The New Sudoku Players' Forum

[urhegyi]

This is the nr 75 of the april 2020 edition of a famous sudoku puzzle magazine from last year.
Mostly the 12 stars level is a combination of swordfish, skyscraper, string-kite, w-wing, UR and x-wing, xy-wing, xyz-wing and xyzw-wing.
Thisone I can solve upto:

Code: Select all

87..9621.251..8.699..12...7.3...2..1.......2.4.28...3.3..2.16.862.9..14.1.56...72

Finding an opening without the use of trial and error took me over 4 hours. And it's none of the ones described above.
Has anyone new fresh ideas?

denksport 75 april 2020.png (15.32 KiB) Viewed 625 times

[jco]

Hello,

Code: Select all

 1   2     3     4     5       6       7     8     9
.---------------+---------------------+-----------------.
| 8   7   d'34  |c345   9       6     | 2      1    45-3| 1
| 2   5     1   |b347  b347     8     |a34     6    9   | 2
| 9   46    346 | 1     2      c345   | 3458   58   7   | 3
|---------------+---------------------+-----------------|
| 57  3     689 | 457   4567    2     | 45789  589  1   | 4
| 57  1689  689 | 3457  134567  34579 | 45789  2    456 | 5
| 4   169   2   | 8     1567    579   | 579    3    56  | 6
|---------------+---------------------+-----------------|
| 3   49    479 | 2     457     1     | 6      59   8   | 7
| 6   2     78  | 9     3578    357   | 1      4   f35  | 8
| 1   489   5   | 6     348    d34    |e39     7    2   | 9
'---------------+---------------------+-----------------'

I describe next the way I saw an elimination after basics. The first thing I noticed was two "34" at r1c3 and r2c7 that seem to ask for a W-Wing. That wing is not there but if r7c3<>3 (being 3 implies immediately r1c9<>3), then (being 4) we have a locked pair 37 at r2c45, so r1c4 would be 4 or 5 (same with r3c6).
In case it is 4, we have r1c3=3, implying r1c9<>3; in case r1c4 is 5, then r3c6 must be 4, which implies that r9c6=3, so r9c7 cannot be 3, which in turn implies r8c9=3, and so we have again r1c9<>3.
After a few attempts, I did not manage to write this as an AIC.

Regards,
jco

Edit: included board, corrected and improved text.

[denis_berthier]

SER 8.3
It can be solved with chains no longer than 4, but it's not an easy one.

(solve "87..9621.251..8.699..12...7.3...2..1.......2.4.28...3.3..2.16.862.9..14.1.56...72")
***********************************************************************************************
*** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = TyW+B+gW+SFin
*** Download from: https://github.com/denis-berthier/CSP-Rules-V2.1
***********************************************************************************************

Code: Select all

Starting non trivial part of solution with the following RESOLUTION STATE:
   8         7         34        345       9         6         2         1         345       
   2         5         1         347       347       8         34        6         9         
   9         46        346       1         2         345       3458      58        7         
   57        3         6789      457       4567      2         45789     589       1         
   57        1689      6789      3457      134567    34579     45789     2         456       
   4         169       2         8         1567      579       579       3         56       
   3         49        479       2         457       1         6         59        8         
   6         2         78        9         3578      357       1         4         35       
   1         489       5         6         348       34        39        7         2       
130 candidates, 642 csp-links and 642 links. Density = 7.66%

whip[1]: c1n7{r5 .} ==> r5c3 ≠ 7, r4c3 ≠ 7
93 g-candidates, 378 csp-glinks and 237 non-csp glinks
biv-chain[4]: r5n1{c5 c2} - c2n8{r5 r9} - r8c3{n8 n7} - r7n7{c3 c5} ==> r5c5 ≠ 7
z-chain[4]: c9n4{r5 r1} - r1n5{c9 c4} - r3c6{n5 n3} - r9c6{n3 .} ==> r5c6 ≠ 4
z-chain[4]: c9n3{r8 r1} - r1n5{c9 c4} - r3c6{n5 n4} - r9c6{n4 .} ==> r8c6 ≠ 3
whip[4]: c9n4{r5 r1} - r1n5{c9 c4} - c4n3{r1 r2} - r2c7{n3 .} ==> r5c4 ≠ 4
g-whip[4]: r9c7{n9 n3} - r2n3{c7 c456} - c6n3{r3 r5} - c6n9{r5 .} ==> r6c7 ≠ 9
t-whip[4]: r6n9{c6 c2} - c3n9{r5 r7} - r7n7{c3 c5} - r8c6{n7 .} ==> r6c6 ≠ 5
finned-x-wing-in-rows: n5{r7 r6}{c5 c8} ==> r4c8 ≠ 5
t-whip-rn[4]: r6n5{c9 c5} - r7n5{c5 c8} - r8n5{c9 c6} - r3n5{c6 .} ==> r5c7 ≠ 5, r4c7 ≠ 5
t-whip[4]: r8c6{n7 n5} - r7n5{c5 c8} - r3n5{c8 c7} - r6c7{n5 .} ==> r6c6 ≠ 7
naked-single ==> r6c6 = 9
naked-triplets-in-a-row: r5{c1 c4 c6}{n7 n5 n3} ==> r5c9 ≠ 5, r5c7 ≠ 7, r5c5 ≠ 5, r5c5 ≠ 3
whip[1]: b6n5{r6c9 .} ==> r6c5 ≠ 5
biv-chain[3]: r5n1{c5 c2} - r6c2{n1 n6} - b6n6{r6c9 r5c9} ==> r5c5 ≠ 6
biv-chain-bn[4]: b9n9{r9c7 r7c8} - b9n5{r7c8 r8c9} - b6n5{r6c9 r6c7} - b6n7{r6c7 r4c7} ==> r4c7 ≠ 9
biv-chain[4]: r1n5{c4 c9} - c7n5{r3 r6} - c7n7{r6 r4} - r4c1{n7 n5} ==> r4c4 ≠ 5
biv-chain[4]: r4c4{n4 n7} - b6n7{r4c7 r6c7} - c7n5{r6 r3} - b2n5{r3c6 r1c4} ==> r1c4 ≠ 4
biv-chain[3]: r3c2{n6 n4} - r1n4{c3 c9} - r5c9{n4 n6} ==> r5c2 ≠ 6
biv-chain[3]: c3n3{r3 r1} - r1n4{c3 c9} - r2c7{n4 n3} ==> r3c7 ≠ 3
finned-x-wing-in-columns: n3{c7 c5}{r2 r9} ==> r9c6 ≠ 3
naked-single ==> r9c6 = 4
whip[1]: b2n4{r2c5 .} ==> r2c7 ≠ 4
stte


Starting from the same PM, it can also be solved using only reversible chains (bivalue-chains and z-chains) but then length has to be up to 7:
biv-chain[4]: r5n1{c5 c2} - c2n8{r5 r9} - r8c3{n8 n7} - r7n7{c3 c5} ==> r5c5 ≠ 7
z-chain[4]: c9n4{r5 r1} - r1n5{c9 c4} - r3c6{n5 n3} - r9c6{n3 .} ==> r5c6 ≠ 4
z-chain[4]: c9n3{r8 r1} - r1n5{c9 c4} - r3c6{n5 n4} - r9c6{n4 .} ==> r8c6 ≠ 3
z-chain[5]: c9n4{r5 r1} - r1n5{c9 c4} - c4n3{r1 r2} - r2c7{n3 n4} - r4n4{c7 .} ==> r5c4 ≠ 4
z-chain[7]: b8n3{r9c5 r9c6} - c6n4{r9 r3} - r2c5{n4 n7} - r7n7{c5 c3} - r8c3{n7 n8} - c2n8{r9 r5} - r5n1{c2 .} ==> r5c5 ≠ 3
z-chain[6]: b2n5{r1c4 r3c6} - r8c6{n5 n7} - r8c3{n7 n8} - r9n8{c2 c5} - c5n3{r9 r8} - c9n3{r8 .} ==> r1c4 ≠ 3
z-chain[3]: b1n4{r3c3 r1c3} - r1n3{c3 c9} - r2c7{n3 .} ==> r3c7 ≠ 4
z-chain[4]: r9c6{n4 n3} - b9n3{r9c7 r8c9} - r1n3{c9 c3} - b1n4{r1c3 .} ==> r3c6 ≠ 4
hidden-single-in-a-column ==> r9c6 = 4
whip[1]: b8n3{r9c5 .} ==> r2c5 ≠ 3
whip[1]: r3n4{c3 .} ==> r1c3 ≠ 4
stte

[Cenoman]

In three steps:

Code: Select all

 +--------------------+--------------------------+----------------------+
 |  8    7      34    | x345    9        6       |  2       1    y345   |
 |  2    5      1     | b347   b347      8       | c34      6     9     |
 |  9    46     346   |  1      2     wAa345     |  3458    58    7     |
 +--------------------+--------------------------+----------------------+
 |  57   3      689   |  457    4567     2       |  45789   589   1     |
 |  57   1689   689   |  3457   134567   34579   |  45789   2     456   |
 |  4    169    2     |  8      1567     579     |  579     3     56    |
 +--------------------+--------------------------+----------------------+
 |  3    49     479   |  2      457      1       |  6       59    8     |
 |  6    2      78    |  9      3578     357     |  1       4    z35    |
 |  1    489    5     |  6      348     B34      |  9-3     7     2     |
 +--------------------+--------------------------+----------------------+

1. Kraken cell (345)r3c6
(3)r3c6 - r2c45 = (3)r2c7
(4)r3c6 - (4=3)r9c6
(5)r3c6 - r1c4 = r1c9 - (5=3)r8c9
=> -3 r9c7; 5 placements & basics

Note: without 3 at r3c6, there would be an ALS W-Wing, so as a single line AIC:
Almost ALS W-Wing: [(3=5)r8c9 - r1c9 = r1c4 - (5*=*43)r39c6] = (3)r3c6 - r2c45 = (3)r2c7 => -3 r9c7

Last two steps are a grouped S-Wing and an ALS M-Wing

Code: Select all

 +--------------------+--------------------------+--------------------+
 |  8    7      34    |Cb345    9        6       |  2      1  Bc45    |
 |  2    5      1     | e347   e347      8       | d34     6    9     |
 |  9    46     346   |  1      2      Da35-4    |  345    8    7     |
 +--------------------+--------------------------+--------------------+
 |  57   3      68    |  457    4567     2       |  4578   9    1     |
 |  57   1689   689   |  3457   134567   3579-4  |  4578   2   A456   |
 |  4    169    2     |  8      1567     579     |  57     3    56    |
 +--------------------+--------------------------+--------------------+
 |  3    49     479   |  2      47       1       |  6      5    8     |
 |  6    2      78    |  9      578      57      |  1      4    3     |
 |  1    48     5     |  6      348     D34      |  9      7    2     |
 +--------------------+--------------------------+--------------------+

2. (5)r3c6 = r1c4 - (5=4)r1c9 - r2c7 = (4)r2c45 => -4 r3c6
3. (4)r5c9 = (4-5)r1c9 = r1c4 - (5=34)r39c6 => -4 r5c6; ste

[denis_berthier]

On seeing Cenoman's 3-step solution, I wondered why nobody has proposed a solution with fewer steps.

There are only 6 anti-backdoors (whether or not you allow whips[1] and Subsets at the end:

Code: Select all

6 BRT-ANTI-BACKDOORS FOUND:
n8r9c5 n8r8c3 n7r7c3 n6r6c9 n4r5c9 n8r5c2

6 W1-ANTI-BACKDOORS FOUND:
n8r9c5 n8r8c3 n7r7c3 n6r6c9 n4r5c9 n8r5c2

6 S-ANTI-BACKDOORS FOUND:
n8r9c5 n8r8c3 n7r7c3 n6r6c9 n4r5c9 n8r5c2



None of them allows a solution with a single whip.


My big surprise came when I looked for the anti-backdoor pairs (too many for listing them here):

Code: Select all

615 BRT-ANTI-BACKDOOR-PAIRS FOUND

622 W1-ANTI-BACKDOOR-PAIRS FOUND
626 S-ANTI-BACKDOOR-PAIRS FOUND

The world of Sudoku remains full of surprises!
Of these pairs, none gives rise to a 2-step solution using only reversible chains (i.e. bivalue-chains or z-chains).

But a lot of them give rise to a 2-step solution using whips. I'll give here only a few of the simplest ones.
They start with the same whip (followed by the same Singles):

Code: Select all

whip[6]: c9n3{r8 r1} - r1n5{c9 c4} - r1n4{c4 c3} - c3n3{r1 r3} - r3c6{n3 n4} - r9c6{n4 .} ==> r9c7 ≠ 3
singles ==> r9c7 = 9, r7c8 = 5, r3c8 = 8, r4c8 = 9, r8c9 = 3

After that, here are a few possibilities allowing an stte solution, the first being the simplest:

Code: Select all

z-chain[5]: r1n5{c4 c9} - c9n4{r1 r5} - c6n4{r5 r9} - r7c5{n4 n7} - r8c6{n7 .} ==> r3c6 ≠ 5
OR
whip[5]: r1c9{n4 n5} - b2n5{r1c4 r3c6} - r8c6{n5 n7} - r7c5{n7 n4} - c6n4{r9 .} ==> r5c9 ≠ 4
OR
whip[5]: r3n5{c7 c6} - r8c6{n5 n7} - r7c5{n7 n4} - c6n4{r9 r5} - c9n4{r5 .} ==> r1c9 ≠ 5
OR
whip[7]: c2n8{r9 r5} - r5n1{c2 c5} - c5n3{r5 r2} - r2c7{n3 n4} - b6n4{r4c7 r5c9} - r5n6{c9 c3} - r4c3{n6 .} ==> r9c5 ≠ 8

[pjb]

Here's 2 steps, but they are not too pretty:

Code: Select all

 8       7      c34     |b345    9      6      | 2      1     b345   
 2       5       1      | 347    347    8      | 34     6      9     
 9      d46     d346    | 1      2     e345    | 3458   58     7     
------------------------+----------------------+---------------------
 57      3       689    | 457    4567   2      | 45789  589    1     
 57      1689    689    | 3457   134567 34579  | 45789  2      456   
 4       169     2      | 8      1567   579    | 579    3      56     
------------------------+----------------------+---------------------
 3       49      479    | 2      457    1      | 6      59     8     
 6       2       78     | 9      578-3  57-3   | 1      4     a35     
 1       489     5      | 6      348   f34     | 9-3    7      2     


(3)r8c9 = (3-5)r1c9 = (5^-4)r1c49 = (4)r1c3 - (4=3)r3c23 - (3|5^=4)r3c6 - (4=3)r9c6 => -3 r8c56, r9c7; giving

Code: Select all

 8       7       34     |b345    9      6      | 2      1     a45     
 2       5       1      | 347    347    8      | 34     6      9     
 9       46      346    | 1      2     c345    | 345    8      7     
------------------------+----------------------+---------------------
 57      3       68     | 457    4567   2      | 4578   9      1     
 57      1689    689    | 3457  g134567 34579  | 4578   2      56-4   
 4       169     2      | 8      1567   579    | 57     3      56     
------------------------+----------------------+---------------------
 3       49      479    | 2     e47     1      | 6      5      8     
 6       2       78     | 9      578   d57     | 1      4      3     
 1       48      5      | 6      348   f34     | 9      7      2     


(4=5)r1c9 - (5)r1c4 = (5)r3c6 - (5=7)r8c6 - (7=4)r7c5 - (4)r9*3c6 = (4)r5c6 => -4 r5c9; stte

Phil

[denis_berthier]

Here's 2 steps, but they are not too pretty:
(3)r8c9 = (3-5)r1c9 = (5^-4)r1c49 = (4)r1c3 - (4=3)r3c23 - (3|5^=4)r3c6 - (4=3)r9c6 => -3 r8c56, r9c7; giving
(4=5)r1c9 - (5)r1c4 = (5)r3c6 - (5=7)r8c6 - (7=4)r7c5 - (4)r9*3c6 = (4)r5c6 => -4 r5c9; stte

As far as I can see, they are re-writings (in an extended illegible AIC notation) of my second solution.

[denis_berthier]

As I mentioned before, this puzzle has few anti-backdoors:

Code: Select all

6 BRT-ANTI-BACKDOORS:
n8r9c5 n8r8c3 n7r7c3 n6r6c9 n4r5c9 n8r5c2
6 W1-ANTI-BACKDOORS:
n8r9c5 n8r8c3 n7r7c3 n6r6c9 n4r5c9 n8r5c2
6 S-ANTI-BACKDOORS:
n8r9c5 n8r8c3 n7r7c3 n6r6c9 n4r5c9 n8r5c2

and none of them gives rise to a 1-elimination solution.

But it has a huge number of anti-backdoor-pairs:

Code: Select all

615 BRT-ANTI-BACKDOOR-PAIRS
622 W1-ANTI-BACKDOOR-PAIRS
626 S-ANTI-BACKDOOR-PAIRS

When this puzzle was proposed, I was busy on other things and checking all these pairs seemed a too large amount of work. Especially as I had found some acceptable 2-step solution with a whip[6].
I've now written some code to automate this and to check all the pairs.
The simplest 2-step solution I've found is the following (better than in my previous post):

Code: Select all

whip[5]: c3n3{r1 r3} - c7n3{r3 r9} - r9c6{n3 n4} - r3c6{n4 n5} - b3n5{r3c7 .} ==> r1c9 ≠ 3
singles ==> r8c9 = 3, r9c7 = 9, r7c8 = 5, r3c8 = 8, r4c8 = 9
whip[5]: r3n5{c7 c6} - r8c6{n5 n7} - r7c5{n7 n4} - c6n4{r9 r5} - c9n4{r5 .} ==> r1c9 ≠ 5
stte

[jco]

Hello,

In the first reply in this thread, I wrote

(...)
"I did not manage to write this as an AIC."
(...)

Not willing to leave something unfinished (at least regarding the writing of the move), I write that move as follows

Code: Select all

 1   2     3     4     5       6       7     8     9
.---------------+---------------------+-----------------.
| 8   7   d'34  |c345   9       6     | 2      1    45-3| 1
| 2   5     1   |b347  b347     8     |a34     6    9   | 2
| 9   46    346 | 1     2      c345   | 3458   58   7   | 3
|---------------+---------------------+-----------------|
| 57  3     689 | 457   4567    2     | 45789  589  1   | 4
| 57  1689  689 | 3457  134567  34579 | 45789  2    456 | 5
| 4   169   2   | 8     1567    579   | 579    3    56  | 6
|---------------+---------------------+-----------------|
| 3   49    479 | 2     457     1     | 6      59   8   | 7
| 6   2     78  | 9     3578    357   | 1      4   f35  | 8
| 1   489   5   | 6     348    d34    |e39     7    2   | 9
'---------------+---------------------+-----------------'

(3)r2c7=(3)r2c45-(3)b2p19=Kraken(4)b2p19 => -3 r1c9

Kraken(4)b2p19
(4)r1c4-(4=3)r1c3
||
(4)r3c6-(4=3)r9c6-r9c7=(3)r8c9

My attempt for a PM representation

Hidden Text: Show

Code: Select all

3r2c7  3r2c45
       3b2p19  4r1c4  4r3c6
3r1c3          4r1c3
                      4r9c6  3r9c6
3r8c9                        3r9c7

Regards,
jco