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Website · by **denis_berthier** » Mon Feb 13, 2023 12:26 pm

.
This should be fun if you want to find several impossible patterns in a single puzzle (with guardians, of course).

Code: Select all: +-------+-------+-------+ ! 1 . 3 ! . . . ! 7 . 9 ! ! . 5 7 ! . . . ! . 3 6 ! ! 9 6 . ! . . . ! 1 5 . ! +-------+-------+-------+ ! . 7 . ! . . 1 ! 3 . 5 ! ! . . . ! 5 . 7 ! 6 9 . ! ! . . . ! . . . ! . 7 1 ! +-------+-------+-------+ ! 6 3 . ! . 9 4 ! . . . ! ! . 9 . ! 2 . 8 ! . . . ! ! . . . ! . . . ! 9 . . ! +-------+-------+-------+ 1.3...7.9.57....3696....15..7...13.5...5.769........7163..94....9.2.8.........9..;54;533 SER = 11.6

Code: Select all: Resolution state after Singles (and whips[1]): +----------------------+----------------------+----------------------+ ! 1 248 3 ! 468 24568 256 ! 7 248 9 ! ! 248 5 7 ! 1489 1248 29 ! 248 3 6 ! ! 9 6 248 ! 3478 23478 23 ! 1 5 248 ! +----------------------+----------------------+----------------------+ ! 248 7 24689 ! 4689 2468 1 ! 3 248 5 ! ! 2348 1248 1248 ! 5 2348 7 ! 6 9 248 ! ! 23458 248 245689 ! 34689 23468 2369 ! 248 7 1 ! +----------------------+----------------------+----------------------+ ! 6 3 1258 ! 17 9 4 ! 258 128 278 ! ! 457 9 145 ! 2 13567 8 ! 45 146 347 ! ! 24578 1248 12458 ! 1367 13567 356 ! 9 12468 23478 ! +----------------------+----------------------+----------------------+ 189 candidates.

by **totuan** » Tue Feb 14, 2023 12:38 pm

After Tridagon this one can solve by Impossible Patterns, but it is also solved by RTs with the same eliminations

Code: Select all: *--------------------------------------------------------------------* | 1 *248 3 | 468 24568 256 | 7 *248 9 | |*248 5 7 | 1489 1248 29 |*248 3 6 | | 9 6 *248 | 3478 23478 23 | 1 5 *248 | |----------------------+----------------------+----------------------| |*248 7 69 | 4689 2468 1 | 3 *248 5 | | 3 1248 *1-248 | 5 248 7 | 6 9 *248 | | 5 *248 69 | 34689 23468 2369 |*248 7 1 | |----------------------+----------------------+----------------------| | 6 3 1258 | 17 9 4 | 258 128 278 | | 47 9 145 | 2 13567 8 | 45 146 347 | | 2478 1248 12458 | 1367 13567 356 | 9 12468 23478 | *--------------------------------------------------------------------*

01: Tridagon(248) => r5c3=1

Code: Select all: *--------------------------------------------------------------------* | 1 248 3 | 468 24568 256 | 7 248 9 | | 248 5 7 | 1489 1248 29 | 248 3 6 | | 9 6 *248A | 3478 23478 23 | 1 5 *248A | |----------------------+----------------------+----------------------| | 248 7 69 | 4689 2468 1 | 3 248 5 | | 3 248 1 | 5 248 7 | 6 9 *248A | | 5 248 69 | 34689 23468 2369 | 248 7 1 | |----------------------+----------------------+----------------------| | 6 3 #258 | 17 9 4 | 258 128 278 | | 7 9 #45 | 2 16 8 | 45 16 3 | |#248 1 #248 | 367 3567 356 | 9 2468 7-248 | *--------------------------------------------------------------------*

02: RT-A(248)r3c39/r5c9, whichever (2|4|8)r3c3 => lead to r9c1 by B7 => r9c9<>248, r9c9=7
For impossible pattern:

Hidden Text: Show

Code: Select all: *--------------------------------------------------------------------* | 1 248 3 | 468 24568 256 | 7 248 9 | |*248B 5 7 | 1 #248 9 |*248B 3 6 | | 9 6 248 | 348 7 23 | 1 5 248 | |----------------------+----------------------+----------------------| |%248 7 69 | 4689 6-248 1 | 3 %248 5 | | 3 #248 1 | 5 @248 7 | 6 9 @248 | | 5 248 69 | 34689 23468 236 |#248B 7 1 | |----------------------+----------------------+----------------------| | 6 3 258 | 7 9 4 | 258 1 28 | | 7 9 45 | 2 1 8 | 45 6 3 | | 248 1 248 | 36 356 356 | 9 248 7 | *--------------------------------------------------------------------*

03: RT-B(248) => whichever (2|4|8)r2c5 => lead to r6c7 => B6 & R5 => r5c2 => r2c5/r4c18 form RT(248) => r4c5<>248, r4c5=6
For impossible pattern:

Hidden Text: Show

Code: Select all: *-----------------------------------------------------------* | 1 248C 3 | 468 5-248 56 | 7 248C 9 | | 248B 5 7 | 1 *248 9 | 248B 3 6 | | 9 6 248 | 348 7 23 | 1 5 248 | |-------------------+-------------------+-------------------| | 248 7 9 | 48 6 1 | 3 #248C 5 | | 3 248 1 | 5 #248 7 | 6 9 %248 | | 5 248 6 | 9 2348 23 | 248B 7 1 | |-------------------+-------------------+-------------------| | 6 3 258 | 7 9 4 | 258 1 28 | | 7 9 45 | 2 1 8 | 45 6 3 | | 248 1 248 | 36 35 56 | 9 248 7 | *-----------------------------------------------------------*

04: By RT-B(248) => whichever(2|4|8)r5c5 => eliminate (2|4|8)r2c5 => eliminate (2|4|8)r6c7
=> whichever(2|4|8)r5c5 lead to r4c8 by B6 => by RT-C(248) => r5c5/r1c28 form RT(248) => r1c5<>248, r1c5=5
For impossible pattern:

Hidden Text: Show

Then ER-6.6 and one more coloring to finish.

Thanks for the puzzle!
totuan

by **Leren** » Wed Feb 15, 2023 3:02 am

totuan wrote : 03: RT-B(248) => whichever (2|4|8)r2c5 => lead to r6c7 .....

I don't know whether this affects the overall logic but this doesn't seem quite right. r2c5 = 2 => r6c6 = 2 via ER(2) Box 5 => r2c5 <> 2. So R2c5 = 48, which seems to work as described, so I think this => r6c7 <> 2.

Leren

by **totuan** » Wed Feb 15, 2023 11:39 am

Leren wrote:
totuan wrote : 03: RT-B(248) => whichever (2|4|8)r2c5 => lead to r6c7 .....

I don't know whether this affects the overall logic but this doesn't seem quite right. r2c5 = 2 => r6c6 = 2 via ER(2) Box 5 => r2c5 <> 2. So R2c5 = 48, which seems to work as described, so I think this => r6c7 <> 2

My path based on RTs (remote triple – three different digits in three cells in the solution).
In tridagon, after fill number in cell that form rectangle – in this one is r5c3 and RT-A, then we have other remote triples RT-B & RT-C.

Code: Select all: *-----------------------------------------------------------* | . C248 . | . . . | . C248 . | |B248 . . | . . . |B248 . . | | . . A248 | . . . | . . A248 | |-------------------+-------------------+-------------------| | 248 . . | . . . | . C248 . | | . . X | . . . | . . A248 | | . 248 . | . . . |B248 . . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | *-----------------------------------------------------------*

My path proves r2c5/r4c18 form remote triple (248) => r4c5<>248
In your case, meant r2c5/r6c7/r5c2<>2 in puzzle solution and after that it also proves r2c5/r4c18 form remote triple (248) with r2c5=48. So, I keep in one move

Prove RT-ABC:

Hidden Text: Show

Code: Select all: For RT-B: suppose, B is not RT => pairs [in this case: pair (48)] *-----------------------------------------------------------* | . 248 . | . . . | . 248 . | | 48 . . | . . . | 48 . . | | . . 248 | . . . | . . 248 | |-------------------+-------------------+-------------------| | 248 . . | . . . | . 248 . | | . . . | . . . | . . 248 | | . 248 . | . . . | 48 . . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | *-----------------------------------------------------------* Let r2c1=4 => *-----------------------------------------------------------* | . 28 . | . . . | . 24 . | | 4 . . | . . . | 8 . . | | . . 28 | . . . | . . 24 | |-------------------+-------------------+-------------------| | 28 . . | . . . | . 28 . | | . . . | . . . | . . 28 | | . 28 . | . . . | 4 . . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | *-----------------------------------------------------------* Let r1c2=2 => *-----------------------------------------------------------* | . 2 . | . . . | . 4 . | | 4 . . | . . . | 8 . . | | . . 8 | . . . | . . 2 | |-------------------+-------------------+-------------------| | 8 . . | . . . | . 2 . | | . . . | . . . | . . 8 | | . 8 . | . . . | 4 . . | |-------------------+-------------------+-------------------| | . . . | . . . | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | *-----------------------------------------------------------* => two 8’s in B4 => impossible, the same for other cases => B must form RT (remote triple) The same for RT-AC

totuan

by **eleven** » Wed Feb 15, 2023 3:32 pm

After the tridagon and RT step above it also can be solved with the remote triple this way:

Code: Select all: +-------------------+-------------------+-------------------+ | 1 248 3 | 468 2458 56 | 7 248 9 | | 248 5 7 | 1 248 9 | 248 3 6 | | 9 6 T8-24 |#48+3 7 a23 | 1 5 T#48+2 | +-------------------+-------------------+-------------------+ | 248 7 9 |#48 6 1 | 3 248 5 | | 3 248 1 | 5 #48+2 7 | 6 9 T#48+2 | | 5 248 6 | 9 2348 *23 | 248 7 1 | +-------------------+-------------------+-------------------+ | 6 3 258 | 7 9 4 | 258 1 8-2 | | 7 9 45 | 2 1 8 | 45 6 3 | | 248 1 248 | 36 35 56 | 9 248 7 | +-------------------+-------------------+-------------------+

RT (T): 4r35c9 => -4r3c3
oddagon 48 (#):
3r3c4 - (3=2)r3c6
2r35c9 (RT) - 2r3c3
2r5c5 - r6c6 = 2r3c6
=> -2r3c3, (RT)r7c9

Then e.g.

Code: Select all: *-----------------------------------------------------------* | 1 24 3 | 468 2458 56 | 7 248 9 | | 24 5 7 | 1 248 9 | 248 3 6 | | 9 6 8 | a34 7 a23 | 1 5 4-2 | |------------------+--------------------+-------------------| | 24 7 9 | b48 6 1 | 3 248 5 | | 3 248 1 | 5 c248 7 | 6 9 d24 | | 5 248 6 | 9 2348 b23 | 248 7 1 | |------------------+--------------------+-------------------| | 6 3 25 | 7 9 4 | 25 1 8 | | 7 9 45 | 2 1 8 | 45 6 3 | | 8 1 24 | 36 35 56 | 9 24 7 | *-----------------------------------------------------------*

(2=34)r3c64 - (3|4=28)b5p91 - (2|8=4)r5c5 - (4=2)r5c9 => -2r3c9, stte

Website · by **denis_berthier** » Thu Feb 16, 2023 4:44 am

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Thanks for your solutions. Sometime, I should code RT.
But, for the time being, here's mine, with the two most frequent impossible 3-digit patterns EL13c290 and EL13c234 (not counting tridagon).
See the http://forum.enjoysudoku.com/how-to-deal-with-large-numbers-of-patterns-t40889.html thread for more details about these two patterns.

The start should now look familiar, with the simplest form of tridagon:

Code: Select all: hidden-pairs-in-a-column: c3{n6 n9}{r4 r6} ==> r6c3≠8, r6c3≠5, r6c3≠4, r6c3≠2, r4c3≠8, r4c3≠4, r4c3≠2 singles ==> r6c1=5, r5c1=3 +-------------------+-------------------+-------------------+ ! 1 248 3 ! 468 24568 256 ! 7 248 9 ! ! 248 5 7 ! 1489 1248 29 ! 248 3 6 ! ! 9 6 248 ! 3478 23478 23 ! 1 5 248 ! +-------------------+-------------------+-------------------+ ! 248 7 69 ! 4689 2468 1 ! 3 248 5 ! ! 3 1248 1248 ! 5 248 7 ! 6 9 248 ! ! 5 248 69 ! 34689 23468 2369 ! 248 7 1 ! +-------------------+-------------------+-------------------+ ! 6 3 1258 ! 17 9 4 ! 258 128 278 ! ! 47 9 145 ! 2 13567 8 ! 45 146 347 ! ! 2478 1248 12458 ! 1367 13567 356 ! 9 12468 23478 ! +-------------------+-------------------+-------------------+ tridagon for digits 2, 4 and 8 in blocks: b4, with cells: r5c3 (target cell), r6c2, r4c1 b6, with cells: r5c9, r6c7, r4c8 b1, with cells: r3c3, r1c2, r2c1 b3, with cells: r3c9, r1c8, r2c7 ==> r5c3≠2,4,8 singles ==> r5c3=1, r9c2=1 naked-pairs-in-a-row: r8{c3 c7}{n4 n5} ==> r8c9≠4, r8c8≠4, r8c5≠5, r8c1≠4 singles ==> r8c1=7, r8c9=3 whip[1]: b8n5{r9c6 .} ==> r9c3≠5

Impossible patterns are looked for at this point. Here are 3, only 2 of which will be effectively used; I kept the first to show a case with the same EL13c290 relation having two instances that share many cells (and also share them with the tridagon) but have different ones in the middle stack.

Code: Select all: OR5-EL13c290 relation for digits: 2, 4 and 8 in cells: (r5c9 r6c5 r6c2 r6c7 r4c5 r4c1 r4c8 r1c2 r1c8 r2c1 r2c7 r3c5 r3c9) with 5 guardians : n3r6c5 n6r6c5 n6r4c5 n3r3c5 n7r3c5 +-------------------------+-------------------------+-------------------------+ ! 1 248# 3 ! 468 24568 256 ! 7 248# 9 ! ! 248# 5 7 ! 1489 1248 29 ! 248# 3 6 ! ! 9 6 248 ! 3478 23478#@ 23 ! 1 5 248# ! +-------------------------+-------------------------+-------------------------+ ! 248# 7 69 ! 4689 2468#@ 1 ! 3 248# 5 ! ! 3 248 1 ! 5 248 7 ! 6 9 248# ! ! 5 248# 69 ! 34689 23468#@ 2369 ! 248# 7 1 ! +-------------------------+-------------------------+-------------------------+ ! 6 3 258 ! 17 9 4 ! 258 128 278 ! ! 7 9 45 ! 2 16 8 ! 45 16 3 ! ! 248 1 248 ! 367 3567 356 ! 9 2468 2478 ! +-------------------------+-------------------------+-------------------------+ This will not be used OR3-EL13c290 relation for digits: 2, 4 and 8 in cells: (r3c9 r2c5 r2c1 r2c7 r1c5 r1c2 r1c8 r4c1 r4c8 r6c2 r6c7 r5c5 r5c9) with 3 guardians : n1r2c5 n5r1c5 n6r1c5 +-------------------------+-------------------------+-------------------------+ ! 1 248# 3 ! 468 24568#@ 256 ! 7 248# 9 ! ! 248# 5 7 ! 1489 1248#@ 29 ! 248# 3 6 ! ! 9 6 248 ! 3478 23478 23 ! 1 5 248# ! +-------------------------+-------------------------+-------------------------+ ! 248# 7 69 ! 4689 2468 1 ! 3 248# 5 ! ! 3 248 1 ! 5 248# 7 ! 6 9 248# ! ! 5 248# 69 ! 34689 23468 2369 ! 248# 7 1 ! +-------------------------+-------------------------+-------------------------+ ! 6 3 258 ! 17 9 4 ! 258 128 278 ! ! 7 9 45 ! 2 16 8 ! 45 16 3 ! ! 248 1 248 ! 367 3567 356 ! 9 2468 2478 ! +-------------------------+-------------------------+-------------------------+ OR2-EL13c234 relation for digits: 2, 4 and 8 in cells: (r3c9 r1c2 r1c8 r2c5 r2c7 r6c2 r6c7 r5c5 r5c2 r5c9 r4c5 r4c1 r4c8) with 2 guardians : n1r2c5 n6r4c5 +----------------------+----------------------+----------------------+ ! 1 248# 3 ! 468 24568 256 ! 7 248# 9 ! ! 248 5 7 ! 1489 1248#@ 29 ! 248# 3 6 ! ! 9 6 248 ! 3478 23478 23 ! 1 5 248# ! +----------------------+----------------------+----------------------+ ! 248# 7 69 ! 4689 2468#@ 1 ! 3 248# 5 ! ! 3 248# 1 ! 5 248# 7 ! 6 9 248# ! ! 5 248# 69 ! 34689 23468 2369 ! 248# 7 1 ! +----------------------+----------------------+----------------------+ ! 6 3 258 ! 17 9 4 ! 258 128 278 ! ! 7 9 45 ! 2 16 8 ! 45 16 3 ! ! 248 1 248 ! 367 3567 356 ! 9 2468 2478 ! +----------------------+----------------------+----------------------+

EL13c234-OR2-whip[2]: OR2{{n6r4c5 | n1r2c5}} - r8c5{n1 .} ==> r9c5≠6, r6c5≠6, r1c5≠6

Ultra-persistency of the ORk relations:

Code: Select all: +-------------------+-------------------+-------------------+ ! 1 248 3 ! 468 2458 256 ! 7 248 9 ! ! 248 5 7 ! 1489 1248 29 ! 248 3 6 ! ! 9 6 248 ! 3478 23478 23 ! 1 5 248 ! +-------------------+-------------------+-------------------+ ! 248 7 69 ! 4689 2468 1 ! 3 248 5 ! ! 3 248 1 ! 5 248 7 ! 6 9 248 ! ! 5 248 69 ! 34689 2348 2369 ! 248 7 1 ! +-------------------+-------------------+-------------------+ ! 6 3 258 ! 17 9 4 ! 258 128 278 ! ! 7 9 45 ! 2 16 8 ! 45 16 3 ! ! 248 1 248 ! 367 357 356 ! 9 2468 2478 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous EL13c290-OR3-relation between candidates n1r2c5 n5r1c5 n6r1c5 has just been eliminated. There remains an EL13c290-OR2-relation between candidates: n1r2c5 n5r1c5

EL13c290-OR2-whip[3]: r8c5{n6 n1} - OR2{{n1r2c5 | n5r1c5}} - r9n5{c5 .} ==> r9c6≠6

Code: Select all: biv-chain[3]: r3c6{n2 n3} - r9c6{n3 n5} - b2n5{r1c6 r1c5} ==> r1c5≠2 whip[7]: b4n4{r6c2 r4c1} - c8n4{r4 r9} - c8n6{r9 r8} - c5n6{r8 r4} - r4n2{c5 c8} - r1n2{c8 c6} - c6n6{r1 .} ==> r1c2≠4 whip[1]: c2n4{r6 .} ==> r4c1≠4 biv-chain[2]: r8n4{c7 c3} - b1n4{r3c3 r2c1} ==> r2c7≠4 whip[6]: c8n6{r9 r8} - c5n6{r8 r4} - c6n6{r6 r1} - r1n5{c6 c5} - r1n4{c5 c4} - r4n4{c4 .} ==> r9c8≠4 whip[6]: r4c1{n2 n8} - r9c1{n8 n4} - b1n4{r2c1 r3c3} - c9n4{r3 r5} - b6n8{r5c9 r6c7} - r2c7{n8 .} ==> r2c1≠2 whip[3]: c1n2{r9 r4} - c2n2{r5 r1} - c8n2{r1 .} ==> r9c9≠2 t-whip[6]: b7n4{r9c3 r9c1} - c1n2{r9 r4} - c1n8{r4 r2} - r2c7{n8 n2} - b6n2{r6c7 r5c9} - c9n4{r5 .} ==> r3c3≠4 hidden-single-in-a-block ==> r2c1=4 whip[3]: c1n8{r9 r4} - c2n8{r5 r1} - c8n8{r1 .} ==> r9c9≠8 z-chain[4]: r3c3{n8 n2} - r3c9{n2 n4} - r9c9{n4 n7} - c5n7{r9 .} ==> r3c5≠8 z-chain[4]: r3n7{c5 c4} - r3n4{c4 c9} - r9c9{n4 n7} - c5n7{r9 .} ==> r3c5≠3, r3c5≠2 t-whip[5]: r3c6{n3 n2} - r2n2{c6 c7} - r1n2{c8 c2} - r6n2{c2 c5} - c5n3{r6 .} ==> r9c6≠3 singles ==> r9c6=5, r1c5=5 finned-x-wing-in-rows: n4{r1 r4}{c8 c4} ==> r6c4≠4

EL13c234-OR2-ctr-whip[5]: c5n8{r6 r2} - r2n1{c5 c4} - c4n9{r2 r4} - r4c3{n9 n6} - OR2{{n1r2c5 n6r4c5 | .}} ==> r6c4≠8
z-chain[6]: c5n4{r6 r3} - r3n7{c5 c4} - r7c4{n7 n1} - r8c5{n1 n6} - r4n6{c5 c3} - r4n9{c3 .} ==> r4c4≠4

The end is easy, in S3+BC4:

Code: Select all: whip[1]: b5n4{r6c5 .} ==> r3c5≠4 singles ==> r3c5=7, r9c5=3 naked-triplets-in-a-row: r6{c2 c5 c7}{n2 n4 n8} ==> r6c6≠2 whip[1]: c6n2{r3 .} ==> r2c5≠2 biv-chain[4]: r4n4{c8 c5} - c5n6{r4 r8} - r9c4{n6 n7} - r9c9{n7 n4} ==> r5c9≠4 biv-chain[4]: c4n3{r6 r3} - r3n4{c4 c9} - r9c9{n4 n7} - r9c4{n7 n6} ==> r6c4≠6 biv-chain[5]: r2c5{n8 n1} - c4n1{r2 r7} - b8n7{r7c4 r9c4} - r9c9{n7 n4} - r3n4{c9 c4} ==> r3c4≠8 biv-chain[3]: b1n2{r1c2 r3c3} - r3n8{c3 c9} - b3n4{r3c9 r1c8} ==> r1c8≠2 biv-chain[3]: b6n4{r6c7 r4c8} - r1c8{n4 n8} - r2c7{n8 n2} ==> r6c7≠2 biv-chain[3]: c2n4{r5 r6} - r6c7{n4 n8} - r5c9{n8 n2} ==> r5c2≠2 biv-chain[3]: r5n2{c9 c5} - r6n2{c5 c2} - b1n2{r1c2 r3c3} ==> r3c9≠2 singles ==> r2c7=2, r2c6=9 whip[1]: r2n8{c5 .} ==> r1c4≠8 t-whip[2]: r1n8{c8 c2} - b4n8{r6c2 .} ==> r4c8≠8 biv-chain[3]: r3c9{n4 n8} - b6n8{r5c9 r6c7} - b6n4{r6c7 r4c8} ==> r1c8≠4 stte

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#116 in 63,137 (or 158,276) T&E(3) min-expands

#116 in 63,137 (or 158,276) T&E(3) min-expands

Re: #116 in 63,137 (or 158,276) T&E(3) min-expands

Re: #116 in 63,137 (or 158,276) T&E(3) min-expands

Re: #116 in 63,137 (or 158,276) T&E(3) min-expands

Re: #116 in 63,137 (or 158,276) T&E(3) min-expands

Re: #116 in 63,137 (or 158,276) T&E(3) min-expands