 ! 14 579 14 ! +----------------+----------------+----------------+ ! 2 3 8 ! 579 5679 4 ! 1579 579 156 ! ! 5 6 4 ! 789 1 2 ! 789 3 79 ! ! 7 9 1 ! 3 56 58 ! 458 2 456 ! +----------------+----------------+----------------+ ! 3 57 6 ! 2 4 579 ! 579 1 8 ! ! 8 4 57 ! 6 579 1 ! 3 579 2 ! ! 9 1 2 ! 578 3 578 ! 6 4 57 ! +----------------+----------------+----------------+ At least one candidate of a previous EL10c28-OR3-relation between candidates n2r2c7 n8r9c6 n4r8c3 has just been eliminated. There remains an EL10c28-OR2-relation between candidates: n2r2c7 n8r9c6

finned-x-wing-in-rows: n9{r8 r1}{c5 c8} ==> r3c8≠9
naked-triplets-in-a-column: c9{r1 r5 r9}{n5 n9 n7} ==> r6c9≠5, r4c9≠5
finned-swordfish-in-columns: n9{c5 c8 c9}{r1 r8 r4} ==> r4c7≠9
biv-chain[3]: c9n5{r9 r1} - c9n9{r1 r5} - c8n9{r4 r8} ==> r8c8≠5
biv-chain[3]: b7n7{r7c2 r8c3} - r8c8{n7 n9} - b8n9{r8c5 r7c6} ==> r7c6≠7
EL10c28-OR2-whip[2]: OR2{{n2r2c7 | n8r9c6}} - c6n7{r9 .} ==> r2c7≠7
z-chain[3]: b9n5{r7c7 r9c9} - c4n5{r9 r3} - c8n5{r3 .} ==> r4c7≠5
z-chain[3]: c8n5{r3 r4} - c4n5{r4 r9} - r8n5{c5 .} ==> r3c3≠5
biv-chain[2]: c9n5{r9 r1} - r3n5{c8 c4} ==> r9c4≠5
x-wing-in-columns: n5{c4 c8}{r3 r4} ==> r4c5≠5
z-chain[2]: b8n5{r9c6 r8c5} - c3n5{r8 .} ==> r2c6≠5
biv-chain[3]: r2c6{n7 n9} - r7n9{c6 c7} - r7n7{c7 c2} ==> r2c2≠7
z-chain[3]: c2n7{r1 r7} - r8n7{c3 c8} - b3n7{r3c8 .} ==> r1c5≠7
z-chain[3]: r1c5{n5 n9} - b3n9{r1c9 r2c7} - c7n2{r2 .} ==> r1c7≠5
z-chain[3]: c3n5{r8 r2} - r1n5{c2 c9} - r9n5{c9 .} ==> r8c5≠5
w1-tte

by **marek stefanik** » Sat Mar 11, 2023 11:28 pm

The 18-cell pattern doesn't seem to be helpful – the solution contains only one of its guardians (1r4c7), but even just placing that digit does nothing for the solve.

Code: Select all: .------------------.-------------------.----------------------. | 139 12357 i3–579| 4 #579 6 | 25789 25789 #579 | | 4 a257 a579 | 1 8 #579 |b#579+2 6 3 | | 6 8 a579 |#579 2 3 | 14 #579 14 | :------------------+-------------------+----------------------: | 2 g34 8 | 579 5679 f4579 | 14579 3579 145679 | | 5 6 h34 |d789 1 e24789 | 24789 23789 479 | | 7 9 1 | 3 56 e2458 | 2458 258 456 | :------------------+-------------------+----------------------: | 39 357 6 | 2 4 #579 |#579 1 8 | | 189 1457 4579 | 6 #579 15789 | 3 #579 2 | | 189 157 2 |c#579+8 3 15789 | 6 4 #579 | '------------------'-------------------'----------------------'

TH 579# internals 2r2c7, 8r9c4
(579=2)b1p569 – (2=8)# – 8r5c4 = (28–4)r56c6 = 4r4c6 – 4r4c2 = (4–3)r5c3 = 3r1c3 => –579r1c3

Code: Select all: .----------------.-------------------.-----------------. | 19 1257 3 | 4 579 6 |A25789 28 C579 | | 4 257 579 | 1 8 579 |A2579 6 3 | | 6 8 579 | 579 2 3 | 14 B579 14 | :----------------+-------------------+-----------------: | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-------------------+-----------------: | 3 57 6 | 2 4 A579 |A579 1 8 | | 18 4 579 | 6 B579 18 | 3 B579 2 | | 189 157 2 |C5789 3 C15789 | 6 4 C579 | '----------------'-------------------'-----------------'

Each of b389 contains one of 579 in A-, B-, and C-marked cells.
Corresponding cells among b39 and b89 all see each other, therefore all boxes have the same parity.
b38 cannot have the same permutation either, as the digit in the B-marked cells would be forced into r2c6 in b2 and eliminated out of c3.
Therefore each box has a different permutation of the same parity, so A-, B-, and C-marked cells each contain each of 579 exactly once.
9r7 \ A => –9r12c7
5c9 \ C => –5r9c46

Code: Select all: .----------------.-----------------.-----------------. |#19 1257 3 | 4 57–9 6 | 2578 28 #579 | | 4 257 579 | 1 8 579 | 257 6 3 | | 6 8 579 | 579 2 3 | 14 #579 14 | :----------------+-----------------+-----------------: | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-----------------+-----------------: | 3 57 6 | 2 4 579 | 579 1 8 | | 18 4 #579 | 6 #579 18 | 3 #579 2 | |#189 157 2 | 789 3 1789 | 6 4 579 | '----------------'-----------------'-----------------'

9r8c1b3 \ r1c58b7 => –9r1c5

Code: Select all: .----------------.-----------------.-----------------. | 19 #1257 3 | 4 #57 6 |A2578 28 #579 | | 4 257 579 | 1 8 579 |A27–5 6 3 | | 6 8 579 | 579 2 3 | 14 579 14 | :----------------+-----------------+-----------------: | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-----------------+-----------------: | 3 #57 6 | 2 4 A579 |A579 1 8 | | 18 4 579 | 6 #579 18 | 3 579 2 | | 189 157 2 | 789 3 1789 | 6 4 579 | '----------------'-----------------'-----------------'

5r17b8 \ c25b3AA => –5r2c7 (A is doubled to sufficiently cover r7c6)

Code: Select all: .----------------.-----------------.-----------------. | 19 1257 3 | 4 57 6 | 2578 28 C59–7| | 4 b257 579 | 1 8 579 |a27 6 3 | | 6 8 579 | 579 2 3 | 14 579 14 | :----------------+-----------------+-----------------: | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-----------------+-----------------: | 3 b57 6 | 2 4 579 | 579 1 8 | | 18 4 579 | 6 579 18 | 3 579 2 | | 189 c15–7 2 |dC789 3 dC1789 | 6 4 dC579 | '----------------'-----------------'-----------------'

(7=2)r2c7 – (2=57)r27c2 – 7r9c2 = 7r9c469 => –7r1c9, 7C \ r9 => –7r9c2

Code: Select all: .----------------.-----------------.-----------------. | 19 1257 3 | 4 d57 6 | 2578 28 c59 | | 4 257 579 | 1 8 579 | 27 6 3 | | 6 8 579 | 579 2 3 | 14 579 14 | :----------------+-----------------+-----------------: | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-----------------+-----------------: | 3 57 6 | 2 4 579 | 579 1 8 | | 18 4 579 | 6 59–7 18 | 3 579 2 | | 189 15 2 |a789 3 a1789 | 6 4 b579 | '----------------'-----------------'-----------------'

7r9c46 = (7–5)r9c9 = 5r1c9 – (5=7)r1c5 => –7r8c5
(Note that (5|9)r2c5 takes r4c8 in c8 and makes a triple with 28 in b5p469, then we can even eliminate 79r4c7 but it doesn't progress the solve.)

Code: Select all: .----------------.-----------------.-----------------. | 19 1257 3 | 4 57 6 | 2578 28 59 | | 4 257 579 | 1 8 f579 | 27 6 3 | | 6 8 579 |e579 2 3 | 14 bB579 14 | :----------------+-----------------+-----------------: | 2 3 8 |d579 5679 4579 | 14579 c579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-----------------+-----------------: | 3 57 6 | 2 4 57–9 | 579 1 8 | | 18 4 579 | 6 aB59 18 | 3 bB579 2 | | 189 15 2 | 789 3 178–9| 6 4 579 | '----------------'-----------------'-----------------'

(9=5)r8c5 – 5r38c8 = 5r4c8 – 5r4c4 = (5–9)r3c4 = 9r2c6 => –9r79c6

Code: Select all: .----------------.-----------------.----------------. | 19 1257 3 | 4 57 6 | 2578 28 59 | | 4 257 579 | 1 8 579 | 27 6 3 | | 6 8 579 | 579 2 3 | 14 579 14 | :----------------+-----------------+----------------: | 2 3 8 | 579 5679 4579 | 1457 579 146 | | 5 6 4 | 789 1 2789 | 278 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-----------------+----------------: | 3 a57 6 | 2 4 57 | 9 1 8 | | 18 4 b579 | 6 59 18 | 3 c57 2 | | 189 1–5 2 | 789 3 178 | 6 4 d57 | '----------------'-----------------'----------------'

(5=7)r7c2 – 7r8c3 = 7r8c8 – (7=5)r9c9 => –5r9c2, stte

Marek

by **yzfwsf** » Sat Mar 11, 2023 11:42 pm

marek stefanik wrote:The 18-cell pattern doesn't seem to be helpful – the solution contains only one of its guardians (1r4c7), but even just placing that digit does nothing for the solve.
Marek

Please give it a try.

Code: Select all: ..34.678...718..36....374.129..6....5........73....6.837..4.........3.......718.3;#5067

by **marek stefanik** » Sun Mar 12, 2023 12:53 am

I don't know if you wanted me to try the puzzle with that pattern in mind, but instead I found a path using TH and an almost firework triple.

Code: Select all: .------------------------.-------------------.----------------------. | 19 125 3 | 4 #259 6 | 7 8 #259 | | 49 245 7 | 1 8 #259 |#259 3 6 | | 689 2568 *25689 |#259 3 7 | 4 #259 1 | :------------------------+-------------------+----------------------: | 2 9 148 | 37 6 458 | 135 1457 457 | | 5 1468 1468 | 37 129 2489 | 1239 12479 2479 | | 7 3 14 | 259 1259 2459 | 6 12459 8 | :------------------------+-------------------+----------------------: | 3 7 125689 | 25689 4 #259 | 1259 12569 #259 | | 14689 124568 1245689 | 25689 #259 3 |#259+1 1245679 24579 | | 469 2456 *24569 |*#259+6 7 1 | 8 *#259+46 3 | '------------------------'-------------------'----------------------'

TH 249#, internals 1r8c7, 6r9c4, 46r9c8
almost firework triple 259* (259r9c3 \ b7)
If r8c7 is from 259, the TH needs a guardian in r9, which locks the firework triple.
The remaining TH guardians get eliminated, so we get RTs, and the digit in r3c3 is eliminated from the triple at the rectangle, i.e. contra.

Edit: Maybe a simpler way to think about it:
The digits in r3c48 are forced into r789c3 in c3, so one of them is forced into r9c48.
That digit is then it's eliminated from the TH 8-loop in either b29 or b38, so the 8-loop needs a guardian.

Therefore –259r8c7, reducing the puzzle to 7.8 skfr.

The finish is also quite nice:

Code: Select all: .------------------.----------------.-------------------. | 1 25 3 | 4 b259 6 | 7 8 259 | | 49 245 7 | 1 8 259 | 259 3 6 | | 68 68 259 | 259 3 7 | 4 c259 1 | :------------------+----------------+-------------------: | 2 9 8 | 37 6 4 | 35 1 57 | | 5 1 6 | 37 a29 8 | 239 2479 2479 | | 7 3 4 |bc259 1 259 | 6 a29 8 | :------------------+----------------+-------------------: | 3 7 1 | 8 4 b259 | 259 6 259 | | 4689 24568 259 | 2569 c259 3 | 1 b24579 24579 | | 469 2456 259 | 2569 7 1 | 8 b2459 3 | '------------------'----------------'-------------------'

Note that r5c5 and r6c8 contain the same digit.
Suppose now (2|5|9)r89c8.
That digit would take r7c6 in r7, then r1c5 in c5.
The digit in r8c5 would take r3c8 in c8.
Both of them would have to take r6c4 in c4, i.e. contra.
Therefore –259r89c8.

Code: Select all: .--------------.---------------.-------------. | 1 25 3 | 4 #259 6 | 7 8 #29 | | 49 245 7 | 1 8 #259 |#29 3 6 | | 68 68 29 |#29 3 7 | 4 #5 1 | :--------------+---------------+-------------: | 2 9 8 | 3 6 4 | 5 1 7 | | 5 1 6 | 7 29 8 | 3 29 4 | | 7 3 4 | 259 1 259 | 6 29 8 | :--------------+---------------+-------------: | 3 7 1 | 8 4 #259 |#29 6 #259 | | 48 48 259 | 6 #259 3 | 1 7 #259 | | 69 26 259 |#259 7 1 | 8 4 3 | '--------------'---------------'-------------'

b38 have the same horizontal parity (propagation via b2), the last digit is different (otherwise it would be eliminated from b9), so all digits are different. –5r8c5, stte

Marek

by **yzfwsf** » Sun Mar 12, 2023 1:11 am

Your solution path is elegant. But my solver uses brute force.
I think this is a technique I should add to my solver.

by **eleven** » Sun Mar 12, 2023 11:25 pm

marek stefanik wrote:
Code: Select all
.----------------.-------------------.-----------------. | 19 1257 3 | 4 579 6 |A25789 28 C579 | | 4 257 579 | 1 8 579 |A2579 6 3 | | 6 8 579 | 579 2 3 | 14 B579 14 | :----------------+-------------------+-----------------: | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2789 | 2789 3 79 | | 7 9 1 | 3 56 2458 | 2458 28 46 | :----------------+-------------------+-----------------: | 3 57 6 | 2 4 A579 |A579 1 8 | | 18 4 579 | 6 B579 18 | 3 B579 2 | | 189 157 2 |C5789 3 C15789 | 6 4 C579 | '----------------'-------------------'-----------------'
Each of b389 contains one of 579 in A-, B-, and C-marked cells.
Corresponding cells among b39 and b89 all see each other, therefore all boxes have the same parity.
b38 cannot have the same permutation either, as the digit in the B-marked cells would be forced into r2c6 in b2 and eliminated out of c3.
Therefore each box has a different permutation of the same parity, so A-, B-, and C-marked cells each contain each of 579 exactly once.

Nice observation!

For me it's simpler to first show the remote triple in B because of b2 and c3 (the same digit can't be in r3c8 and r8c5).

Then it can be used, that if there is a remote triple marked XYZ here and none of the 3 digits can go to the #-marked cells, then the A- and C-marked cells also must contain a remote triple.

Code: Select all: *---------------------------------------- | A # # | A A A | | # X # | # Y # | | # # C | C C C | |-------------------+-------------------+ | A # C | . . . | | A Z C | . . . | | A # C | . . . | |-------------------+-------------------+

It is easy to verify.

Hidden Text: Show

by **totuan** » Tue Mar 14, 2023 8:32 am

yzfwsf wrote:
Code: Select all
...4.6...4..18..6368..23...2.8......56..1....7913.......624..18...6..3.2..2.3.64.

18 Cells Impossible Pattern
Code: Select all
....1.....11..1.....11...1....1..11....1....1..........1...11....1.1..1..........

Please ask the experts to identify if this puzzle cannot be solved using the Impossible Pattern forcing chain.

My path for this one – a bit different at start

Hidden Text: Show

Code: Select all: *-----------------------------------------------------------------------------* | 139 12357 3579 | 4 *579 6 | 25789# 25789# *579 | | 4 257 579 | 1 8 *579 |*579+2 6 3 | | 6 8 579 |*579 2 3 | 14 *579 14 | |-------------------------+-------------------------+-------------------------| | 2 34 8 | 579 5679 4579 | 14579 3579 145679 | | 5 6 34 | 79+8 1 24789# | 24789# 379+28 479 | | 7 9 1 | 3 56 2458# | 2458 258# 456 | |-------------------------+-------------------------+-------------------------| | 39 357 6 | 2 4 *579 |*579 1 8 | | 189 1457 4579 | 6 *579 15789 | 3 *579 2 | | 189 157 2 |*5789 3 15789 | 6 4 *579 | *-----------------------------------------------------------------------------*

Tridagon (579) * marked cells => (2)r2c7=(8)r9c4
MUG (28) # marked cells => (2)r2c7=(8)r5c4=(28)r5c8
01: Present as diagram: => r2c7=2, some singles

Code: Select all: (2)r2c7* || (28-3)r5c8=(3-4)r5c3=r4c2-(4=579)r247c6-(579=18)r89c6-(8)r9c4==(2)r2c7* || (8)r5c4-(8)r9c5==(2)r2c7*

Code: Select all: *--------------------------------------------------------------------* | 1 2 3 | 4 57-9 6 |a579 8 a579 | | 4 57 579 | 1 8 579 | 2 6 3 | | 6 8 579 | 579 2 3 | 14 b579 14 | |----------------------+----------------------+----------------------| | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2 | 789 3 79 | | 7 9 1 | 3 56 458 | 458 2 46 | |----------------------+----------------------+----------------------| | 3 57 6 | 2 4 579 | 579 1 8 | | 8 4 57 | 6 d579 1 | 3 c579 2 | | 9 1 2 | 578 3 578 | 6 4 57 | *--------------------------------------------------------------------*

02: 9’s abcd => r1c5<>9

Code: Select all: *--------------------------------------------------------------------* | 1 2 3 | 4 a57 6 |g579 8 h9-57 | | 4 57 579 | 1 8 579 | 2 6 3 | | 6 8 579 | 579 2 3 | 14 57 14 | |----------------------+----------------------+----------------------| | 2 3 8 | 579 5679 4579 | 14579 579 146 | | 5 6 4 | 789 1 2 | 789 3 79 | | 7 9 1 | 3 56 458 | 458 2 46 | |----------------------+----------------------+----------------------| | 3 57 6 | 2 4 579 |f579 1 8 | | 8 4 57 | 6 b579 1 | 3 e579 2 | | 9 1 2 |c578 3 c578 | 6 4 d57 | *--------------------------------------------------------------------*

Without 9r8c5 => almost RP (57)abcd => r1c9<>57
03: [almost RP (57) abcd]=(9)r8c5-r8c8=r7c7-r1c7=r1c9 => r1c9<>57, some singles

Code: Select all: *-----------------------------------------------------------* | 1 2 3 | 4 57 6 | 57 8 9 | | 4 57 579 | 1 8 579 | 2 6 3 | | 6 8 579 |e579 2 3 | 14 57 14 | |-------------------+-------------------+-------------------| | 2 3 8 |d579 5679 4579 | 1459 c59 146 | | 5 6 4 |a89 1 2 |b89 3 7 | | 7 9 1 | 3 56 458 | 458 2 46 | |-------------------+-------------------+-------------------| | 3 57 6 | 2 4 59 | 79 1 8 | | 8 4 57 | 6 59 1 | 3 79 2 | | 9 1 2 | 78 3 78 | 6 4 5 | *-----------------------------------------------------------*

04: (9)r5c4=r5c7-(9=5)r4c8-r4c4=r3c4 => r3c4<>9, stte

yzfwsf wrote:Please give it a try.
Code: Select all
..34.678...718..36....374.129..6....5........73....6.837..4.........3.......718.3;#5067

Nice path from marek!
I had not found other way based on impossible patterns for this one at start, but thinking how to present marek’s path that is not by word

Thanks for your puzzle!
totuan

by **eleven** » Tue Mar 14, 2023 2:15 pm

totuan wrote: ... thinking how to present marek’s path that is not by word

My way:
Let xyz be 259 in whatever order.
259r9c348 & (TH)1r8c7 = xr9c12 - r123c12 = (x-y|z)r3c3 [& *yzr3c48] = yz23c12 - (y|z)r9c12 = yzr9c348 - *RTr39c48 = (TH)1r8c7

Either 259 are in r9c348, leaving 1 r8c7 as the only TH guardian, or one, say x, is not in r9c348, then it is in r9c12 and r3c3, yz in r3c48 and one of y and z in r9c48. Since this cell together with r3c48 is no remote triple of the TH rectangle (all are yz), 1 must be in r8c7.

by **yzfwsf** » Tue Mar 14, 2023 3:36 pm

yzfwsf wrote:Your solution path is elegant. But my solver uses brute force.
I think this is a technique I should add to my solver.

II have implemented this technique in the solver, but it is likely to be used for very few puzzles.

by **marek stefanik** » Sat Mar 18, 2023 10:31 am

totuan wrote:
Code: Select all
*-----------------------------------------------------------------------------* | 139 12357 3579 | 4 *579 6 | 25789# 25789# *579 | | 4 257 579 | 1 8 *579 |*579+2 6 3 | | 6 8 579 |*579 2 3 | 14 *579 14 | |-------------------------+-------------------------+-------------------------| | 2 34 8 | 579 5679 4579 | 14579 3579 145679 | | 5 6 34 | 79+8 1 24789# | 24789# 379+28 479 | | 7 9 1 | 3 56 2458# | 2458 258# 456 | |-------------------------+-------------------------+-------------------------| | 39 357 6 | 2 4 *579 |*579 1 8 | | 189 1457 4579 | 6 *579 15789 | 3 *579 2 | | 189 157 2 |*5789 3 15789 | 6 4 *579 | *-----------------------------------------------------------------------------*
Tridagon (579) * marked cells => (2)r2c7=(8)r9c4
MUG (28) # marked cells => (2)r2c7=(8)r5c4=(28)r5c8

Very nice!
b6p57 are actually both part of the MUG, so you can simplify it even further.

Marek

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