The New Sudoku Players' Forum

by **totuan** » Fri Dec 23, 2022 2:59 pm

Just checked some puzzles on mith’s list, but – for me, the below puzzle is really quite hard.

"min_expand_20221106_basics_special_only_full"

Code: Select all: .....6......18..368...2...4.45......3.8....41719.......3.6..4.8...83..12.8..4.36. 100;27050;1 5;5;n579;b2p267+b3p348+b8p267+b9p249 - line 816

For challenging - Enjoy & Good Luck...!

totuan

by **marek stefanik** » Sat Dec 24, 2022 1:45 am

Really quite hard, the way I found uses almost the entire toolkit.

Code: Select all: .--------------------.---------------------.--------------------. | 12459 2579 12347 | 34579 #579 6 | 125789 25789 #579 | | 2459 2579 247 | 1 8 #4579 |#2579 3 6 | | 8 5679 1367 |#3579 2 3579 | 1579 #579 4 | :--------------------+---------------------+--------------------: | 26 4 5 | 2379 1679 123789 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 | 2345 56 23458 | 2568 258 35 | :--------------------+---------------------+--------------------: | 1259 3 127 | 6 #1579 12579 | 4 #579 8 | | 4569 5679 467 | 8 3 #579 |#579 1 2 | | 1259 8 127 |#2579 4 12579 | 3 6 #579 | '--------------------'---------------------'--------------------'

TH 579#

Code: Select all: .--------------------.---------------------.--------------------. | 12459 #2579 12347 | 34579 579 6 | 125789 25789 579 | |#2459 #2579 #247 | 1 8 #4579 |#2579 3 6 | | 8 #5679 1367 | 3579 2 3579 | 1579 579 4 | :--------------------+---------------------+--------------------: | 26 4 5 | 2379 1679 123789 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 | 2345 56 23458 | 2568 258 35 | :--------------------+---------------------+--------------------: | 1259 3 127 | 6 1579 12579 | 4 579 8 | | 4569 #5679 467 | 8 3 579 | 579 1 2 | | 1259 8 127 | 2579 4 12579 | 3 6 579 | '--------------------'---------------------'--------------------'

almost firework triple: 579r2c2 \ 579b1 + r2c267 r8c2
For the TH, if 4r2c6 or 2r2c7 is a true guardian, the FT is locked.
The other rectangle guardian is then false and r8c2 becomes a 5|7|9 which has to appear in b1p24568 and is eliminated out of the TH rectangle.
Hence another guardian is needed (3r3c4, 1r7c5, or 2r9c4). Therefore one of r379c6 becomes a TH external. With r258c6 we get 4r2c6 = 2r5c6.

Code: Select all: .--------------------.---------------------.--------------------. | 14–259d2579 134–27| 34579 579 6 | 125789 25789 579 | |d2459 d2579 d247 | 1 8 a4579 | 2579 3 6 | | 8 d5679 136–7 | 3579 2 3579 | 1579 579 4 | :--------------------+---------------------+--------------------: | 26 4 5 | 2379 1679 1238–79| 26789 2789 379 | | 3 c26 8 | 579–2 5679 b2579 | 5679–2 4 1 | | 7 1 9 | 2345 56 2348–5 | 2568 258 35 | :--------------------+---------------------+--------------------: | 1259 3 127 | 6 1579 12579 | 4 579 8 | | 4569 579–6 467 | 8 3 579 | 579 1 2 | | 1259 8 127 | 2579 4 12579 | 3 6 579 | '--------------------'---------------------'--------------------'

4r2c6 = 2r5c6 – (2=6)r5c2 – (6=24579)b1p24568 – Loop => –2r5c47, –6r8c2, –2579b1p1379
Remember, there is a 5|7|9 in r8c6, in r(3|7|9)c6 and now even in r(2|5)c6. Together they form a triple. Hence –579r46c6.
Let A be the 5|7|9 in r379c6. Note that there can only be one because of the triple. We have proven (at least) one of 3r3c4, 1r7c5, and 2r9c4 to be true.

Code: Select all: .-----------------.--------------------.--------------------. | 14 2579 134 | 4579–3 579 6 | 125789 25789 579 | | 2459 2579 27–4| 1 8 α4579 | 2579 3 6 | | 8 5679 16–3|a3579 2 d3579 | 1579 579 4 | :-----------------+--------------------+--------------------: |δ26 4 5 | 2379 1679 1238 | 26789 2789 379 | | 3 γ26 8 | 579 5679 β2579 | 5679 4 1 | | 7 1 9 | 2345 56 2348 | 2568 258 35 | :-----------------+--------------------+--------------------: | 1259 3 127 | 6 b1579 c12579 | 4 579 8 | |ε4569 579 ζ467 | 8 3 579 | 579 1 2 | | 1259 8 127 |b2579 4 c12579 | 3 6 579 | '-----------------'--------------------'--------------------'

3r3c4 = (1|2)b8p27 – (12=A)r79c6 – (A=3)r3c6 – Loop => –3r1c4, –3r3c3
4r2c6 = 2r5c6 – 2r5c2 = (2–6)r4c1 = (6–4)r8c1 = 4r8c3 => –4r2c3, stte

Added: Xsudo input: Show

Marek

by **eleven** » Sat Dec 24, 2022 1:11 pm

Code: Select all: +-------------------------+-------------------------+-------------------------+ |c12459 2579 b12347 | 34579 579 6 | 125789 25789 579 | | 2459 2579 a247 | 1 8 4579 | 2579 3 6 | | 8 5679 b1367 | 3579 2 3579 | 1579 579 4 | +-------------------------+-------------------------+-------------------------+ | 26 4 5 | 2379 1679 123789 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 | 2345 56 23458 | 2568 258 35 | +-------------------------+-------------------------+-------------------------+ | 1259 3 a127 | 6 1579 12579 | 4 579 8 | | 4569 5679 467 | 8 3 579 | 579 1 2 | | 1259 8 a127 | 2579 4 12579 | 3 6 579 | +-------------------------+-------------------------+-------------------------+

(4=271)r279c3 - r13c3 = 1r1c1 => -4r1c1

Code: Select all: +-------------------------+-------------------------+-------------------------+ | 1259 2579 34 |#34 579 6 | 125789 25789 579 | | 2459 2579 247 | 1 8 #4579 | 2579 3 6 | | 8 5679 1367 | 3579 2 #3579 | 1579 579 4 | +-------------------------+-------------------------+-------------------------+ | 26 4 5 | 2379 1679 38-1279 | 26789 2789 379 | | 3 26 8 | 2579 5679 *2579 | 25679 4 1 | | 7 1 9 | 2345 56 348-25 | 2568 258 35 | +-------------------------+-------------------------+-------------------------+ | 1259 3 127 | 6 1579 *12579 | 4 579 8 | | 4569 5679 467 | 8 3 *579 | 579 1 2 | | 1259 8 127 | 2579 4 *12579 | 3 6 579 | +-------------------------+-------------------------+-------------------------+

Because of 34r1c4 r23c6 cannot be 34 => 12579 in r235789, -1279r4c6,-25r6c6

Code: Select all: +-------------------------+--------------------------+-------------------------+ | 1259 2579 34 | a34 579 6 | 125789 25789 579 | | 2459 2579 247 | 1 8 4579 | 2579 3 6 | | 8 5679 1367 | 579-3 2 3579 | 1579 579 4 | +-------------------------+--------------------------+-------------------------+ | 26 4 5 |d 2379 1 c38 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 |db2345 56 c348 | 2568 258 35 | +-------------------------+--------------------------+-------------------------+ | 1259 3 127 | 6 579 12579 | 4 579 8 | | 4569 5679 467 | 8 3 579 | 579 1 2 | | 1259 8 127 | 2579 4 12579 | 3 6 579 | +-------------------------+--------------------------+-------------------------+

(3=4)r1c4 - r6c4 = (48-3)r46c6 = 3r46c4 => -3r3c4

Now it's easily solved

Code: Select all: +-------------------------+-------------------------+-------------------------+ | 1259 #2579 34 | 34 #579 6 | 125789 25789 #579 | |#2459 #2579 #247 | 1 8 #4579 |#2579 3 6 | | 8 #5679 1367 |#579 2 3579 | 1579 #579 4 | +-------------------------+-------------------------+-------------------------+ | 26 4 5 | 2379 1 38 | 26789 2789 379 | | 3 #26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 | 2345 56 348 | 2568 258 35 | +-------------------------+-------------------------+-------------------------+ | 1259 3 127 | 6 #579 12579 | 4 #579 8 | | 4569 #5679 467 | 8 3 #579 |#579 1 2 | | 1259 8 127 |#579+2 4 12579 | 3 6 #579 | +-------------------------+-------------------------+-------------------------+

Impossible pattern in #-marked cells => 2r9c4, stte

Hidden Text: Show

Suppose -2r9c4, then either 4r2c6 or 2r2c7

Code: Select all: *------------------------------------------------------* | . 2579 . | . 579 . | . . 579 | | 2459 2579 247 | . . 579+4 | 579+2 . . | | . 5679 . | 579 . . | . 579 . | |------------------+-----------------+-----------------| | . . . | . . . | . . . | | . 26 . | . . . | . . . | | . . . | . . . | . . . | |------------------+-----------------+-----------------| | . . . | . 579 . | . 579 . | | . 5679 . | . . 579 | 579 . . | | . . . | 579 . . | . . 579 | *------------------------------------------------------*

(a) 4r2c6
-> -(4=25796) b1p24568, 62r53c2 => 2r2c13, -6r8c2

Code: Select all: *------------------------------------------------------* | . 579 . | . 579 . | . . 579 | | 259 579 27 | . . 4 |x579 . . | | . 6 . | 579 . . | . 579 . | |------------------+-----------------+-----------------| | . . . | . . . | . . . | | . 2 . | . . . | . . . | | . . . | . . . | . . . | |------------------+-----------------+-----------------| | . . . | . 579 . | . 579 . | | . x579 . | . . z579 |y579 . . | | . . . | 579 . . | . . 579 | *------------------------------------------------------*

xr2c7 -> (RT)xr8c2 - r12c2 = r2c13, contrad.

(b)2r2c7:

Code: Select all: *------------------------------------------------------* | . 5792 . | . 579 . | . . 579 | | 459 579 47 | . . 4579 | 2 . . | | . 5679 . | 579 . . | . 579 . | |------------------+-----------------+-----------------| | . . . | . . . | . . . | | . 26 . | . . . | . . . | | . . . | . . . | . . . | |------------------+-----------------+-----------------| | . . . | . 579 . | . 579 . | | . 5796 . | . . 579 | 579 . . | | . . . | 579 . . | . . 579 | *------------------------------------------------------*

r2c6 can't be 4: 24r2c67 -> 579r2c123,26r13c2, contra to 26r5c2

Code: Select all: *------------------------------------------------------* | . 5792 . | . 579 . | . . 579 | | 459 579 47 | . . x579 | 2 . . | | . 5679 . | 579 . . | . 579 . | |------------------+-----------------+-----------------| | . . . | . . . | . . . | | . 26 . | . . . | . . . | | . . . | . . . | . . . | |------------------+-----------------+-----------------| | . . . | . 579 . | . 579 . | | . x579+6 . | . . y579 |z579 . . | | . . . | 579 . . | . . 579 | *------------------------------------------------------*

r58c2 can't be 26 (only 1 digit can be missing in ALS b1p24568) => -6r8c2
r8c2 can't be x: RT xr2c6 -> 1 digit missing in b1p24568 -> 26r135c2, contrad.

by **totuan** » Sun Dec 25, 2022 3:03 am

.
Thanks for your nice path – marek & eleven!

Code: Select all: *-----------------------------------------------------------------------------* |a1259-4 2579 b34-127 |e34-579 579 6 | 125789 25789 579 | | 2459 2579 247 | 1 8 4579 | 2579 3 6 | | 8 5679 b1367 | 579-3 2 h3579 | 1579 579 4 | |-------------------------+-------------------------+-------------------------| | 26 4 5 | 2379 1679 g38-1279 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 |f2345 56 g348-25 | 2568 258 35 | |-------------------------+-------------------------+-------------------------| | 1259 3 127 | 6 1579 12579 | 4 579 8 | |d4569 5679 c467 | 8 3 579 | 579 1 2 | | 1259 8 127 | 2579 4 12579 | 3 6 579 | *-----------------------------------------------------------------------------*

My path for this one – it’s not really nice

01: (1)r1c1=(13-6)r13c3=(6-4)r6c3=r8c1 => r1c1<>4, pair (34)r1c34
02: (3=4)r1c4-r6c4=(48-3)r46c6=r3c6 => Loop, r3c4<>3, r4c6<>1279, r6c6<>25 => r4c5=1

Code: Select all: *-----------------------------------------------------------------------------* | 1259 A2579 34 | 34 *579 6 | 125789 25789 *579 | | 2459 2579 247 | 1 8 4579 |*2579 3 6 | | 8 5679 167-3 |*579 2 *3579 | 1579 *579 4 | |-------------------------+-------------------------+-------------------------| | 26 4 5 | 2379 1 38 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 | 2345 56 348 | 2568 258 35 | |-------------------------+-------------------------+-------------------------| | 1259 3 127 | 6 *579 12579 | 4 *579 8 | | 4569 *5679 467 | 8 3 *579 |*579 1 2 | | 1259 8 127 |*2579 4 12579 | 3 6 *579 | *-----------------------------------------------------------------------------*

Impossible pattern (579) * marked cells => (26)r18c2=(3)r3c6=(2)r2c7=(2)r9c4

Hidden Text: Show

03: Present as diagram: => r3c3<>3, some singles

Code: Select all: (26)r158c2-(6)r3c2=r3c3* || (2)r9c4-r79c6=r5c6--(2=6)r5c2-r3c2=r3c3* || | || (2)r1c2-- || || (2)r2c7—--(2)r2c123 || || (3)r3c6* (2-1)r1c1=r3c3*

Code: Select all: *-----------------------------------------------------------------------------* | 1259 2579 3 | 4 *579 6 |#125789 #25789 *579 | | 2459 2579 247 | 1 8 *579 |*2579 3 6 | | 8 5679 167 |*579 2 3 | 1579 *579 4 | |-------------------------+-------------------------+-------------------------| | 26 4 5 | 2379 1 8 | 2679 279 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 |#235 56 4 |#2568 #258 35 | |-------------------------+-------------------------+-------------------------| | 1259 3 127 | 6 *579 12579 | 4 *579 8 | | 4569 5679 467 | 8 3 *579 |*579 1 2 | | 1259 8 127 |*2579 4 12579 | 3 6 *579 | *-----------------------------------------------------------------------------*

Tridagon (579) * marked cells => (2)r2c7=(2)r9c4
UR (28)r16c78 => (2)r2c7=(2)r6c4
04: (2)r2c7==r6c4-r9c4==(2)r2c7 => r2c7=2

Code: Select all: *--------------------------------------------------------------------* | 1259 A2579 3 | 4 #579 6 | 15789 5789 #579 | | 459 579 47 | 1 8 #579 | 2 3 6 | | 8 5679 67-1 |#579 2 3 |#1579 #579 4 | |----------------------+----------------------+----------------------| | 26 4 5 | 2379 1 8 | 679 279 379 | | 3 26 8 | 2579 5679 2579 | 5679 4 1 | | 7 1 9 | 235 56 4 | 568 258 35 | |----------------------+----------------------+----------------------| | 159 3 127 | 6 #579 12579 | 4 #579 8 | | 4569 #5679 467 | 8 3 #579 |#579 1 2 | | 159 8 127 |#2579 4 12579 | 3 6 #579 | *--------------------------------------------------------------------*

Impossible pattern (579) # marked cells => (26)r18c2=(2)r9c4=(1)r3c7

Hidden Text: Show

05: Present as diagram: => r3c3<>1, some singles

Code: Select all: (1)r3c7* || (2)r9c4-r79c6=r5c6-(2=6)r5c2-r3c2=r3c3* || (26)r158c2-(6)r3c2=r3c3*

Code: Select all: *-----------------------------------------------------------* | 1 2 3 | 4 579 6 | 579 8 579 | | 4 59 7 | 1 8 59 | 2 3 6 | | 8 59 6 | 579 2 3 | 1 579 4 | |-------------------+-------------------+-------------------| | 2 4 5 | 379 1 8 | 6 79 379 | | 3 6 8 | 2579 579 27 | 579 4 1 | | 7 1 9 | 35 6 4 | 8 2 35 | |-------------------+-------------------+-------------------| | 59 3 *12 | 6 579 *12 | 4 579 8 | | 6 7 4 | 8 3 59 | 59 1 2 | | 59 8 *12 | 2579 4 *12+7 | 3 6 579 | *-----------------------------------------------------------*

06: UR (12)r79c36 => r9c6=7, stte

Merry Christmas & Happy New Year to All!

totuan

by **shye** » Sun Dec 25, 2022 12:16 pm

my own path is similar to both marek & eleven, this puzzle is crazy!!

1) ALS h-wing

Hidden Text: Show

2) CNL

Hidden Text: Show

3) tridagon & firework fusion

Hidden Text: Show

by **eleven** » Mon Dec 26, 2022 10:36 am

Hm, without r5c2 i can't see a firework triple in the marked cells for 2r2c7.
What am i missing ?

Website · by **denis_berthier** » Tue Dec 27, 2022 7:06 am

.
Just seen this. A really hard one.

Using only all the ORk chains ORk g-chains plus their forcing counterparts, the puzzle is not solved with max-length = 11. I didn't try longer lengths.

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 12459 2579 12347 ! 34579 579 6 ! 125789 25789 579 ! ! 2459 2579 247 ! 1 8 4579 ! 2579 3 6 ! ! 8 5679 1367 ! 3579 2 3579 ! 1579 579 4 ! +----------------------+----------------------+----------------------+ ! 26 4 5 ! 2379 1679 123789 ! 26789 2789 379 ! ! 3 26 8 ! 2579 5679 2579 ! 25679 4 1 ! ! 7 1 9 ! 2345 56 23458 ! 2568 258 35 ! +----------------------+----------------------+----------------------+ ! 1259 3 127 ! 6 1579 12579 ! 4 579 8 ! ! 4569 5679 467 ! 8 3 579 ! 579 1 2 ! ! 1259 8 127 ! 2579 4 12579 ! 3 6 579 ! +----------------------+----------------------+----------------------+ 200 candidates. 195 g-candidates, 1049 csp-glinks and 608 non-csp glinks OR5-anti-tridagon[12] for digits 5, 7 and 9 in blocks: b2, with cells: r1c5, r2c6, r3c4 b3, with cells: r1c9, r2c7, r3c8 b8, with cells: r7c5, r8c6, r9c4 b9, with cells: r7c8, r8c7, r9c9 with 5 guardians: n4r2c6 n2r2c7 n3r3c4 n1r7c5 n2r9c4

There's an easy start, allowing to get rid of 2 guardians:

Code: Select all: z-chain[4]: c3n3{r1 r3} - b1n1{r3c3 r1c1} - r1n4{c1 c4} - r1n3{c4 .} ==> r1c3≠2, r1c3≠7 t-whip[4]: c3n4{r2 r8} - c3n6{r8 r3} - c3n3{r3 r1} - b1n1{r1c3 .} ==> r1c1≠4 hidden-pairs-in-a-row: r1{n3 n4}{c3 c4} ==> r1c4≠9, r1c4≠7, r1c4≠5, r1c3≠1 z-chain[4]: c6n8{r6 r4} - c6n3{r4 r3} - r1c4{n3 n4} - r6n4{c4 .} ==> r6c6≠5, r6c6≠2 t-whip[4]: r1c4{n3 n4} - c6n4{r2 r6} - c6n8{r6 r4} - c6n3{r4 .} ==> r3c4≠3 At least one candidate of a previous Trid-OR5-relation has just been eliminated. There remains a Trid-OR4-relation between candidates: n4r2c6 n2r2c7 n1r7c5 n2r9c4 +----------------------+----------------------+----------------------+ ! 1259 2579 34 ! 34 579 6 ! 125789 25789 579 ! ! 2459 2579 247 ! 1 8 4579 ! 2579 3 6 ! ! 8 5679 1367 ! 579 2 3579 ! 1579 579 4 ! +----------------------+----------------------+----------------------+ ! 26 4 5 ! 2379 1679 123789 ! 26789 2789 379 ! ! 3 26 8 ! 2579 5679 2579 ! 25679 4 1 ! ! 7 1 9 ! 2345 56 348 ! 2568 258 35 ! +----------------------+----------------------+----------------------+ ! 1259 3 127 ! 6 1579 12579 ! 4 579 8 ! ! 4569 5679 467 ! 8 3 579 ! 579 1 2 ! ! 1259 8 127 ! 2579 4 12579 ! 3 6 579 ! +----------------------+----------------------+----------------------+ t-whip[4]: c6n8{r4 r6} - r6n4{c6 c4} - r1c4{n4 n3} - b5n3{r4c4 .} ==> r4c6≠1, r4c6≠2, r4c6≠7, r4c6≠9 hidden-single-in-a-block ==> r4c5=1 At least one candidate of a previous Trid-OR4-relation has just been eliminated. There remains a Trid-OR3-relation between candidates: n4r2c6 n2r2c7 n2r9c4 +----------------------+----------------------+----------------------+ ! 1259 2579 34 ! 34 579 6 ! 125789 25789 579 ! ! 2459 2579 247 ! 1 8 4579 ! 2579 3 6 ! ! 8 5679 1367 ! 579 2 3579 ! 1579 579 4 ! +----------------------+----------------------+----------------------+ ! 26 4 5 ! 2379 1679 38 ! 26789 2789 379 ! ! 3 26 8 ! 2579 5679 2579 ! 25679 4 1 ! ! 7 1 9 ! 2345 56 348 ! 2568 258 35 ! +----------------------+----------------------+----------------------+ ! 1259 3 127 ! 6 579 12579 ! 4 579 8 ! ! 4569 5679 467 ! 8 3 579 ! 579 1 2 ! ! 1259 8 127 ! 2579 4 12579 ! 3 6 579 ! +----------------------+----------------------+----------------------+

Now, the tridagon part, with 2 eliminations by an OR3-g-whips[8]
Trid-OR3-whip[3]: r6n2{c8 c4} - OR3{{n2r9c4 n2r2c7 | n4r2c6}} - r6n4{c6 .} ==> r4c7≠2
Trid-OR3-whip[3]: r6n2{c8 c4} - OR3{{n2r9c4 n2r2c7 | n4r2c6}} - r6n4{c6 .} ==> r5c7≠2
whip[6]: b1n1{r1c1 r3c3} - c7n1{r3 r1} - r1n8{c7 c8} - r1n2{c8 c2} - r5c2{n2 n6} - r3n6{c2 .} ==> r1c1≠9
whip[6]: b1n1{r1c1 r3c3} - c7n1{r3 r1} - r1n8{c7 c8} - r1n2{c8 c2} - r5c2{n2 n6} - r3n6{c2 .} ==> r1c1≠5
whip[6]: b8n1{r7c6 r9c6} - c6n2{r9 r5} - b4n2{r5c2 r4c1} - r1c1{n2 n1} - r7n1{c1 c3} - r7n2{c3 .} ==> r7c6≠9
whip[6]: b8n1{r7c6 r9c6} - c6n2{r9 r5} - b4n2{r5c2 r4c1} - r1c1{n2 n1} - r7n1{c1 c3} - r7n2{c3 .} ==> r7c6≠7
whip[6]: b8n1{r7c6 r9c6} - c6n2{r9 r5} - b4n2{r5c2 r4c1} - r1c1{n2 n1} - r7n1{c1 c3} - r7n2{c3 .} ==> r7c6≠5
Trid-OR3-ctr-gwhip[8]: c1n4{r8 r2} - c3n4{r1 r8} - r8n6{c3 c2} - b7n7{r8c2 r789c3} - r2c3{n7 n2} - c2n2{r2 r5} - b5n2{r5c6 r456c4} - OR3{{n2r9c4 n2r2c7 n4r2c6 | .}} ==> r8c1≠9, r8c1≠5

Code: Select all: +----------------------+----------------------+----------------------+ ! 12 2579 34 ! 34 579 6 ! 125789 25789 579 ! ! 2459 2579 247 ! 1 8 4579 ! 2579 3 6 ! ! 8 5679 1367 ! 579 2 3579 ! 1579 579 4 ! +----------------------+----------------------+----------------------+ ! 26 4 5 ! 2379 1 38 ! 6789 2789 379 ! ! 3 26 8 ! 2579 5679 2579 ! 5679 4 1 ! ! 7 1 9 ! 2345 56 348 ! 2568 258 35 ! +----------------------+----------------------+----------------------+ ! 1259 3 127 ! 6 579 12 ! 4 579 8 ! ! 46 5679 467 ! 8 3 579 ! 579 1 2 ! ! 1259 8 127 ! 2579 4 12579 ! 3 6 579 ! +----------------------+----------------------+----------------------+

Now, the puzzle is easily solved with replacement:

Code: Select all: ***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE ***** RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens. Trying in block 9 +----------------------+----------------------+----------------------+ ! 12 2579 34 ! 34 579 6 ! 125789 25789 579 ! ! 24579 2579 24579 ! 1 8 4579 ! 2579 3 6 ! ! 8 5679 135679 ! 579 2 3579 ! 1579 579 4 ! +----------------------+----------------------+----------------------+ ! 26 4 579 ! 23579 1 38 ! 56789 25789 3579 ! ! 3 26 8 ! 2579 5679 2579 ! 5679 4 1 ! ! 579 1 579 ! 234579 5679 348 ! 256789 25789 3579 ! +----------------------+----------------------+----------------------+ ! 12579 3 12579 ! 6 579 12 ! 4 9 8 ! ! 46 5679 45679 ! 8 3 579 ! 7 1 2 ! ! 12579 8 12579 ! 2579 4 12579 ! 3 6 5 ! +----------------------+----------------------+----------------------+ whip[1]: c2n7{r1 .} ==> r2c1≠7, r2c3≠7, r3c3≠7 whip[1]: r5n7{c4 .} ==> r4c4≠7, r6c4≠7, r6c5≠7 z-chain[3]: c5n9{r5 r1} - c9n9{r1 r6} - r5n9{c7 .} ==> r4c4≠9 z-chain[3]: c5n9{r5 r1} - c9n9{r1 r4} - b4n9{r4c3 .} ==> r6c4≠9 z-chain[5]: c4n5{r4 r3} - r3c8{n5 n7} - r1c9{n7 n9} - c5n9{r1 r6} - c5n6{r6 .} ==> r5c5≠5 z-chain[5]: c4n5{r4 r3} - r3c8{n5 n7} - r1c9{n7 n9} - c5n9{r1 r5} - c5n6{r5 .} ==> r6c5≠5 z-chain[3]: c5n9{r5 r1} - c5n5{r1 r7} - r8c6{n5 .} ==> r5c6≠9 whip[5]: c5n5{r1 r7} - r8c6{n5 n9} - b2n9{r3c6 r1c5} - r1c9{n9 n7} - r3c8{n7 .} ==> r3c4≠5 whip[1]: c4n5{r6 .} ==> r5c6≠5 biv-chain[4]: r5c6{n7 n2} - r5c2{n2 n6} - r3n6{c2 c3} - r3n3{c3 c6} ==> r3c6≠7 whip[6]: r3c4{n9 n7} - r3c8{n7 n5} - b2n5{r3c6 r1c5} - b8n5{r7c5 r8c6} - c2n5{r8 r2} - r2n7{c2 .} ==> r2c6≠9 z-chain[4]: b2n9{r3c6 r1c5} - c5n5{r1 r7} - r8c6{n5 n9} - c2n9{r8 .} ==> r3c3≠9 whip[6]: b2n9{r3c6 r1c5} - c5n5{r1 r7} - r8c6{n5 n9} - c2n9{r8 r2} - r2n7{c2 c6} - r3c4{n7 .} ==> r3c7≠9 t-whip[7]: r2n7{c2 c6} - r5c6{n7 n2} - r5c2{n2 n6} - r3n6{c2 c3} - r3n3{c3 c6} - c6n5{r3 r8} - r8c2{n5 .} ==> r2c2≠9 whip[5]: r3c8{n5 n7} - r1c9{n7 n9} - b2n9{r1c5 r3c4} - c2n9{r3 r8} - r8c6{n9 .} ==> r3c6≠5 finned-x-wing-in-columns: n5{c6 c2}{r8 r2} ==> r2c3≠5, r2c1≠5 z-chain[3]: c1n7{r9 r6} - c1n5{r6 r7} - r7c5{n5 .} ==> r7c3≠7 z-chain[4]: r8n9{c3 c6} - b8n5{r8c6 r7c5} - c1n5{r7 r6} - b4n9{r6c1 .} ==> r9c3≠9 z-chain[5]: b2n5{r2c6 r1c5} - b2n7{r1c5 r3c4} - r3c8{n7 n5} - r2n5{c7 c2} - r2n7{c2 .} ==> r2c6≠4 At least one candidate of a previous Trid-OR3-relation has just been eliminated. There remains a Trid-OR2-relation between candidates: n2r2c7 n2r9c4 +-------------------+-------------------+-------------------+ ! 12 2579 34 ! 34 579 6 ! 12589 2578 79 ! ! 249 257 249 ! 1 8 57 ! 259 3 6 ! ! 8 5679 1356 ! 79 2 39 ! 15 57 4 ! +-------------------+-------------------+-------------------+ ! 26 4 579 ! 235 1 38 ! 5689 2578 379 ! ! 3 26 8 ! 2579 679 27 ! 569 4 1 ! ! 579 1 579 ! 2345 69 348 ! 25689 2578 379 ! +-------------------+-------------------+-------------------+ ! 1257 3 125 ! 6 57 12 ! 4 9 8 ! ! 46 569 4569 ! 8 3 59 ! 7 1 2 ! ! 1279 8 127 ! 279 4 1279 ! 3 6 5 ! +-------------------+-------------------+-------------------+ singles ==> r1c4=4, r1c3=3, r3c6=3, r4c6=8, r6c6=4 whip[1]: c6n9{r9 .} ==> r9c4≠9 Trid-OR2-whip[2]: OR2{{n2r9c4 | n2r2c7}} - c3n2{r2 .} ==> r9c1≠2 biv-chain[3]: r7c6{n1 n2} - r9c4{n2 n7} - r7n7{c5 c1} ==> r7c1≠1 Trid-OR2-whip[3]: OR2{{n2r2c7 | n2r9c4}} - b5n2{r6c4 r5c6} - c2n2{r5 .} ==> r2c1≠2 z-chain[3]: c3n4{r8 r2} - r2c1{n4 n9} - b4n9{r6c1 .} ==> r8c3≠9 biv-chain[3]: b7n9{r8c2 r9c1} - r2c1{n9 n4} - r8c1{n4 n6} ==> r8c2≠6 naked-pairs-in-a-row: r8{c2 c6}{n5 n9} ==> r8c3≠5 biv-chain[2]: c5n5{r1 r7} - r8n5{c6 c2} ==> r1c2≠5 z-chain[3]: r8c2{n9 n5} - b1n5{r2c2 r3c3} - r3n6{c3 .} ==> r3c2≠9 hidden-single-in-a-row ==> r3c4=9 naked-pairs-in-a-column: c5{r1 r7}{n5 n7} ==> r5c5≠7 biv-chain[3]: r5n5{c7 c4} - r5n7{c4 c6} - r2c6{n7 n5} ==> r2c7≠5 hidden-pairs-in-a-row: r2{n5 n7}{c2 c6} ==> r2c2≠2 x-wing-in-rows: n5{r2 r8}{c2 c6} ==> r3c2≠5 biv-chain[3]: c2n2{r1 r5} - r5c6{n2 n7} - b2n7{r2c6 r1c5} ==> r1c2≠7 naked-triplets-in-a-block: b1{r1c2 r2c1 r2c3}{n2 n9 n4} ==> r1c1≠2 singles ==> r1c1=1, r3c7=1 z-chain[2]: b4n5{r6c1 r4c3} - r3n5{c3 .} ==> r6c8≠5 biv-chain[3]: c3n1{r9 r7} - r7c6{n1 n2} - r9c4{n2 n7} ==> r9c3≠7 whip[1]: c3n7{r4 .} ==> r6c1≠7 biv-chain[3]: c1n2{r4 r7} - c1n7{r7 r9} - r9c4{n7 n2} ==> r4c4≠2 biv-chain[2]: r4n2{c8 c1} - c2n2{r5 r1} ==> r1c8≠2 whip[1]: c8n2{r6 .} ==> r6c7≠2 Trid-OR2-whip[3]: OR2{{n2r2c7 | n2r9c4}} - b5n2{r6c4 r5c6} - c2n2{r5 .} ==> r1c7≠2 stte

Website · by **denis_berthier** » Tue Dec 27, 2022 7:12 am

.
Note about using the other contradictory pattern, as in totuan's solution.
This pattern eliminates 2 candidates: n3r3c3 and n1r3c3. The rule for this pattern is not coded in SudoRules, but I can simulate the 2 eliminations at the start:

Code: Select all: (bind ?*simulated-eliminations* (create$ (nrc-to-label 3 3 3) (nrc-to-label 1 3 3)))

The solution is then quite easy:

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 1 2579 3 ! 4 579 6 ! 25789 25789 579 ! ! 2459 2579 247 ! 1 8 579 ! 2579 3 6 ! ! 8 5679 67 ! 579 2 3 ! 1 579 4 ! +-------------------+-------------------+-------------------+ ! 26 4 5 ! 2379 1 8 ! 2679 279 379 ! ! 3 26 8 ! 2579 5679 2579 ! 25679 4 1 ! ! 7 1 9 ! 235 56 4 ! 2568 258 35 ! +-------------------+-------------------+-------------------+ ! 259 3 127 ! 6 579 12579 ! 4 579 8 ! ! 4569 5679 467 ! 8 3 579 ! 579 1 2 ! ! 259 8 127 ! 2579 4 12579 ! 3 6 579 ! +-------------------+-------------------+-------------------+ 151 candidates. OR2-anti-tridagon[12] for digits 5, 7 and 9 in blocks: b2, with cells: r1c5, r2c6, r3c4 b3, with cells: r1c9, r2c7, r3c8 b8, with cells: r7c5, r8c6, r9c4 b9, with cells: r7c8, r8c7, r9c9 with 2 guardians: n2r2c7 n2r9c4

Trid-OR2-whip[2]: OR2{{n2r9c4 | n2r2c7}} - c3n2{r2 .} ==> r9c1≠2
Trid-OR2-whip[2]: OR2{{n2r2c7 | n2r9c4}} - b5n2{r4c4 .} ==> r5c7≠2
Trid-OR2-whip[2]: OR2{{n2r2c7 | n2r9c4}} - r6n2{c4 .} ==> r4c7≠2
Trid-OR2-whip[3]: OR2{{n2r2c7 | n2r9c4}} - r4n2{c4 c8} - r6n2{c7 .} ==> r2c1≠2
Trid-OR2-whip[3]: OR2{{n2r2c7 | n2r9c4}} - b5n2{r4c4 r5c6} - c2n2{r5 .} ==> r2c3≠2
end in S2

So, it seems that the other pattern is more crucial here than the anti-tridagon.

by **marek stefanik** » Tue Dec 27, 2022 11:02 pm

eleven wrote:Hm, without r5c2 i can't see a firework triple in the marked cells for 2r2c7.
What am i missing ?

Code: Select all: .--------------------.---------------------.--------------------. | 12459 2579 12347 | 34579 579 6 | 125789 25789 579 | | 2459 #2579 247 | 1 8 #4579 |#2579 3 6 | | 8 5679 1367 | 3579 2 3579 | 1579 579 4 | :--------------------+---------------------+--------------------: | 26 4 5 | 2379 1679 123789 | 26789 2789 379 | | 3 26 8 | 2579 5679 2579 | 25679 4 1 | | 7 1 9 | 2345 56 23458 | 2568 258 35 | :--------------------+---------------------+--------------------: | 1259 3 127 | 6 1579 12579 | 4 579 8 | | 4569 #5679 467 | 8 3 579 | 579 1 2 | | 1259 8 127 | 2579 4 12579 | 3 6 579 | '--------------------'---------------------'--------------------'

The fireworks are on 579. There is no 579 candidate in r5c2.
Each of 579 has to appear in one of the #-marked cells (otherwise it would be forced into r2c13 and r13c2).
Therefore if 2r2c7, the remaining three cells form a triple.
Hope it helps.

Marek

by **eleven** » Tue Dec 27, 2022 11:20 pm

Ah thanks, with 2r1c2 there is a hidden triple 579 in r238c2, which also eliminates the 6r38c2 (and locks 4 to r2c139).

[added]So the point is, that outside of the cross in b1 the 3 digits can only be in 3 cells in c2 and r2.
If one is in r8c2, the other 2 are bound to r123c2 and cannot be in r2c13.
if none is in r8c2, there is a triple in r123c2, and none in r2c13.

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