The New Sudoku Players' Forum

[Leren]

Code: Select all

*-----------*
|...|.8.|.4.|
|1..|.59|..2|
|.4.|2.6|...|
|---+---+---|
|...|...|3.1|
|...|8.1|.96|
|9..|5..|7..|
|---+---+---|
|..1|.2.|4.8|
|6..|..8|9.5|
|...|61.|..3|
*-----------*
....8..4.1...59..2.4.2.6.........3.1...8.1.969..5..7....1.2.4.86....89.5...61...3

[pjb]

Code: Select all

 235    d2359    259-3  | 1      8     f37     | 6      4     e79     
 1       67      67     | 4      5      9      | 8      3      2     
 38      4       389    | 2      37     6      | 1      5      79     
------------------------+----------------------+---------------------
 458    b568     4568   | 7      9      24     | 3      28     1     
 2347    237     2347   | 8      34     1      | 5      9      6     
 9       1      a38     | 5      6      2-3    | 7      28     4     
------------------------+----------------------+---------------------
 357     357     1      | 9      2      57     | 4      6      8     
 6       27      247    | 3      47     8      | 9      1      5     
 458    c589     4589   | 6      1      45     | 2      7      3     


(3=8)r6c3 - (8)r4c2 = (8-9)r9c2 = (9)r1c2 - (9=7)r1c9 - (7=3)r1c6 => -3 r1c3, r6c6; stte

Phil

[denis_berthier]

.

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------+----------------+----------------+
   ! 235  2359 2359 ! 1    8    37   ! 6    4    79   !
   ! 1    67   67   ! 4    5    9    ! 8    3    2    !
   ! 38   4    389  ! 2    37   6    ! 1    5    79   !
   +----------------+----------------+----------------+
   ! 458  568  4568 ! 7    9    24   ! 3    28   1    !
   ! 2347 237  2347 ! 8    34   1    ! 5    9    6    !
   ! 9    1    38   ! 5    6    23   ! 7    28   4    !
   +----------------+----------------+----------------+
   ! 357  357  1    ! 9    2    57   ! 4    6    8    !
   ! 6    27   247  ! 3    47   8    ! 9    1    5    !
   ! 458  589  4589 ! 6    1    45   ! 2    7    3    !
   +----------------+----------------+----------------+

1) normal solution, hardest pattern = bivalue-chain[3]
finned-x-wing-in-columns: n4{c6 c1}{r9 r4} ==> r4c3 ≠ 4
finned-x-wing-in-rows: n3{r6 r1}{c6 c3} ==> r3c3 ≠ 3
biv-chain[3]: r8c2{n2 n7} - c1n7{r7 r5} - c1n2{r5 r1} ==> r1c2 ≠ 2
biv-chain[3]: c2n8{r4 r9} - b7n9{r9c2 r9c3} - r3c3{n9 n8} ==> r4c3 ≠ 8, r6c3 ≠ 8
stte


2) real one-step solutions: (no Subset applied before)
There are 19 W1-anti-backdoors: n4r9c6 n9r9c2 n7r8c5 n4r8c3 n2r8c2 n5r7c6 n2r6c8 n3r6c6 n8r6c3 n4r5c5 n8r4c8 n2r4c6 n8r4c2 n7r3c9 n3r3c5 n9r3c3 n8r3c1 n9r1c9 n7r1c6
17 of which give rise to 1-step solutions with a chain of length ≤ 10

Here are the simplest ones:

biv-chain[5]: r1c6{n3 n7} - r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} - r6c3{n8 n3} ==> r6c6 ≠ 3, r1c3 ≠ 3 (this is pjb's chain)
stte

biv-chain[5]: r6n3{c3 c6} - r1c6{n3 n7} - r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} ==> r6c3 ≠ 8
stte

biv-chain[5]: r1n7{c9 c6} - c6n3{r1 r6} - r6c3{n3 n8} - c2n8{r4 r9} - c2n9{r9 r1} ==> r1c9 ≠ 9
stte

biv-chain[5]: r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} - r6c3{n8 n3} - c6n3{r6 r1} ==> r1c6 ≠ 7
stte

whip[5]: c6n3{r1 r6} - r6c3{n3 n8} - c2n8{r4 r9} - r9n9{c2 c3} - r3c3{n9 .} ==> r3c5 ≠ 3
stte

biv-chain[6]: r4c8{n2 n8} - c2n8{r4 r9} - c2n9{r9 r1} - r1c9{n9 n7} - r1c6{n7 n3} - r6c6{n3 n2} ==> r6c8 ≠ 2, r4c6 ≠ 2
stte

biv-chain[6]: r6c8{n8 n2} - r6c6{n2 n3} - r1c6{n3 n7} - r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} ==> r4c8 ≠ 8, r6c3 ≠ 8
stte

biv-chain[6]: r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} - r6c3{n8 n3} - b5n3{r6c6 r5c5} - r3c5{n3 n7} ==> r3c9 ≠ 7, r1c6 ≠ 7
stte

biv-chain[6]: r3c9{n9 n7} - b2n7{r3c5 r1c6} - c6n3{r1 r6} - r6c3{n3 n8} - c2n8{r4 r9} - b7n9{r9c2 r9c3} ==> r3c3 ≠ 9
stte

biv-chain[7]: r4c6{n4 n2} - r4c8{n2 n8} - c2n8{r4 r9} - b7n9{r9c2 r9c3} - r3n9{c3 c9} - r3n7{c9 c5} - r8c5{n7 n4} ==> r9c6 ≠ 4, r5c5 ≠ 4
stte

whip[6]: c6n7{r7 r1} - r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} - r6c3{n8 n3} - c6n3{r6 .} ==> r8c5 ≠ 7
stte

whip[6]: c6n7{r7 r1} - r1c9{n7 n9} - c2n9{r1 r9} - c2n8{r9 r4} - r6c3{n8 n3} - c6n3{r6 .} ==> r7c6 ≠ 5
stte

whip[6]: c5n3{r5 r3} - c6n3{r1 r6} - r6c3{n3 n8} - c2n8{r4 r9} - r9n9{c2 c3} - r3c3{n9 .} ==> r5c5 ≠ 4
stte

[Cenoman]

Code: Select all

 +-----------------------+-----------------+-----------------+
 |  235   f2359  c2359#  |  1    8    37*  |  6    4    7-9  |
 |  1      67     67     |  4    5    9    |  8    3    2    |
 | b38#    4      38-9*  |  2   a37*  6    |  1    5   a79   |
 +-----------------------+-----------------+-----------------+
 |  458   e568    4568   |  7    9    24   |  3    28   1    |
 |  2347   237   c2347#  |  8    34   1    |  5    9    6    |
 |  9      1     d38*    |  5    6    23*  |  7    28   4    |
 +-----------------------+-----------------+-----------------+
 |  357    357    1      |  9    2    57   |  4    6    8    |
 |  6      27     247    |  3    47   8    |  9    1    5    |
 |  458   e589    4589   |  6    1    45   |  2    7    3    |
 +-----------------------+-----------------+-----------------+

5-link oddagon (3)r36, c36, b2 having three guardians:
(9=73)r3c59 - r3c1 == r15c3 - (3=8)r6c3 - (89)r49c2 = (9)r1c2 => -9 r1c9, r3c3; ste

[Ngisa]

Code: Select all

+----------------------+---------------+---------------+
| 235     2359    2359 | 1    8     37 | 6    4     79 |
| 1       67      67   | 4    5     9  | 8    3     2  |
| 38      4       38-9 | 2   b37    6  | 1    5    a79 |
+----------------------+---------------+---------------+
| 458    f568     4568 | 7    9    e24 | 3   e28    1  |
| 2347    237     2347 | 8    34    1  | 5    9     6  |
| 9       1       38   | 5    6     23 | 7    28    4  |
+----------------------+---------------+---------------+
| 357     357     1    | 9    2     57 | 4    6     8  |
| 6       27      247  | 3   c47    8  | 9    1     5  |
| 458    g589    h4589 | 6    1    d45 | 2    7     3  |
+----------------------+---------------+---------------+


(9=7)r3c9 - r3c5 = (7-4)r8c5 = r9c6 - (4=28)r4c68 - (8)r4c2 = (8-9)r9c2 = (9)r9c3 => - 9r3c3; stte

Clement

[jco]

After basics

Code: Select all

.------------------------------------------.
| 235   2359  2359 | 1  8   37 | 6  4   79 |
| 1     67    67   | 4  5   9  | 8  3   2  |
| 38    4    c389  | 2 b37  6  | 1  5  b79 |
|------------------+-----------+-----------|
| 458  f568   4568 | 7  9   24 | 3  28  1  |
| 247-3 27-3  247-3| 8 a34  1  | 5  9   6  |
| 9     1    g38   | 5  6   2-3| 7  28  4  |
|------------------+-----------+-----------|
| 357   357   1    | 9  2   57 | 4  6   8  |
| 6     27    247  | 3  47  8  | 9  1   5  |
| 458  e589  d4589 | 6  1   45 | 2  7   3  |
'------------------------------------------'

(3)r5c5=(3-79)r3c59=r3c3-r9c3=(9-8)r9c2=r4c2-(8=3)r6c3 => -3 r5c123, r6c6; ste

[denis_berthier]

5-link oddagon (3)r36, c36, b2 having three guardians:
(9=73)r3c59 - r3c1 == r15c3 - (3=8)r6c3 - (89)r49c2 = (9)r1c2 => -9 r1c9, r3c3; ste

What's your definition of an oddagon? (What does the "==" sign mean? What is the 73r3c59 in an oddagon?)
Starting from the same PM and activating only oddagons, I do get oddagons, but not yours:

oddagon[5]: r1n3{c1 c6},c6n3{r1 r6},r6n3{c6 c3},c3n3{r6 r3},b1n3{r3c3 r1c1} ==> r3c3 ≠ 3
oddagon[5]: r4n4{c3 c6},c6n4{r4 r9},r9n4{c6 c1},c1n4{r9 r5},b4n4{r5c1 r4c3} ==> r4c3 ≠ 4
oddagon[9]: b2n3{r1c6 r3c5},r3n3{c5 c1},r3c1{n3 n8},r3n8{c1 c3},c3n8{r3 r4},b4n8{r4c3 r6c3},r6c3{n8 n3},r6n3{c3 c6},c6n3{r6 r1} ==> r4c3 ≠ 8
oddagon[11]: r3n3{c1 c5},c5n3{r3 r5},b5n3{r5c5 r6c6},r6n3{c6 c3},r6c3{n3 n8},c3n8{r6 r9},r9n8{c3 c2},c2n8{r9 r4},b4n8{r4c2 r4c1},c1n8{r4 r3},r3c1{n8 n3} ==> r9c3 ≠ 8
This is still not enough to solve the puzzle.

[eleven]

5-link oddagon (3)r36, c36, b2 having three guardians:
(9=73)r3c59 - r3c1 == r15c3 - (3=8)r6c3 - (89)r49c2 = (9)r1c2 => -9 r1c9, r3c3; ste

What's your definition of an oddagon? (What does the "==" sign mean? What is the 73r3c59 in an oddagon?)

Hi Denis,

Cenoman's oddagon is exactly your first one.

Unfortunately you do not denote the guardians (or extra candidates), which you can see in the notation 3r3c1 == 3r15c3 (the 3's are taken from the nodes before and after in the chain), namely 3r3c1, r1c3, r5c3 . As you write, a direct elimination is -3r3c3.
But using the link (either 3r3c1 or 3r15c3) in a chain, you can deduce, that 9 in r1c9, r3c3 can be eliminated (too).

[denis_berthier]

5-link oddagon (3)r36, c36, b2 having three guardians:
(9=73)r3c59 - r3c1 == r15c3 - (3=8)r6c3 - (89)r49c2 = (9)r1c2 => -9 r1c9, r3c3; ste

What's your definition of an oddagon? (What does the "==" sign mean? What is the 73r3c59 in an oddagon?)

Hi Denis,
Cenoman's oddagon is exactly your first one. [...] As you write, a direct elimination is -3r3c3.

Hi eleven,
1) Unfortunately, this "direct" elimination - which for me is the only real oddagon elimination - is not even mentioned in cenoman's oddagon.
2) Recalling my oddagon:
oddagon[5]: r1n3{c1 c6},c6n3{r1 r6},r6n3{c6 c3},c3n3{r6 r3},b1n3{r3c3 r1c1} ==> r3c3 ≠ 3,
and cenoman's:
5-link oddagon (3)r36, c36, b2 - which I understand as using CSP-Variables r3n3, r6n3, c3n3, c6n3 and b2n3
we don't even use the same CSP-Variables.

With these two points, how could the two oddagons be the same?

Unfortunately you do not denote the guardians (or extra candidates), which you can see in the notation 3r3c1 == 3r15c3 (the 3's are taken from the nodes before and after in the chain), namely 3r3c1, r1c3, r5c3 .

I never mention z-candidates in any of my chains. They play no essential role in the proof. If Z was True, all of them would be False; and we could proceed as if they didn't exist.
The only effect of mentioning them directly in the chain is to introduce confusion.

But using the link (either 3r3c1 or 3r15c3) in a chain, you can deduce, that 9 in r1c9, r3c3 can be eliminated (too).

This ad hoc conclusion goes beyond the oddagon rule. As a result it's hard to maintain this is a 1-step solution.

[eleven]

Hi Denis,

as Cenoman pointed out in a private message, you are right, it is not the same oddagon, just the same guardians. Sorry, that i am still not good in reading your notation.

And i just wanted to answer your question, hopefully you now can understand this (here and there) often used notation.
I did not want to start a discussion, your z-candidates have been debated to death for me.

[denis_berthier]

Hi Denis,
as Cenoman pointed out in a private message, you are right, it is not the same oddagon, just the same guardians. Sorry, that i am still not good in reading your notation.

Hi eleven
The CSP-Variables are explicit in my chain notation. In Cenoman's notation, he has to mention them above the chain because his chain notation is unable to express them.

hopefully you now can understand this (here and there) often used notation.

Don't worry about my understanding of the real AIC notation, in spite of its inconsistencies. I wouldn't claim the same about all of its unlimited number of idiomatic extensions and variants. AFAIK, the == is not part of the AIC notation.