The New Sudoku Players' Forum

[StrmCkr]

SE: 8.5 ~ this puzzle was posted on reddit

....9.6..3......8.7...8.......7.8....9....2..5..3.9......8...354......67.6.......

after basics + iW ring & finned sword fish
i reached this stage

Code: Select all

+-------------------------+------------------------+-------------------------+
| 128    12458    12458   | 1245    9       37     | 6      12457    1234    |
| 3      1245     69      | 12456   14567   124567 | 14579  8        1249    |
| 7      1245     69      | 12456   8       123456 | 13459  12459    1249    |
+-------------------------+------------------------+-------------------------+
| 126    1234     1234    | 7       12456   8      | 13459  1459     1469    |
| 16(8)  9        (37)    | (1456)  (1456)  (1456) | 2      -145(7)  146(38) |
| 5      (12478)  (12478) | 3       (1246)  9      | 1478   (14)     1468    |
+-------------------------+------------------------+-------------------------+
| 129    127      127     | 8       1467    1246   | 149    3        5       |
| 4      12358    12358   | 1259    135     125    | 189    6        7       |
| 189    6        13578   | 1459    13457   1457   | 1489   1249     12489   |
+-------------------------+------------------------+-------------------------+

(7)r5c8=(378)r5c139 - (78=1246)r6c2358 - (6 = 145)r5c456 => r5c8 <> 145

ahs has 7 @ r5c8 or its a hidden locked set of 378,

inference the aals(124678) into a locked set, which also inferences the als(1456) into a locked set which sees r5c8 : thus its either "7" or never "1,4,5"

what else can we do on this grid? {and 2ndly not exactly sure on how we use eureka to notate AHS ? as this isn't covered in the wiki entry }

Edit typo for aahs to aals

[Leren]

Some Eureka notation for an almost AHS could go like ... - 7 r5c8 =(378 - 146) r5c137 .....

Leren

[P.O.]

i'm not sure if you are interested in solutions with chains or templates but just in case:
basics:

Hidden Text: Show

Code: Select all

intersections:
((((9 0) (2 3 1) (1 2 4 5 6 9)) ((9 0) (3 3 1) (1 2 4 5 6 9)))
 (((6 0) (2 3 1) (1 2 4 5 6 9)) ((6 0) (3 3 1) (1 2 4 5 6 9)))
 (((3 0) (8 5 8) (1 2 3 5)) ((3 0) (9 5 8) (1 2 3 4 5 7)))
 (((2 0) (9 8 9) (1 2 4 9)) ((2 0) (9 9 9) (1 2 4 8 9)))
 (((2 0) (4 5 5) (1 2 4 5 6)) ((2 0) (6 5 5) (1 2 4 6))))

QUINTE BOX: ((1 1 1) (1 2 8)) ((1 2 1) (1 2 4 5 8)) ((1 3 1) (1 2 4 5 8)) ((2 2 1) (1 2 4 5)) ((3 2 1) (1 2 4 5))
(((2 3 1) (1 2 4 5 6 9)) ((3 3 1) (1 2 4 5 6 9)))


Code: Select all

128     12458   12458   1245    9       123457  6       12457   1234             
3       1245    69      12456   14567   124567  14579   8       1249             
7       1245    69      12456   8       123456  13459   12459   12349           
126     1234    1234    7       12456   8       13459   1459    13469           
168     9       13478   1456    1456    1456    2       1457    13468           
5       12478   12478   3       1246    9       1478    147     1468             
129     127     127     8       1467    12467   149     3       5               
4       12358   12358   1259    135     125     189     6       7               
189     6       13578   1459    13457   1457    1489    1249    12489           

7r6c7 => r6c59 <> 6
 r6c7=7 - r5n7{c8 c3} - r5n3{c3 c9} |- b6n8{r5c9 r6c9}
                                    |- r5n8{c9 c1} - r5n6{c1 c456}
=> r6c7 <> 7
ste.


the puzzle is in 4-template and is solved with this one combination (3 6 7 8)

Hidden Text: Show

Code: Select all

....9.6..3......8.7...8.......7.8....9....2..5..3.9......8...354......67.6.......

Initialization:
#VT: (3872 176 6 816 138 13 6 13 16)
Cells: nil nil nil nil nil nil nil nil nil
Candidates: nil (14 59 68 73 75 76 77 78) (69 78) nil nil (30 39 48) (60) nil (57 66 75)
                                                                               
128     12458   12458   1245    9       123457  6       12457   1234             
3       1245    124569  12456   14567   124567  14579   8       1249             
7       1245    124569  12456   8       123456  13459   12459   12349           
126     1234    1234    7       12456   8       13459   1459    13469           
168     9       13478   1456    1456    1456    2       1457    13468           
5       12478   12478   3       1246    9       1478    147     1468             
129     127     127     8       1467    1246    149     3       5               
4       12358   12358   1259    135     125     189     6       7               
189     6       13578   1459    13457   1457    1489    1249    12489           

1: (3 6 7 8)

(3 6 7 8): 14 instances
.8...76.33...6.78.7.6.83...6..7.83....36...78.783....6...876.3..3....867867.3....
.8...76.33...6.78.7.6.83...6..7.83..8.36...7..7.3..8.6...876.3..38....67.67.3...8
.8...76.33..6..78.7.6.83......7.83.66.3....78.7836.......876.3..3....867867.3....
.8...76.33..6..78.7.6.83......7683..6.3....78.783....6...876.3..3....867867.3....
.8...76.33..6..78.7.6.83...6..7.83..8.3....76.7.36...8...876.3..38....67.67.3.8..
.8...76.33..6..78.7.6.83...6..7.83..8.3....76.7.36.8.....876.3..38....67.67.3...8
.8...76.33..6..78.7.6.83...6..7.83....3.6..78.783....6...876.3..3....867867.3....
.8...76.33..6..78.7.6.83...6..7.83..8.3.6..7..7.3..8.6...876.3..38....67.67.3...8
.8...76.33.6...78.7..683......7.83.66.3....78.7836.......876.3..3....867867.3....
.8...76.33.6...78.7..683......7683..6.3....78.783....6...876.3..3....867867.3....
.8...76.33.6...78.7..683...6..7.83..8.3....76.7.36...8...876.3..38....67.67.3.8..
.8...76.33.6...78.7..683...6..7.83..8.3....76.7.36.8.....876.3..38....67.67.3...8
.8...76.33.6...78.7..683...6..7.83....3.6..78.783....6...876.3..3....867867.3....
.8...76.33.6...78.7..683...6..7.83..8.3.6..7..7.3..8.6...876.3..38....67.67.3...8

........33.............3.........3....3.........3............3..3...........3....

......6......6......6......6...........6.............6.....6..........6..6.......
......6.....6.......6..............66............6.........6..........6..6.......
......6.....6.......6..........6....6................6.....6..........6..6.......
......6.....6.......6......6................6....6.........6..........6..6.......
......6.....6.......6......6............6............6.....6..........6..6.......
......6....6.........6.............66............6.........6..........6..6.......
......6....6.........6.........6....6................6.....6..........6..6.......
......6....6.........6.....6................6....6.........6..........6..6.......
......6....6.........6.....6............6............6.....6..........6..6.......

.....7.........7..7...........7............7..7...........7............7..7......

.8..............8.....8.........8...........8..8.........8...........8..8........
.8..............8.....8.........8...8................8...8.......8............8..
.8..............8.....8.........8...8..............8.....8.......8..............8

#VT: (3872 176 1 816 138 9 1 3 16)
Cells: nil nil (9 24 34 39 65 77) nil nil (60) (6 16 44 47 59 75) (2) nil
SetVC: ( n8r1c2   n7r1c6   n3r1c9   n7r2c7   n3r3c6   n3r4c7
         n3r5c3   n7r5c8   n7r6c2   n7r7c5   n6r7c6   n3r8c2
         n7r9c3   n3r9c5   n4r7c7   n5r8c3   n5r3c7   n5r4c8
         n8r8c7   n8r9c1   n8r6c3   n9r8c4   n9r9c7   n9r4c9
         n9r7c1   n1r6c7   n4r6c8   n6r6c9   n1r8c5   n2r8c6
         n8r5c9   n2r6c5   n5r2c2   n5r1c4   n9r3c8   n9r2c3
         n4r9c4   n5r9c6   n4r1c3   n5r5c5   n6r3c3   n4r5c6
         n4r3c9   n4r4c2   n1r2c6   n2r2c9   n2r3c4   n6r4c5
         n1r5c4   n1r9c9   n1r1c8   n6r2c4   n4r2c5   n1r3c2
         n6r5c1   n2r7c2   n1r7c3   n2r9c8   n2r1c1   n1r4c1
         n2r4c3 )
2 8 4   5 9 7   6 1 3
3 5 9   6 4 1   7 8 2
7 1 6   2 8 3   5 9 4
1 4 2   7 6 8   3 5 9
6 9 3   1 5 4   2 7 8
5 7 8   3 2 9   1 4 6
9 2 1   8 7 6   4 3 5
4 3 5   9 1 2   8 6 7
8 6 7   4 3 5   9 2 1


[eleven]

Code: Select all

+-------------------------+--------------------------+--------------------------+
| 128     12458   12458   |  1245    9       123457  |  6       12457   1234    |
| 3       1245    69      |  12456   14567   124567  |  14579   8       1249    |
| 7       1245    69      |  12456   8       123456  |  13459   12459   12349   |
+-------------------------+--------------------------+--------------------------+
| 126     1234    1234    |  7       12456   8       |  13459   1459    13469   |
|b168     9      b13478   | c1456   c1456   c1456    |  2       1457   b13468   |
| 5      d12478  d12478   |  3      d1246    9       | d148-7   147    d1468    |
+-------------------------+--------------------------+--------------------------+
| 129     127     127     |  8       1467    12467   |  149     3       5       |
| 4       12358   12358   |  1259    135     125     |  189     6       7       |
| 189     6       13578   |  1459    13457   1457    |  1489    1249    12489   |
+-------------------------+--------------------------+--------------------------+

7r6c23 = (738)r5c391 & 6r5c456 - (8|6)r6c235 = 68r6c79 => -7r6c7, ste

[yzfwsf]

Code: Select all

,-------------------,----------------------,---------------------,
| 128  12458  12458 | 1245   9      37     | 6      12457  1234  |
| 3    1245   69    | 12456  14567  124567 | 14579  8      1249  |
| 7    1245   69    | 12456  8      123456 | 13459  12459  1249  |
:-------------------+----------------------+---------------------:
| 126  1234   1234  | 7      12456  8      | 13459  1459   1469  |
| 168  9      37    | 1456   1456   1456   | 2      1457   13468 |
| 5    12478  12478 | 3      1246   9      | 1478   14     1468  |
:-------------------+----------------------+---------------------:
| 129  127    127   | 8      1467   1246   | 149    3      5     |
| 4    12358  12358 | 1259   135    125    | 189    6      7     |
| 189  6      13578 | 1459   13457  1457   | 1489   1249   12489 |
'-------------------'----------------------'---------------------'

Region Forcing Chain: Each 8 in r6 true in turn will all lead to: r5c8<>1,r5c8<>4,r5c8<>5,r1c8,r5c3,r6c7<>7
8r6c2 - (8=14567)r5c14568
8r6c3 - (8=14567)r5c14568
(8-7)r6c7 = 7r5c8
(8-6)r6c9 = 6r6c5 - (6=1457)r5c4568 ;stte

[Cenoman]

First of all, your solution with AIC steps only is very nice !
I rewrite it my way, in two steps:
1. (7)r1c8 = (7-3)r1c6 = r1c9 - r5c9 = (3-7)r5c3 = (7)r5c8 Loop
=> -7 r6c8, + other loop eliminations (-1245 r1c6, -3 r34c9, -148 r5c3)
2. (7)r5c8 = (378)r5c319 - (7|8=1246)r6c1258 - (6=145)r5c456 => -145 r5c8; ste
-7r5c8 is the only elimination needed to validate your second step.

inference the aahs(124678) into a locked set

To me, r61258 is not an AAHS, but an AALS, the reason why I write it (7|8=1246)r6c1258 in the chain.
Before talking of notations, i'd like recall the definitions I use:
An ALS: n cells, subset of a sector, containing n+1 digits. Only one digit can be eliminated, leaving a naked subset.
An AALS contains n+2 digits (it is therefore an almost-almost naked subset)
An AHS: p cells, subset of a sector, containing n-1 locked digits within itself. Only one digit can be True, besides the locked digits, forming a hidden subset.
An AAHS contains n-2 locked digits.
I don't see 2 locked digits within r6c1258 (only the 2s are locked herein) On the contrary I see 6 digits in four cells forming diverse NQ with two spoilers => AALS.

So the third strong link has to be written (7|8=1246)r6c1258, with the OR symbol '|'
Otherwise (78=1246)r6c2358 would mean: whether NQ(1246)r6c2358 OR (78)r62358, term which means to me 7r6c2358 AND 8r6c2358. NQ(1246) is False if at least 1 out of the 4 digits is false, in which case 7 AND 8 can't be simultaneously True.
OK, all this is nitpicking

not exactly sure on how we use eureka to notate AHS ?

Concerning the AHS (378)r5c319, your writing and notations are Eureka compatible.
In the present puzzle, the AHS is used to provide weak links to candidates that are out of its sector. When used for internal weak links, I'd usually write e.g. (738-6)r5c139, were such a link useful. With the following rules: linking digits at the right and left side, bystanders (locked digits) appended to the left side linking digit. The bystanders could be written at the right side as well without any change to the logic.

If you would focus on the 7,8 being @r6c13, you could use the notation (3,7,8)r6c931 where the commas mean that the digits are ordered the same as the cells behind.
Here, such a notation is not required, since the digits have only one arrangement possible in the cells.

what else can we do on this grid?

As already shown by other players, the grid can be solved in one step:

Code: Select all

 +------------------------+---------------------------+--------------------------+
 |  128   12458   12458   |  1245    9       123457   |  6       1245-7  1234    |
 |  3     1245    69      |  12456   14567   124567   | z14579   8       1249    |
 |  7     1245    69      |  12456   8       123456   |  13459   12459   12349   |
 +------------------------+---------------------------+--------------------------+
 |  126   1234    1234    |  7       12456   8        |  13459   1459    13469   |
 |  168   9      a13478   | B1456   B1456   B1456     |  2     Ba1457   a13468   |
 |  5     12478   12478   |  3      A1246    9        | z1478    147    A1468    |
 +------------------------+---------------------------+--------------------------+
 |  129   127     127     |  8       1467    12467    |  149     3       5       |
 |  4     12358   12358   |  1259    135     125      | y189     6       7       |
 |  189   6       13578   |  1459    13457   1457     | y1489    1249   x12489   |
 +------------------------+---------------------------+--------------------------+

Kraken column (8)r569c9
(837)r5c389
(86)r6c59 - (6=1457)r5c4568
(8)r9c9 - r89c7 = (87)r26c7
=> -7 r1c8; ste

[StrmCkr]

Typo for aahs correct to als, missed that on my review.

My bad good catch.

My deffintion for ahs dof x is diffrent (where x is 1 for ahs)

N digits in n+x cells

Then I use ahs as strong links just like regular aic constructs.
With diffrent Elim trigs

Just e at sure how to list the ahs as aic chains are
(digits) cell =cell for each node.

And this is backwards comparitivly.
(cell) digit=digit

I'll look over your chain after work for extra reviews insite.

The pipe | is that also " and or" for aic?, for clairity so I Can update my eureka Notation write up plus my new solver output.. +for the reddit wiki I am managing.

Also wasn't aware of the digit order seperation by Commons for placment as a visual aid, intresting my code just lists it numerical incremental order.

[marek stefanik]

I see I'm late for this one.
It's interesting to see something this complex within just two rows.

These are all the candidates Eleven used, only 6 of them can be true:

Code: Select all

+-------------------------+------------------------+-------------------------+
| 126    1234     1234    | 7       12456   8      | 13459  1459     1469    |
| 1(68)  9        (37)    | 145(6)  145(6)  145(6) | 2      1457     14(368) |
| 5      124(78)  124(78) | 3       124(6)  9      | 14(78) 14       14(68)  |
+-------------------------+------------------------+-------------------------+

Note the hidden grouped s-wing:

Code: Select all

+-------------------------+------------------------+-------------------------+
| 126    1234     1234    | 7       12456   8      | 13459  1459     1469    |
| 1(68)  9        37      | 145(6)  145(6)  145(6) | 2      1457     13468   |
| 5      1247(8)  1247(8) | 3       124(6)  9      | 1478   14       14(68)  |
+-------------------------+------------------------+-------------------------+

At most 3 of these candidates can be true: S = (6r56b5 8r6b4 + r5c1 r6c9) / 2.

6 truths: 368r5 678r6
7 links: 7b4 S[3] + r5c39 r6c7
=> –7r5c3

At first, I had actually found a different way to express the 6-link.
Both ways allow to better express different patterns, so I created this puzzle for comparison:

Code: Select all

..4...6..31..6..8.7.......4...7.8....9..1.2..5..3...9......3...4.81....5.6..8...7
,------------------,---------------------,--------------------,
| 289  258   4     | 2589   23579  12579 | 6     12357  1239  |
| 3    1     259   | 2459   6      24579 | 579   8      29    |
| 7    258   2569  | 2589   2359   1259  | 1359  1235   4     |
:------------------+---------------------+--------------------:
| 126  234   1236  | 7      2459   8     | 1345  13456  136   |
| 68   9     367   | 456    1      456   | 2     34567  368   |
| 5    2478  1267  | 3      24     246   | 1478  9      168   |
:------------------+---------------------+--------------------:
| 129  257   12579 | 24569  24579  3     | 1489  1246   12689 |
| 4    237   8     | 1      279    2679  | 39    236    5     |
| 129  6     12359 | 2459   8      2459  | 1349  1234   7     |
'------------------'---------------------'--------------------'

(8.4 skfr, the expected step gets it down to 7.1)

Marek

[denis_berthier]

.

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 128    12458  12458  ! 1245   9      123457 ! 6      12457  1234   !
   ! 3      1245   124569 ! 12456  14567  124567 ! 14579  8      1249   !
   ! 7      1245   124569 ! 12456  8      123456 ! 13459  12459  12349  !
   +----------------------+----------------------+----------------------+
   ! 126    1234   1234   ! 7      12456  8      ! 13459  1459   13469  !
   ! 168    9      13478  ! 1456   1456   1456   ! 2      1457   13468  !
   ! 5      12478  12478  ! 3      1246   9      ! 1478   147    1468   !
   +----------------------+----------------------+----------------------+
   ! 129    127    127    ! 8      1467   12467  ! 149    3      5      !
   ! 4      12358  12358  ! 1259   135    125    ! 189    6      7      !
   ! 189    6      13578  ! 1459   13457  1457   ! 1489   1249   12489  !
   +----------------------+----------------------+----------------------+
265 candidates.

1) Simplest-first solution in Z6, using only a few short elementary reversible chains:

Code: Select all

hidden-pairs-in-a-column: c3{n6 n9}{r2 r3} ==> r3c3≠5, r3c3≠4, r3c3≠2, r3c3≠1, r2c3≠5, r2c3≠4, r2c3≠2, r2c3≠1
finned-swordfish-in-rows: n7{r5 r1 r9}{c3 c8 c6} ==> r7c6≠7
biv-chain[4]: r1n3{c6 c9} - r5n3{c9 c3} - r5n7{c3 c8} - r1n7{c8 c6} ==> r1c6≠1, r1c6≠2, r1c6≠4, r1c6≠5
biv-chain[4]: r1n3{c9 c6} - r1n7{c6 c8} - r5n7{c8 c3} - r5n3{c3 c9} ==> r3c9≠3, r4c9≠3
biv-chain[4]: r1n7{c8 c6} - r1n3{c6 c9} - r5n3{c9 c3} - r5n7{c3 c8} ==> r6c8≠7
biv-chain[4]: r5n3{c3 c9} - r1n3{c9 c6} - r1n7{c6 c8} - r5n7{c8 c3} ==> r5c3≠1, r5c3≠4, r5c3≠8
z-chain[4]: c8n7{r1 r5} - r5c3{n7 n3} - b6n3{r5c9 r4c7} - c7n5{r4 .} ==> r1c8≠5
z-chain[5]: r5c3{n7 n3} - r9n3{c3 c5} - r9n7{c5 c6} - r1n7{c6 c8} - r5n7{c8 .} ==> r6c3≠7, r7c3≠7
biv-chain[3]: b4n7{r6c2 r5c3} - r5n3{c3 c9} - r5n8{c9 c1} ==> r6c2≠8
biv-chain[4]: c2n8{r1 r8} - c2n3{r8 r4} - r5n3{c3 c9} - r5n8{c9 c1} ==> r1c1≠8
z-chain[5]: b3n3{r3c7 r1c9} - r1c6{n3 n7} - b3n7{r1c8 r2c7} - c7n5{r2 r4} - c7n3{r4 .} ==> r3c7≠9, r3c7≠4, r3c7≠1
z-chain[6]: r7c3{n1 n2} - r7c1{n2 n9} - r9c1{n9 n8} - r5n8{c1 c9} - r5n3{c9 c3} - c3n7{r5 .} ==> r9c3≠1
z-chain[6]: r7c3{n1 n2} - r7c1{n2 n9} - r9c1{n9 n8} - r5n8{c1 c9} - b6n3{r5c9 r4c7} - c2n3{r4 .} ==> r8c2≠1
z-chain[6]: r5c3{n3 n7} - b6n7{r5c8 r6c7} - b6n8{r6c7 r6c9} - c9n6{r6 r4} - c1n6{r4 r5} - r5n8{c1 .} ==> r5c9≠3
stte

2) Solution with fewer steps still based on elementary reversible chains, allowing a slightly longer one (in Z7)

Code: Select all

hidden-pairs-in-a-column: c3{n6 n9}{r2 r3} ==> r3c3≠5, r3c3≠4, r3c3≠2, r3c3≠1, r2c3≠5, r2c3≠4, r2c3≠2, r2c3≠1
biv-chain[4]: r1n3{c6 c9} - r5n3{c9 c3} - r5n7{c3 c8} - r1n7{c8 c6} ==> r1c6≠5, r1c6≠1, r1c6≠2, r1c6≠4
biv-chain[4]: r5n3{c3 c9} - r1n3{c9 c6} - r1n7{c6 c8} - r5n7{c8 c3} ==> r5c3≠4, r5c3≠1, r5c3≠8
z-chain[7]: r5n8{c9 c1} - c1n6{r5 r4} - c9n6{r4 r6} - b6n8{r6c9 r6c7} - c7n7{r6 r2} - r1n7{c8 c6} - r1n3{c6 .} ==> r5c9≠3
stte

.

[eleven]

Code: Select all

 *--------------------------------------------------------------------------*
 |  289   258   4       |  2589    2357   12579   |  6      12357   1239    |
 |  3     1     259     |  2459    6      24579   |  579    8       29      |
 |  7     258   6       |  2589    235    1259    |  1359   1235    4       |
 |----------------------+-------------------------+-------------------------|
 |  126   234   123     |  7       9      8       | c1345  c13456  c136     |
 | b68    9    b37      | b456     1     b456     |  2     b3467   b368     |
 |  5     478   17      |  3       24    a246     |  1478   9       18-6    |
 |----------------------+-------------------------+-------------------------|
 |  129   257   12579   |  24569   2457   3       |  1489   1246    12689   |
 |  4     237   8       |  1       27     2679    |  39     236     5       |
 |  129   6     12359   |  2459    8      2459    |  1349   1234    7       |
 *--------------------------------------------------------------------------*

6r6c6 = 6r5c46,8374r5c1938 - (3|4=156)r4c789 => -6r6c9

Code: Select all

 *-------------------------------------------------------------------*
 |  289   258   4      |  289   35   1279   |  6      12357   1239   |
 |  3     1     259    |  249   6    2479   |  579    8       29     |
 |  7     258   6      |  289   35   129    |  1359   1235    4      |
 |---------------------+--------------------+------------------------|
 |  126   234   123    |  7     9    8      |  1345   1345    136    |
 |  68    9     37     |  45    1    45     |  2      37      68     |
 |  5     478   17     |  3     2    6      |  1478   9       18     |
 |---------------------+--------------------+------------------------|
 | a129   57    57     |  6     4    3      |  189    12      1289   |
 |  4    b23    8      |  1     7   b29     |  39     6       5      |
 | a129   6     123-9  | c259   8   c259    |  1349   1234    7      |
 *-------------------------------------------------------------------*

(9=12)r79c1 - r8c2 = (2-9)r8c6 = 9r9c46 => -9r9c3

[P.O.]

Marek’s puzzle
basics:

Hidden Text: Show

Code: Select all

( n6r3c3   n9r4c5 )

intersections:
((((5 0) (5 4 5) (4 5 6)) ((5 0) (5 6 5) (4 5 6)))
 (((2 0) (6 5 5) (2 4)) ((2 0) (6 6 5) (2 4 6))))


Code: Select all

289    258    4      2589   2357   12579  6      12357  1239           
3      1      259    2459   6      24579  579    8      29             
7      258    6      2589   235    1259   1359   1235   4               
126    234    123    7      9      8      1345   13456  136             
68     9      37     456    1      456    2      3467   368             
5      478    17     3      24     246    1478   9      168             
129    257    12579  24569  2457   3      1489   1246   12689           
4      237    8      1      27     2679   39     236    5               
129    6      12359  2459   8      2459   1349   1234   7             

3r5c389 => r5c8 <> 4
 r5c3=3 - r5n7{c3 c8}
 r5c9=3 - r5n8{c9 c1} - c1n6{r5 r4} - 145r4c789


basics:

Hidden Text: Show

Code: Select all

intersections:
((((4 0) (5 4 5) (4 5 6)) ((4 0) (5 6 5) (4 5 6)))
 ( n6r7c4   n6r8c8   n4r7c5   n7r8c5   n6r6c6   n2r6c5 )
 (((5 0) (9 4 8) (2 5 9)) ((5 0) (9 6 8) (2 5 9)))
 (((5 0) (1 5 2) (3 5)) ((5 0) (3 5 2) (3 5))))

PAIR ROW: ((5 3 4) (3 7)) ((5 8 6) (3 7)) 
(((5 9 6) (3 6 8)))

QUAD ROW: ((7 1 7) (1 2 9)) ((7 7 9) (1 8 9)) ((7 8 9) (1 2)) ((7 9 9) (1 2 8 9))
(((7 2 7) (2 5 7)) ((7 3 7) (1 2 5 7 9)))


Code: Select all

289    258    4      289    35     1279   6      12357  1239           
3      1      259    249    6      2479   579    8      29             
7      258    6      289    35     129    1359   1235   4               
126    234    123    7      9      8      1345   1345   136             
68     9      37     45     1      45     2      37     68             
5      478    17     3      2      6      1478   9      18             
129    57     57     6      4      3      189    12     1289           
4      23     8      1      7      29     39     6      5               
129    6      1239   259    8      259    1349   1234   7           

b7n3{r9c3 r8c2} - r8c7{n3 n9} - r7n9{c79 c1} => r9c3 <> 9
ste.


[StrmCkr]

2) Solution with fewer steps still based on elementary reversible chains, allowing a slightly longer one (in Z7)

z chains are memory chains and aren't reverse-able as they aren't bi directional Xor logic constructs{a.i.c}. - i will not enter a debate on this again its an old hat we both never will see eye to eye on chains that can retain information as "truth" as its moves directional from an implication point. I do get how they work and have helped others code them in the past.

{a.i.c} aren't 1 way directional nor are they implication networks.

How you class this path easier compared to pure a.i.c methods:
{.. when your methods are your own creations nor have they ever been standardized to match what the rest of us implore here...}

when basics & this [ als + aic ]class move i did resulted in singles to the end for this grid.

D.M a message would be a better method to discuss this off to the side

[denis_berthier]

2) Solution with fewer steps still based on elementary reversible chains, allowing a slightly longer one (in Z7)

z chains are memory chains and aren't reverse-able as they aren't bi directional Xor logic constructs{a.i.c}. - i will not enter a debate on this again

There's no debate.
z-chains have no memory ("memory" is related to t-candidates, not to z-candidates).
z-chains are reversible - this has been proven in [CRT] and [PBCS]. It seems you just don't understand the trivial proof or maybe not even the definition of a z-chain.

How you class this path easier compared to pure a.i.c methods:
{.. when your methods are your own creations nor have they ever been standardized to match what the rest of us implore here...}

Nothing has been standardised in Sudoku. You can implore as long as you want, I will not change my nrc notation (because that's your real problem: notation) for the inconsistent AIC one (for which, BTW, there's not a single comprehensive reference).
About my path being easier, I didn't say a word about this. At first sight, it seems eleven's path also uses only chains of length 7 (but I'm not sure, due to the notation).
.

[marek stefanik]

My step is very similar to Eleven's.

Again, (almost) the same candidates are used:

Code: Select all

+----------------------+-------------------------+-------------------------+
|  126   234   123     |  7       9      8       |  1345   13456   136     |
|  (68)  9     (37)    |  45(6)   1      45(6)   |  2      3467    (368)   |
|  5     4(78) 1(7)    |  3       24     24(6)   |  14(78) 9       1(68)   |
+----------------------+-------------------------+-------------------------+

This time note the hidden y-wing:

Code: Select all

+----------------------+-------------------------+-------------------------+
|  126   234   123     |  7       9      8       |  1345   13456   136     |
|  68    9     (37)    |  456     1      456     |  2      3467    6(38)   |
|  5     48(7) 1(7)    |  3       24     246     |  14(78) 9       168     |
+----------------------+-------------------------+-------------------------+

Only 3 of these candidates can be true: Y = (3r5 7r6b4 8b6 + r5c39 r6c7) / 2

6 truths: 678r6 + r5c139
7 links: 6r5b5 8b4 Y[3] + r6c9
=> –6r5c46

Theory rant: Show

Both patterns use the same candidates, but it's not obvious from one that the other exists, unlike say:

Code: Select all

xy-wing 7r6c23 - (7=3)r5c3 – (3=8)r5c9 – (8=7)r6c7 – 7r6c23 => –7r6c23
l3-wing 7r6c23 = (7–3)r5c3 = (3–8)r5c9 = (8–7)r6c7 – 7r6c23 => –7r6c7

which use the same strong/weak links, the only difference is which links are strong.

We cannot take the multilink used in one of them and use it in the other, unless we allow repeated truths/links.

Out of curiosity, I tried to express the 6-link directly (i.e. without nesting), there is only one way:
(3r5 6r5566b5 7r66b4 8r566b4 + r5c13399 r6c79) / 3
(We need all of 8r5b4 + r5c1 otherwise we can get 7 true candidates, the rest follows easily.)

There is only one pattern which would translate to it:
6 truths: 6r5 678r6 + r5c39
8 links: 3r5 6b5 7b4 8r5b4 + r5c1 r6c79
=> -8r5c1,
which places 6r5c9 only into the truths, so it would only work without it:

Code: Select all

+---------------+---------------+---------------+
| .    .    .   | .    .    .   | .    .    .   |
| 6+-8 .    37  | 6+   6+   6+  | .    .    38  |  6r5 + r5c39
| 78+  78+  78+ | 6+   6+   6+  | 78+  .    68+ |  678r6
+---------------+---------------+---------------+
Kraken 8r6:
| 8r6c123
| (8–7)r6c7 = 7r6c123 – (7=3)r5c3 – (3=8)r5c9
| (8–6)r6c9 = 6r6c456 – 6r5c456 = 6r5c1
=> –8r5c1

Which is at best a cheesy way to save a step:
7r6c7 = 7r6c123 – (7=3)r5c3 – (3=8)r5c9 => –8r6c7
8r6c123 = (8–6)r6c9 = 6r6c456 – 6r5c456 = 6r5c1 => –8r5c1

On z-chains, as much as I don't like using them in manual solutions (picking a candidate as a potential elimination and closemindedly looking for a contradiction is something I generally avoid until I get desperate), I have to say that they do have some nice properties.
They are reversible, for some (imo) sane notion of reversibility:

Code: Select all

xyz-wing 123r1c1, 12r1c4, 13r2c1 => -1r1c23
z-chain[3]: r1c4{n1 n2} – r1c1{n2 n3} – r2c1{n3 .} => –1r1c23
z-chain[3]: r2c1{n1 n3} – r1c1{n3 n2} – r1c4{n2 .} => –1r1c23

The elimination stands outside the chain, the . is just a matter of notation.

This form of reversibility can be useful in a computer search.
Say you are looking for the shortest z-chain.
When you find a partial z-chain[k] and you have a partial z-chain[k-1] with the same target s.t. their last rlcs are linked, you can combine them into a z-chain[2k-1] which you know is the shortest possible (if the second z-chain has also length k, you need to find possibly all partial z-chains[k] to prove whether there is a z-chain[2k-1]).

They have other nice properties, too.
You never have to consider multiple partial z-chains with the same target and last rlc, you just take one of the shortest ones.
You can efficiently check if a target t can be eliminated by a z-chain without caring about the length, you just start with {t} and for each candidate c add singles you get by placing c and t, until you get two candidates which are linked or the set is closed.

If the abomination that whips are had any of this...

About my path being easier, I didn't say a word about this. At first sight, it seems eleven's path also uses only chains of length 7 (but I'm not sure, due to the notation).

6 actually.

Marek

[denis_berthier]

On z-chains, as much as I don't like using them in manual solutions (picking a candidate as a potential elimination and closemindedly looking for a contradiction is something I generally avoid until I get desperate), I have to say that they do have some nice properties.

z-candidates can be consided as "guardians" in z-chains in the same way as they are in oddagons. If you don't like z-chains, then you don't like "closeminded" oddagons.

They are reversible, for some (imo) sane notion of reversibility...
They have other nice properties, too.

All these properties and more have been stated and proved in [CRT] and [PBCS] some 15 years ago.

If the abomination that whips are...

Hey, this has been the 21st century for 23+ years, in case you didn't notice.

.