The New Sudoku Players' Forum

[P.O.]

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. . .   6 . .   . . 9
8 6 .   . 4 .   . 7 .
. . .   . . 3   5 . .
6 3 .   . 1 .   . 4 .
. . .   . . .   . . .
2 4 .   . 8 .   . 3 .
4 9 .   . 2 .   . 5 .
. . .   . . 1   2 . .
. . .   7 . .   . . 8

...6....986..4..7......35..63..1..4..........24..8..3.49..2..5......12.....7....8


[jco]

I found a solution with two chains (I did not search further for a one-stepper).

Code: Select all

.-----------------------------------------------------------------------------.
| 1357    1257    123457  | 6       57      2578    | 1348    128     9       |
| 8       6       12359   | 1259    4       259     | 13      7       123     |
| 179     127     12479   | 1289    79      3       | 5       1268    1246    |
|-------------------------+-------------------------+-------------------------|
| 6       3       5789    | 259     1       2579    | 789     4       257     |
| 1579    1578    15789   |d23459 ea36-579 b245679  | 16789   12689   12567   |
| 2       4       1579    | 59      8      b5679    | 1679    3       1567    |
|-------------------------+-------------------------+-------------------------|
| 4       9       1367-8  |c38      2      c68      | 1367    5       1367    |
| 357     578     35678   | 4589-3  3569    1       | 2       69      3467    |
| 135     125     12356   | 7       3569    459-6   | 13469   169     8       |
'-----------------------------------------------------------------------------'

1. Loop (6)r5c5 = (6)r56c6 - (6=83)r7c46 - r5c4 = (3-6)r5c5

=> -6 r9c6, -8 r7c3, -3 r8c4, -579 r5c5 [& 2 LC eliminations]
---

Code: Select all

.-----------------------------------------------------------------------------.
| 1357    1257    123457  | 6       57      258     | 1348    128     9       |
| 8       6       12359   | 1259    4       259     | 13      7       123     |
| 179     127     12479   |a1289    79      3       | 5      h126-8  g1246    |
|-------------------------+-------------------------+-------------------------|
| 6       3       5789    | 259     1       2579    | 789     4       257     |
| 1579    1578    15789   |b23459   36      245679  | 16789   12689   12567   |
| 2       4       1579    | 59      8       5679    | 1679    3       1567    |
|-------------------------+-------------------------+-------------------------|
| 4       9       1367    |b38      2       68      | 1367    5       1367    |
| 357     578     35678   |c459     3569    1       | 2       69     f3467    |
| 135     125     12356   | 7       3569   d459     |e13469   169     8       |
'-----------------------------------------------------------------------------'

2. (8)r3c4 = (8-34)r75c4 = (4)r8c4 - r9c6 = r9c7 - r8c9 = (4-6)r3c9 = (6)r3c8 => -8 r7c8; ste

Thanks for the puzzle!

[pjb]

Code: Select all

 1357    1257    123457 | 6       57     2578   | 1348   128    9     
 8       6       12359  | 1259    4      259    | 13     7      123   
 179     127     12479  |g1289    79     3      | 5     f1268  e1246   
------------------------+-----------------------+---------------------
 6       3       5789   | 259     1      2579   | 789    4      257   
 1579    1578    15789  |b2459-3 a35679  245679 | 16789  12689  12567 
 2       4       1579   | 59      8      5679   | 1679   3      1567   
------------------------+-----------------------+---------------------
 4       9       13678  |h38      2      68     | 1367   5      1367   
 357     578     35678  |c34589   569-3  1      | 2      69    d3467   
 135     125     12356  | 7       569-3  4569   | 13469  169    8     


(3)r5c5 = (3-4)r5c4 = (4)r8c4 - (4)r8c9 = (4-6)r3c9 = (6-8)r3c8 = (8)r3c4 - (8=3)r7c4 => -3 r5c4, r89c5; btte
or better
(8)r3c4 = (8-6)r3c8 = (6-4)r3c9 = (4)r8c9 - (4)r8c4 = (4-3)r5c4 = (3-6)r5c5 = (6)r89c5 - (6=8)r7c6 => -8 r1c6, r78c4; stte

Thanks, Phil

[RSW]

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 +------------------+---------------------+-------------------+
 | 1357 1257 123457 | 6     57     257-8  |a1348 a128   9     |
 | 8    6    12359  | 1259  4      259    |a13    7    a123   |
 | 179  127  12479  | 1289  79     3      | 5     1268  1246  |
 +------------------+---------------------+-------------------+
 | 6    3    5789   |d259   1     d2579   | 789   4     257   |
 | 1579 1578 15789  | 23459 35679 d245679 | 16789 12689 12567 |
 | 2    4    1579   |d59    8     d5679   | 1679  3     1567  |
 +------------------+---------------------+-------------------+
 | 4    9    13678  | 38    2     e68     | 1367  5     1367  |
 | 357  578  35678  | 34589 3569   1      | 2     69    3467  |
 | 135  125  12356  | 7     3569  c4569   |b13469 169   8     |
 +------------------+---------------------+-------------------+


(8=1234)b3p1246 - (4)r9c7 = (4)r9c6 - (4=25796)b5p13679 - (6=8)r7c6 => -8r1c6; stte

[denis_berthier]

.
SER = 7.3

Code: Select all

Resolution state after Singles (and whips[1]):
   +----------------------+----------------------+----------------------+
   ! 1357   1257   123457 ! 6      57     2578   ! 1348   128    9      !
   ! 8      6      12359  ! 1259   4      259    ! 13     7      123    !
   ! 179    127    12479  ! 1289   79     3      ! 5      1268   1246   !
   +----------------------+----------------------+----------------------+
   ! 6      3      5789   ! 259    1      2579   ! 789    4      257    !
   ! 1579   1578   15789  ! 23459  35679  245679 ! 16789  12689  12567  !
   ! 2      4      1579   ! 59     8      5679   ! 1679   3      1567   !
   +----------------------+----------------------+----------------------+
   ! 4      9      13678  ! 38     2      68     ! 1367   5      1367   !
   ! 357    578    35678  ! 34589  3569   1      ! 2      69     3467   !
   ! 135    125    12356  ! 7      3569   4569   ! 13469  169    8      !
   +----------------------+----------------------+----------------------+
218 candidates

The simplest-first solution, in W4, involves many elimination steps.
Notice that:
- the number of candidates after whips[1] is very large for a puzzle in W4 (more than 3 standard deviations above the mean, see http://forum.enjoysudoku.com/relation-between-difficulty-and-number-of-candidates-t39836-1.html);
- there is not a single Single before all these eliminations.

Code: Select all

z-chain[4]: b2n8{r1c6 r3c4} - r7c4{n8 n3} - r5n3{c4 c5} - c5n7{r5 .} ==> r1c6≠7
whip[1]: c6n7{r6 .} ==> r5c5≠7
z-chain[4]: r7c6{n6 n8} - r7c4{n8 n3} - r5n3{c4 c5} - c5n6{r5 .} ==> r9c6≠6
t-whip[4]: r7c6{n8 n6} - c5n6{r9 r5} - r5n3{c5 c4} - r7c4{n3 .} ==> r7c3≠8, r8c4≠8
t-whip[4]: r7c4{n3 n8} - r7c6{n8 n6} - c5n6{r9 r5} - r5n3{c5 .} ==> r8c4≠3
t-whip[4]: b9n4{r9c7 r8c9} - c4n4{r8 r5} - c4n3{r5 r7} - b9n3{r7c7 .} ==> r9c7≠1, r9c7≠6, r9c7≠9
whip[1]: b9n9{r9c8 .} ==> r5c8≠9
biv-chain[3]: r2c7{n1 n3} - r9c7{n3 n4} - b3n4{r1c7 r3c9} ==> r3c9≠1
biv-chain[4]: r3n6{c8 c9} - b3n4{r3c9 r1c7} - r9c7{n4 n3} - r2c7{n3 n1} ==> r3c8≠1
t-whip[4]: r5n3{c5 c4} - r7c4{n3 n8} - r7c6{n8 n6} - c5n6{r9 .} ==> r5c5≠5, r5c5≠9
t-whip[4]: r3n8{c8 c4} - c4n1{r3 r2} - r2c7{n1 n3} - r2c9{n3 .} ==> r3c8≠2
biv-chain[4]: r3c8{n6 n8} - c4n8{r3 r7} - c4n3{r7 r5} - r5c5{n3 n6} ==> r5c8≠6
whip[4]: r1c5{n5 n7} - r3c5{n7 n9} - r2n9{c6 c3} - r2n5{c3 .} ==> r1c6≠5
biv-chain[3]: c8n2{r5 r1} - r1c6{n2 n8} - r3n8{c4 c8} ==> r5c8≠8
whip[1]: c8n8{r3 .} ==> r1c7≠8
naked-triplets-in-a-column: c7{r1 r2 r9}{n4 n1 n3} ==> r7c7≠3, r7c7≠1, r6c7≠1, r5c7≠1
whip[1]: c7n1{r2 .} ==> r1c8≠1, r2c9≠1
naked-pairs-in-a-row: r1{c6 c8}{n2 n8} ==> r1c3≠2, r1c2≠2
x-wing-in-rows: n1{r6 r7}{c3 c9} ==> r9c3≠1, r5c9≠1, r5c3≠1, r3c3≠1, r2c3≠1, r1c3≠1
biv-chain[4]: c4n3{r5 r7} - c4n8{r7 r3} - r1c6{n8 n2} - c8n2{r1 r5} ==> r5c4≠2
biv-chain[4]: r1c6{n2 n8} - r7n8{c6 c4} - c4n3{r7 r5} - b5n4{r5c4 r5c6} ==> r5c6≠2
whip[1]: r5n2{c9 .} ==> r4c9≠2
biv-chain[4]: b3n6{r3c9 r3c8} - c8n8{r3 r1} - c8n2{r1 r5} - b6n1{r5c8 r6c9} ==> r6c9≠6
biv-chain[4]: c4n1{r3 r2} - r2c7{n1 n3} - r2c9{n3 n2} - r1n2{c8 c6} ==> r3c4≠2
biv-chain[4]: r7c4{n3 n8} - b2n8{r3c4 r1c6} - r1c8{n8 n2} - r2c9{n2 n3} ==> r7c9≠3
hidden-pairs-in-a-block: b9{n3 n4}{r8c9 r9c7} ==> r8c9≠7, r8c9≠6
whip[1]: r8n7{c3 .} ==> r7c3≠7
biv-chain[3]: r8c9{n3 n4} - c4n4{r8 r5} - c4n3{r5 r7} ==> r8c5≠3
biv-chain[4]: b8n3{r9c5 r7c4} - c4n8{r7 r3} - c4n1{r3 r2} - r2c7{n1 n3} ==> r9c7≠3
stte

Surprisingly, there is a 1-step solution in BC5:

Code: Select all

biv-chain[5]: r7c4{n3 n8} - r3n8{c4 c8} - b3n6{r3c8 r3c9} - c9n4{r3 r8} - c4n4{r8 r5} ==> r5c4≠3
w1-tte

[Mauriès Robert]

Hi all,
Another one-step resolution is as follows (with an anti-track):

(-4r3c9)->4r8c9->[ 4r5c4->3r5c5->6r56c6->8r7c6->3r7c4 ]->3r2c9->1r2c7->1r3c4->... => 8c4 empty => r3c9=4, stte.



Robert

[P.O.]

thank you for your answers;
i always find it interesting to read the somewhat complicated solutions; while some can be called classics in a way, some, especially those involving large sets of candidates like those of RSW, are very surprising;
my solution is to expand both legs of a xor relationship:

Code: Select all

1357    1257    123457  6       57      2578    1348    128     9               
8       6       12359   1259    4       259     13      7       123             
179     127     12479  B12+89   79      3       5      A12+68  a124+6             
6       3       5789    259     1       2579    789     4       257             
1579    1578    15789  c2×3+459 35679   245679  16789   12689   12567           
2       4       1579    59      8       5679    1679    3       1567             
4       9       13678  C+38     2       68      1367    5       1367             
357     578     35678  c3-4589  3569    1       2       69     b3+467             
135     125     12356   7       3569    4569    13469   169     8 

    / c8 - r3n8{c8 c4} - r7c4{n8 n3}
r3n6                                》r5c4 <> 3
    \ c9 - c9n4{r3 r8} - c4n4{r8 r5}
   
single: n3r5c5
intersection: c6n6{r5r6} => r7c6 r9c6 <> 6
ste.


[RSW]

those involving large sets of candidates like those of RSW, are very surprising

I don't deliberately try to find large subsets, but that's the way it works out sometimes. My primary goal is to find a one step solution using the shortest/simplest AIC possible, but the shortest is not necessarily the simplest, and the simplest is not necessarily the shortest. So, it's a bit of a compromise and a somewhat subjective decision as to what is the best solution.

Thank you for posting these puzzles.

[denis_berthier]

Code: Select all

    / c8 - r3n8{c8 c4} - r7c4{n8 n3}
r3n6                                》r5c4 <> 3
    \ c9 - c9n4{r3 r8} - c4n4{r8 r5}
 

This is a network presentation of the bivalue-chain in my answer.

[P.O.]

hi Denis,
a chain start by a xor relationship between two candidates (A,B) then only one leg is developed A<--XOR-->B-->(bi...) each (bi) tested against (A) for elimination; i am experimenting developing both legs (ai...)<--A<--XOR-->B-->(bi...) and comparing both sets of links (ai,bi); this is the way it was found:

Code: Select all

Roots: (((6 0) (3 8 3) (1 2 6 8)) ((6 0) (3 9 3) (1 2 4 6)))
Elimination: ((((5 4 5) (2 3 4 5 9))) (3))

((6 0) (3 8 3) (1 2 6 8))
((8 1 10) (3 4 2) (1 2 8 9))
((3 2 9) (7 4 8) (3 8))

((6 0) (3 9 3) (1 2 4 6))
((4 1 10) (8 9 9) (3 4 6 7))
((4 2 1) (5 4 5) (2 3 4 5 9)) 


I have also a bivalue-chain: c4n4{r5 r8} - c9n4{r8 r3} - r3n6{c9 c8} - r3n8{c8 c4} - r7c4{n8 n3} => r5c4 <> 3

Code: Select all

depth: 4  candidate: 3  from start
 
((4 0) (5 4 5) (2 3 4 5 9))       A   -||||
((4 0) (8 4 8) (3 4 5 8 9))       B    ||||
((4 1 1) (3 9 3) (1 2 4 6))       b1  -||||
((6 2 10) (3 8 3) (1 2 6 8))      b2  --||| } nothing
((8 3 10) (3 4 2) (1 2 8 9))      b3  ---||
((3 4 9) (7 4 8) (3 8))           b4  ----| elimination found.


[denis_berthier]

Hi P.O.
Yes, I understand. But in the present case, the network presentation is more complicated.