The New Sudoku Players' Forum

by **eleven** » Sun Jul 18, 2021 12:52 am

Code: Select all: +-------+-------+-------+ | 6 . . | . . . | 7 . . | | 7 . . | . 2 9 | 4 . . | | 4 . . | . . . | . 5 3 | +-------+-------+-------+ | . . . | . . 4 | . 9 . | | 9 . . | . . . | 8 . . | | . 6 8 | 2 . . | . . 7 | +-------+-------+-------+ | . 9 . | . 6 . | . . . | | 8 . 2 | 4 7 . | . . . | | . . . | . . . | 2 . . | +-------+-------+-------+

ER 9.3 (not a one-stepper for me)

by **pjb** » Sun Jul 18, 2021 11:09 pm

Code: Select all: 6 1358 1359 | 135 4 135 | 7 128 1289 7 1358 135 | 1356 2 9 | 4 168 168 4 2 19 | 1678 18 67 | 169 5 3 ------------------------+----------------------+--------------------- 2 1357 1357 | 135678 1358 4 | 1356 9 156 9 13457 13457 | 13567 135 67 | 8 1236 1256 135 6 8 | 2 9 135 | 135 4 7 ------------------------+----------------------+--------------------- 135 9 47 | 135 6 2 | 135 78 48 8 135 2 | 4 7 135 | 13569 136 1569 135 47 6 | 9 135 8 | 2 137 145

My guess is it has to do with all the 135's.
My solver gets it in one, but the method uses T&E.
The 135's at r7c147 can be in final solution 135, 153, 315, 351, 513, or 531. Each of these arrangements is assigned in turn, and tested for contradiction. It turns out that all except 513 cause contradictions, and assigning this solves in singles. I'm sure Eleven has an elegant solution not involving T&E, and I await it's revelation. A MUG perhaps?

Phil

by **jco** » Mon Jul 19, 2021 1:27 am

Probably there is a much shorter way (probably missed some key idea).

Calling the correct digits: in r6c1=X, in r7c1=Y, in r9c1=Z, we must have

Code: Select all: .-----------------------------------------------------------. | 6 YZ8 YZ9 | YZ 4 X | 7 128 1289 | | 7 YZ8 X | YZ6 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 169 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XYZ678 XZ8-Y 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XYZ67 *XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 *YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 YZ | YZ69 136 1569 | | Z 47 6 | 9 *XY 8 | 2 137 145 | '-----------------------------------------------------------'

1. XYZ-Wing r9c5,r5c5,r6c6 => -Y r4c5

Code: Select all: .-----------------------------------------------------------. | 6 YZ8 YZ9 |*YZ 4 X | 7 128 1289 | | 7 YZ8 X | YZ6 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 169 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XZ678-Y XZ8 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XZ67-Y XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 *YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 |*XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 *YZ | YZ69 136 1569 | | Z 47 6 | 9 XY 8 | 2 137 145 | '-----------------------------------------------------------'

2. W-Wing(r6c6,r8c6,r7c4,r1c4) => -Y r45c4

Code: Select all: .-----------------------------------------------------------. | 6 YZ8 YZ-9 | YZ 4 X | 7 128 1289 | | 7 YZ8 X | YZ6 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 6-19 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XZ678 XZ8 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XZ67 XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 YZ | YZ69 136 1569 | | Z 47 6 | 9 XY 8 | 2 137 145 | '-----------------------------------------------------------'

3. (9=1)r3c3-(1=8)r3c5-(8)r4c5=[XY-Wing(r8c6,r9c5,r4c5)]-(Z=Y)r6c6-(Y=ZX)r67c7-(X)r4c7=[X-Wing(X) r45c45]-(X=ZY6)r172c4-(6=81)r2c89-(18=29)r1c89 (Edit: correction for this step in my third post)

=> X<>1, -(19) r3c7, -9r1c3 (+6 r3c7)

So, with X<>1, after some singles,

Code: Select all: .-----------------------------------------------------------. | 6 8 YZ | YZ 4 X | 7 2 9 | | 7 YZ X | 6 2 9 | 4 a18 b18 | | 4 2 9 | 18 18 7 | 6 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XZ78 XZ8 4 | XYZ 9 6 | | 9 YZ47 YZ47 | XZ7 XYZ 6 | 8 3-1 2 | | X 6 8 | 2 9 eYZ |fYZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 dYZ | 9 6 c15 | | Z 47 6 | 9 XY 8 | 2 137 145 | '-----------------------------------------------------------'

4. (1)r2c8=r2c9-r8c9=r8c6-r6c6=r6c7 => -1 r5c8 [+3r5c8, +3 r7c7]

Code: Select all: .-----------------------------------------------------------. | 6 8 YZ | YZ 4 X | 7 2 9 | | 7 YZ X | 6 2 9 | 4 18 18 | | 4 2 9 | 18 18 7 | 6 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XZ78 XZ8 4 | XYZ 9 6 | | 9 YZ47 YZ47 | XZ7 XYZ 6 | 8 3 2 | | X 6 8 | 2 9 YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | 3 78 48 | | 8 X 2 | 4 7 YZ | 9 6 15 | | Z 47 6 | 9 XY 8 | 2 17 145 | '-----------------------------------------------------------'

From row 7, we have Y<>3. Now, Noticing the places where X is the correct
digit in blocks 1 and 2, we infer (looking at b3) that X=3.

Code: Select all: .-----------------------------------------------------------. | 6 8 Z | Y 4 3 | 7 2 9 | | 7 Y 3 | 6 2 9 | 4 18 18 | | 4 2 9 | 18 18 7 | 6 5 3 | |-------------------+-------------------+-------------------| | 2 Z7 Y7 | X78 Z8 4 | YZ 9 6 | | 9 Z47 Y47 | X7 YZ 6 | 8 3 2 | | 3 6 8 | 2 9 YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | Z 6 2 | 3 78 48 | | 8 3 2 | 4 7 YZ | 9 6 15 | | Z 47 6 | 9 3 8 | 2 17 145 | '-----------------------------------------------------------'

Looking at NP(18)r2c89, we infer that Y<>1, so Y=5.
So, X=3,Y=5 and Z=1, and singles to the end.

Edit: Added the very last part after step 4 that was missing.

by **999_Springs** » Mon Jul 19, 2021 1:48 am

jco wrote:Probably there is a much shorter way (probably missed some key idea).

Calling the correct digits: in r6c1=X, in r7c1=Y, in r9c1=Z, we must have

Code: Select all
.-----------------------------------------------------------. | 6 YZ8 YZ9 | YZ 4 X | 7 128 1289 | | 7 YZ8 X | YZ6 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 169 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XYZ678 XYZ8 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XYZ67 XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 YZ | YZ69 136 1569 | | Z 47 6 | 9 XY 8 | 2 137 145 | '-----------------------------------------------------------'

at this point you miss a turbot fish (skyscraper) in Z's in rows 6 and 7 that gives r8c6=Y and it's all singles to the end from there.

this is basically how i solved it by hand with no pencilmarks (except the subsets) at the same time as you were writing your post. here is my solution

basics to here

Code: Select all: +--------+-------+--------+ | 6 . . | . 4 . | 7 . . | | 7 . . | . 2 9 | 4 . . | | 4 2 . | . . 67| . 5 3 | +--------+-------+--------+ | 2 . . | . . 4 | . 9 . | | 9 . . | . . 67| 8 . . | | . 6 8 | 2 9 . | . 4 7 | +--------+-------+--------+ | . 9 47| . 6 2 | . 78 48| | 8 . 2 | 4 7 . | . . . | | . 47 6 | 9 . 8 | 2 . . | +--------+-------+--------+

write A,B,C to fill in the missing cells in row 7 that must be 1,3,5 (i saw pjb's post and used it as a hint). as a manual aid to non-pencilmark solving, mark any cell that must be 1,3,5 with O

Code: Select all: +--------+-------+--------+ | 6 . . | O 4 O | 7 . . | | 7 . O | . 2 9 | 4 . . | | 4 2 . | . . 67| . 5 3 | +--------+-------+--------+ | 2 . . | . . 4 | . 9 . | | 9 . . | . O 67| 8 . . | | O 6 8 | 2 9 O | O 4 7 | +--------+-------+--------+ | A 9 47| B 6 2 | C 78 48| | 8 O 2 | 4 7 O | . . . | | O 47 6 | 9 O 8 | 2 . . | +--------+-------+--------+

turbot fish C in r6 and b8: r9c1=/=C, singles to here

Code: Select all: +--------+-------+--------+ | 6 . . | A 4 C | 7 . . | | 7 . O | . 2 9 | 4 . . | | 4 2 . | . . 67| . 5 3 | +--------+-------+--------+ | 2 . . | . . 4 | . 9 . | | 9 . . | . O 67| 8 . . | | C 6 8 | 2 9 B | A 4 7 | +--------+-------+--------+ | A 9 47| B 6 2 | C 78 48| | 8 C 2 | 4 7 A | . . . | | B 47 6 | 9 C 8 | 2 . . | +--------+-------+--------+

r3c5=B is a hidden single in b2 => B=1, singles to here

Code: Select all: +--------+-------+--------+ | 6 8 1 | A 4 C | 7 2 9 | | 7 A C | 6 2 9 | 4 1 8 | | 4 2 9 | 8 1 7 | 6 5 3 | +--------+-------+--------+ | 2 7 A | C 8 4 | 1 9 6 | | 9 1 4 | 7 A 6 | 8 C 2 | | C 6 8 | 2 9 1 | A 4 7 | +--------+-------+--------+ | A 9 7 | 1 6 2 | C 8 4 | | 8 C 2 | 4 7 A | 9 6 1 | | 1 4 6 | 9 C 8 | 2 7 A | +--------+-------+--------+

A=5 and C=3 done

pic of solution drawn on notes app on phone with one finger: Show

by **jco** » Mon Jul 19, 2021 2:48 am

999_Springs wrote:
jco wrote:Probably there is a much shorter way (probably missed some key idea).

Calling the correct digits: in r6c1=X, in r7c1=Y, in r9c1=Z, we must have

Code: Select all
.-----------------------------------------------------------. | 6 YZ8 YZ9 | YZ 4 X | 7 128 1289 | | 7 YZ8 X | YZ6 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 169 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XYZ678 XYZ8 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XYZ67 XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 YZ | YZ69 136 1569 | | Z 47 6 | 9 XY 8 | 2 137 145 | '-----------------------------------------------------------'

at this point you miss a turbot fish (skyscraper) in Z's in rows 6 and 7 that gives r8c6=Y and it's all singles to the end from there.

this is basically how i solved it by hand with no pencilmarks (except the subsets) at the same time as you were writing your post. here is my solution (..)

Very nice!! Thanks for your comments. I had the feeling I was missing something, and that increased when I finished step 3 (took me a long time to find it to get X<>1, and then prolonging the chain to get that additional eliminations).
It was a lot of fun anyway!
Regards,

Website · by **denis_berthier** » Mon Jul 19, 2021 5:05 am

.
There is a standard simplest-fist solution in W13 or B12, with nothing noticeable in the resolution path.

There is also an alternative solution in 4 steps using Forcing-T&E.

As SER 9.3 means an extremely hard puzzle, I take the liberty of applying trivial steps

Code: Select all: hidden-pairs-in-a-block: b7{n4 n7}{r7c3 r9c2} ==> r9c2 ≠ 5, r9c2 ≠ 3, r9c2 ≠ 1, r7c3 ≠ 5, r7c3 ≠ 3, r7c3 ≠ 1 hidden-pairs-in-a-column: c6{n6 n7}{r3 r5} ==> r5c6 ≠ 5, r5c6 ≠ 3, r5c6 ≠ 1, r3c6 ≠ 8, r3c6 ≠ 1 hidden-single-in-a-column ==> r9c6 = 8 naked-triplets-in-a-row: r7{c1 c4 c7}{n5 n3 n1} ==> r7c9 ≠ 5, r7c9 ≠ 1, r7c8 ≠ 3, r7c8 ≠ 1

before the real starting point of both solutions.

Code: Select all: Resolution state after Singles, whips[1] and Subsets have been applied: +----------------------+----------------------+----------------------+ ! 6 1358 1359 ! 135 4 135 ! 7 128 1289 ! ! 7 1358 135 ! 1356 2 9 ! 4 168 168 ! ! 4 2 19 ! 1678 18 67 ! 169 5 3 ! +----------------------+----------------------+----------------------+ ! 2 1357 1357 ! 135678 1358 4 ! 1356 9 156 ! ! 9 13457 13457 ! 13567 135 67 ! 8 1236 1256 ! ! 135 6 8 ! 2 9 135 ! 135 4 7 ! +----------------------+----------------------+----------------------+ ! 135 9 47 ! 135 6 2 ! 135 78 48 ! ! 8 135 2 ! 4 7 135 ! 13569 136 1569 ! ! 135 47 6 ! 9 135 8 ! 2 137 145 ! +----------------------+----------------------+----------------------+

1) See the simplest-first solution in W13: Show

biv-chain[3]: c6n6{r5 r3} - b2n7{r3c6 r3c4} - c4n8{r3 r4} ==> r4c4 ≠ 6
whip[1]: r4n6{c9 .} ==> r5c8 ≠ 6, r5c9 ≠ 6
t-whip[4]: r3c3{n1 n9} - r1n9{c3 c9} - r1n2{c9 c8} - r1n8{c8 .} ==> r1c2 ≠ 1
whip[6]: c5n3{r5 r9} - c8n3{r9 r8} - c8n6{r8 r2} - c4n6{r2 r3} - c4n7{r3 r4} - c4n8{r4 .} ==> r5c4 ≠ 3
whip[11]: r4n8{c4 c5} - c5n5{r4 r9} - c5n3{r9 r5} - c5n1{r5 r3} - r3c3{n1 n9} - c7n9{r3 r8} - c9n9{r8 r1} - c9n2{r1 r5} - r5c8{n2 n1} - c4n1{r5 r7} - c7n1{r7 .} ==> r4c4 ≠ 5
whip[12]: r5n6{c4 c6} - b5n7{r5c6 r4c4} - c4n8{r4 r3} - r3n6{c4 c7} - c7n9{r3 r8} - c9n9{r8 r1} - c9n2{r1 r5} - r5c8{n2 n3} - r5c5{n3 n5} - r6c6{n5 n3} - r8n3{c6 c2} - c1n3{r7 .} ==> r5c4 ≠ 1
t-whip[9]: r4n7{c3 c4} - r4n8{c4 c5} - r3c5{n8 n1} - c4n1{r3 r7} - b5n1{r4c4 r6c6} - b5n3{r6c6 r5c5} - b8n3{r9c5 r8c6} - c8n3{r8 r9} - r9n7{c8 .} ==> r5c2 ≠ 7
whip[12]: r4n8{c4 c5} - r3c5{n8 n1} - r3c3{n1 n9} - r1n9{c3 c9} - r1n2{c9 c8} - r1n1{c8 c3} - c4n1{r1 r7} - b5n1{r4c4 r6c6} - b5n3{r6c6 r5c5} - r5c8{n3 n1} - c7n1{r4 r8} - r8n9{c7 .} ==> r4c4 ≠ 7
whip[1]: b5n7{r5c6 .} ==> r5c3 ≠ 7
hidden-pairs-in-a-row: r5{n6 n7}{c4 c6} ==> r5c4 ≠ 5
whip[13]: r3c3{n1 n9} - r1n9{c3 c9} - c9n2{r1 r5} - r5c8{n2 n3} - r5c5{n3 n5} - r5c2{n5 n4} - r9n4{c2 c9} - r9n5{c9 c1} - r6n5{c1 c7} - r6n1{c7 c6} - c1n1{r6 r7} - b8n1{r7c4 r9c5} - r9n3{c5 .} ==> r5c3 ≠ 1
whip[13]: r4n8{c4 c5} - r3c5{n8 n1} - r3c3{n1 n9} - c7n9{r3 r8} - c9n9{r8 r1} - r1n2{c9 c8} - r1n1{c8 c3} - c6n1{r1 r8} - c2n1{r8 r5} - r5c8{n1 n3} - b5n3{r5c5 r6c6} - r8n3{c6 c2} - c1n3{r7 .} ==> r4c4 ≠ 1
whip[13]: c2n7{r4 r9} - r9n4{c2 c9} - r7n4{c9 c3} - r5c3{n4 n3} - r6c1{n3 n1} - b7n1{r7c1 r8c2} - c6n1{r8 r1} - c4n1{r3 r7} - b9n1{r7c7 r9c8} - c8n3{r9 r8} - r8c6{n3 n5} - b5n5{r6c6 r5c5} - c9n5{r5 .} ==> r4c2 ≠ 5
whip[7]: r5n4{c3 c2} - r9n4{c2 c9} - r7n4{c9 c3} - c3n7{r7 r4} - b4n5{r4c3 r6c1} - r9n5{c1 c5} - b5n5{r4c5 .} ==> r5c3 ≠ 3
whip[12]: r1n2{c8 c9} - c9n8{r1 r7} - r7n4{c9 c3} - r5c3{n4 n5} - r5c9{n5 n1} - r2c9{n1 n6} - r4c9{n6 n5} - c5n5{r4 r9} - b5n5{r5c5 r6c6} - r6n1{c6 c1} - r9n1{c1 c8} - r2c8{n1 .} ==> r1c8 ≠ 8
whip[12]: b1n1{r1c3 r2c2} - r3c3{n1 n9} - c7n9{r3 r8} - c9n9{r8 r1} - r1n2{c9 c8} - b3n1{r1c8 r3c7} - c7n6{r3 r4} - r4c9{n6 n5} - r6c7{n5 n3} - r6c1{n3 n5} - b5n5{r6c6 r5c5} - r9n5{c5 .} ==> r4c3 ≠ 1
whip[1]: c3n1{r3 .} ==> r2c2 ≠ 1
whip[12]: c1n5{r9 r6} - c6n5{r6 r1} - c4n5{r2 r7} - r9n5{c5 c9} - r5n5{c9 c5} - r5c3{n5 n4} - r7c3{n4 n7} - r4c3{n7 n3} - b5n3{r4c5 r6c6} - r6c7{n3 n1} - r7c7{n1 n3} - r8n3{c7 .} ==> r8c2 ≠ 5
whip[1]: b7n5{r9c1 .} ==> r6c1 ≠ 5
whip[2]: r8n5{c9 c6} - r6n5{c6 .} ==> r7c7 ≠ 5
whip[5]: b5n1{r5c5 r6c6} - c1n1{r6 r7} - r8c2{n1 n3} - r8c6{n3 n5} - r7n5{c4 .} ==> r9c5 ≠ 1
whip[3]: b8n1{r7c4 r8c6} - b7n1{r8c2 r9c1} - r6n1{c1 .} ==> r7c7 ≠ 1
singles ==> r7c7 = 3, r5c8 = 3, r5c9 = 2, r1c8 = 2
whip[2]: r9n3{c5 c1} - r6n3{c1 .} ==> r4c5 ≠ 3, r8c6 ≠ 3

2) Solution in 4 steps:

FORCING[3]-T&E(W1) applied to trivalue candidates n1r3c7, n6r3c7 and n9r3c7 :
===> 4 candidates eliminated in the three cases: n1r1c2 n6r4c4 n6r5c8 n6r5c9

Code: Select all: +-------------------+-------------------+-------------------+ ! 6 358 1359 ! 135 4 135 ! 7 128 1289 ! ! 7 1358 135 ! 1356 2 9 ! 4 168 168 ! ! 4 2 19 ! 1678 18 67 ! 169 5 3 ! +-------------------+-------------------+-------------------+ ! 2 1357 1357 ! 13578 1358 4 ! 1356 9 156 ! ! 9 13457 13457 ! 13567 135 67 ! 8 123 125 ! ! 135 6 8 ! 2 9 135 ! 135 4 7 ! +-------------------+-------------------+-------------------+ ! 135 9 47 ! 135 6 2 ! 135 78 48 ! ! 8 135 2 ! 4 7 135 ! 13569 136 1569 ! ! 135 47 6 ! 9 135 8 ! 2 137 145 ! +-------------------+-------------------+-------------------+

FORCING[3]-T&E(W1) applied to trivalue candidates n1r6c1, n1r6c6 and n1r6c7 :
===> 1 candidates eliminated in the three cases: n1r7c7

Code: Select all: +-------------------+-------------------+-------------------+ ! 6 358 1359 ! 135 4 135 ! 7 128 1289 ! ! 7 1358 135 ! 1356 2 9 ! 4 168 168 ! ! 4 2 19 ! 1678 18 67 ! 169 5 3 ! +-------------------+-------------------+-------------------+ ! 2 1357 1357 ! 13578 1358 4 ! 1356 9 156 ! ! 9 13457 13457 ! 13567 135 67 ! 8 123 125 ! ! 135 6 8 ! 2 9 135 ! 135 4 7 ! +-------------------+-------------------+-------------------+ ! 135 9 47 ! 135 6 2 ! 35 78 48 ! ! 8 135 2 ! 4 7 135 ! 13569 136 1569 ! ! 135 47 6 ! 9 135 8 ! 2 137 145 ! +-------------------+-------------------+-------------------+

FORCING[3]-T&E(W1) applied to trivalue candidates n1r8c6, n3r8c6 and n5r8c6 :
===> 10 candidates eliminated in the three cases: n1r1c6 n1r3c7 n1r4c9 n1r5c8 n1r5c9 n1r6c1 n1r8c2 n1r8c7 n1r8c8 n1r9c5

Code: Select all: +-------------------+-------------------+-------------------+ ! 6 358 1359 ! 135 4 35 ! 7 128 1289 ! ! 7 1358 135 ! 1356 2 9 ! 4 168 168 ! ! 4 2 19 ! 1678 18 67 ! 69 5 3 ! +-------------------+-------------------+-------------------+ ! 2 1357 1357 ! 13578 1358 4 ! 1356 9 56 ! ! 9 13457 13457 ! 13567 135 67 ! 8 23 25 ! ! 35 6 8 ! 2 9 135 ! 135 4 7 ! +-------------------+-------------------+-------------------+ ! 135 9 47 ! 135 6 2 ! 35 78 48 ! ! 8 35 2 ! 4 7 135 ! 3569 36 1569 ! ! 135 47 6 ! 9 35 8 ! 2 137 145 ! +-------------------+-------------------+-------------------+

FORCING[3]-T&E(W1) applied to trivalue candidates n1r6c6, n3r6c6 and n5r6c6 :
===> 10 values decided in the three cases: n6r3c7 n7r3c6 n6r5c6 n6r2c4 n9r8c7 n2r1c8 n2r5c9 n1r5c2 n8r1c2 n7r5c4
===> 58 candidates eliminated in the three cases: n3r1c2 n5r1c2 n3r1c3 n5r1c3 n3r1c4 n1r1c8 n8r1c8 n1r1c9 n2r1c9 n8r1c9 n1r2c2 n8r2c2 n1r2c3 n1r2c4 n3r2c4 n5r2c4 n6r2c8 n6r2c9 n1r3c4 n6r3c4 n7r3c4 n6r3c6 n9r3c7 n1r4c2 n3r4c2 n5r4c2 n1r4c3 n3r4c3 n1r4c4 n5r4c4 n7r4c4 n3r4c5 n5r4c5 n6r4c7 n3r5c2 n4r5c2 n5r5c2 n7r5c2 n1r5c3 n7r5c3 n1r5c4 n3r5c4 n5r5c4 n6r5c4 n1r5c5 n3r5c5 n7r5c6 n2r5c8 n5r5c9 n3r6c7 n3r7c1 n3r8c6 n3r8c7 n5r8c7 n6r8c7 n9r8c9 n3r9c8 n1r9c9

Code: Select all: +-------------+-------------+-------------+ ! 6 8 19 ! 15 4 35 ! 7 2 9 ! ! 7 35 35 ! 6 2 9 ! 4 18 18 ! ! 4 2 19 ! 8 18 7 ! 6 5 3 ! +-------------+-------------+-------------+ ! 2 7 57 ! 38 18 4 ! 135 9 56 ! ! 9 1 345 ! 7 5 6 ! 8 3 2 ! ! 35 6 8 ! 2 9 135 ! 15 4 7 ! +-------------+-------------+-------------+ ! 15 9 47 ! 135 6 2 ! 35 78 48 ! ! 8 35 2 ! 4 7 15 ! 9 36 156 ! ! 135 47 6 ! 9 35 8 ! 2 17 45 ! +-------------+-------------+-------------+

stte

Note that this is the first result of applying Forcing-T&E. Others tries might lead to fewer steps (ask Robert). But I'm not a great fan of FTE.

by **jco** » Mon Jul 19, 2021 11:59 am

Hello,

jco wrote: (...)
Code: Select all
.-----------------------------------------------------------. | 6 YZ8 YZ9 | YZ 4 X | 7 128 1289 | | 7 YZ8 X | YZ6 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 169 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XZ678 XZ8 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XZ67 XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 YZ | YZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 | XZ 6 2 | XZ 78 48 | | 8 X 2 | 4 7 YZ | YZ69 136 1569 | | Z 47 6 | 9 XY 8 | 2 137 145 | '-----------------------------------------------------------'

3. (9=1)r3c3-(1=8)r3c5-(8)r4c5=[XY-Wing(r8c6,r9c5,r4c5)]-(Z=Y)r6c6-(Y=ZX)r67c7-(X)r4c7=[X-Wing(X) r45c45]-(X=ZY6)r172c4-(6=81)r2c89-(18=29)r1c89
=> X<>1, -(19) r3c7, -9r1c3 (+6 r3c7)
(...)

Looking again at this third step, I see that there is a flaw in the part
... (X)r4c7=[X-Wing(X) r45c45] ...
because 1,3,5 appear at rows 4 and 5. But there is no need for the X-Wing.
So the third step should be

Code: Select all: .-----------------------------------------------------------. | 6 YZ8 YZ9 |fYZ 4 X | 7 128 1289 | | 7 YZ8 X |fYZ6 2 9 | 4 g168 g168 | | 4 2 19 | 1678 a18 67 | 169 5 3 | |-------------------+-------------------+-------------------| | 2 YZ7 YZ7 | XZ678 cbXZ8 4 | XYZ6 9 156 | | 9 YZ47 YZ47 | XZ67 XYZ 67 | 8 1236 1256 | | X 6 8 | 2 9 dYZ |eYZ 4 7 | |-------------------+-------------------+-------------------| | Y 9 47 |fXZ 6 2 |eXZ 78 48 | | 8 X 2 | 4 7 cYZ | YZ69 136 1569 | | Z 47 6 | 9 cXY 8 | 2 137 145 | '-----------------------------------------------------------'

3. (9=1)r3c3-(1=8)r3c5-(8)r4c5=[XY-Wing(r8c6,r9c5,r4c5)]-(Z=Y)r6c6-(Y=ZX)r67c7-(X=ZY6)r712c4-(6=81)r2c89-(18=29)r1c89
=> X<>1, -(19) r3c7, -9r1c3 (+6 r3c7)
Labels "a" to "g" only for the un-prolonged part of the chain (that gets X<>1).
Of course other steps remain the same.

by **eleven** » Mon Jul 19, 2021 7:29 pm

Code: Select all: +--------------------------+-------------------------+-------------------------+ | 6 1358 1359 | 135 4 135 | 7 128 1289 | | 7 1358 135 | 1356 2 9 | 4 168 168 | | 4 2 19 | 1678 18 67 | 169 5 3 | +--------------------------+-------------------------+-------------------------+ | 2 1357 1357 | 135678 T1358 4 | 1356 9 156 | | 9 13457 13457 | 13567 T135 67 | 8 1236 1256 | |+#135 6 8 | 2 9 T#135 |*135 4 7 | +--------------------------+-------------------------+-------------------------+ | -135 9 47 | 135 6 2 |B135 78 48 | | 8 #135 2 | 4 7 #135 |*13569 *136 *1569 | | -135 47 6 | 9 +135 8 | 2 *137 *145 | +-------------------------+-------------------------+-------------------------+

The digit x (one of 135) in r7c7 (eliminating it in the starred cells r6c7,r89c789) leaves 3 strong links for x in r689.
The skyscraper in r68 eliminates it in r79c1, forcing x in r6c1, and with the link in r9 in r9c5.
Thus this x of the 3 digits is eliminated from both r45c5 and r6c6, leaving a triple, and r4c5 must be 8.
(If we can show, that m digits in n cells of a unit cannot all be there, 1+n-m of the other candidates must be there)

Code: Select all: *-----------------------------------------------------------------* | 6 8 1 | 35 4 da35 | 7 2 9 | | 7 35 35 | 6 2 9 | 4 18 18 | | 4 2 9 | 8 1 7 | 6 5 3 | |-----------------------+--------------------+--------------------| | 2 1357 357 | 1357 8 4 | 135 9 6 | | 9 13457 3457 | 1357 35 6 | 8 13 2 | | a135 6 8 | 2 9 1-35 | 135 4 7 | |-----------------------+--------------------+--------------------| | b135 9 47 | 135 6 2 | 13 78 48 | | 8 c135 2 | 4 7 d135 | 9 6 15 | | b135 47 6 | 9 35 8 | 2 137 145 | *-----------------------------------------------------------------*

35r1c6,rc61 = 1r6c1 - r79c1 = r8c2 - (1=35)r18c6 => -35r6c6, stte

by **Cenoman** » Mon Jul 19, 2021 9:56 pm

JCO's method (replace 135 with xyz everywhere except in one sector), and assuming r6c1=x, r6c6=y, r6c7=z, i.e. using solution in row 6 instead of column 1, leads to the puzzle solution with singles to the end.

by **marek stefanik** » Mon Jul 19, 2021 10:37 pm

Very cool puzzle and some beautiful solutions proposed.

The one digit solution is one of the coolest things I've seen in a sudoku, though reusing Xr7c7 => Xr9c5 to finish the puzzle seems a bit more thematic.

I also think there is a flaw in the chain, it doesn't cover the case where r1c6 and r6c1 contain the same digit (which we actually know is true from step 1).

by **jco** » Mon Jul 19, 2021 11:36 pm

Only now I noticed that the very ending in my resolution after step 4 was missing (but there was no further step after the fourth). Added.
I also enjoyed the puzzle and reading all solutions and comments made.

Website · by **denis_berthier** » Tue Jul 20, 2021 4:16 am

.
The XYZ trick by jco and eleven is very smart.

I remember seeing it before in another puzzle. Does anyone remember where?

by **jco** » Tue Jul 20, 2021 1:30 pm

I recall mith had a puzzle and later showed a solution using that idea.
I would have trouble finding it in my records (of all mith puzzles!), but I found it here by searching for "ABC" (first searched for XYZ, but of course that was a bad idea).
In there, mith mentions a link for a previous puzzle with that idea.
Had I chose that letters (ABC) instead of XYZ, I would not have gotten into "wing mode" and would not have missed that SS.
It is remarkable that (in the current puzzle) there is a solution without need of any chain (ste for start!), which is the one found by Cenoman.

Edit: another link. The oldest reference I have found ("ABC" search) is this one (eleven).

Website · by **denis_berthier** » Tue Jul 20, 2021 3:35 pm

jco, thanks for the reference
I had searched for XYZ but I didn't think of trying ABC.

Now that I re-read this old thread, I remember the discussion there about the unknowable complexity of the variable change.

jco wrote:It is remarkable that (in the current puzzle) there is a solution without need of any chain (ste for start!), which is the one found by Cenoman.

Contrary to the puzzle in the other thread, your change of variables doesn't imply any loss of information. On the contrary, as the original puzzle is in W13 and the resulting one can be solved by much more elementary techniques (in 999_Spring or Cenoman's variants), there must be some information input - which is provided, as in the other thread, by the transformation.

The New Sudoku Players' Forum

Enjoy

Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy

Re: Enjoy